October 05, 2025
We study a class of mixed local and nonlocal \(p\)-Laplace equations with prototype \[-\Delta_p u + (-\Delta_p)^s u = f \quad \text{in } \Omega,\] where \(\Omega \subset \mathbb{R}^n\) is bounded and open. We provide sufficient condition on \(f\) to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov–Safonov covering lemma and expansion of positivity: one via the John–Nirenberg lemma, the other via the Bombieri–Giusti lemma. To our knowledge, these approaches are new, even for \(p = 2\) with \(f \equiv 0\), and include a new proof of the reverse Hölder inequality for supersolutions. Further, we establish Harnack inequality for solutions by first deriving a local boundedness result, together with a tail estimate and an initial weak Harnack inequality.
We study a class of mixed local and nonlocal partial and integro-differential equations of the form \[\label{meqn} -\mathrm{div}(\mathcal{A}(x, \nabla u)) + \mathcal{L}u = f \quad \text{in } \Omega,\tag{1}\] where \(\Omega \subset \mathbb{R}^n\) is a bounded open set. We assume \(0 < s < 1 < p < n\) and \(f \in L_{\mathrm{loc}}^{\frac{q}{p}}(\Omega)\) with \(q > n\). The map \(\mathcal{A} : \Omega \times \mathbb{R}^n \to \mathbb{R}^n\) is a Carathéodory function satisfying the growth and coercivity conditions \[\label{A} |\mathcal{A}(x,\xi)| \leq C_2 |\xi|^{p-1}, \quad \mathcal{A}(x,\xi) \xi \geq C_1 |\xi|^p,\tag{2}\] for almost every \(x \in \Omega\) and all \(\xi \in \mathbb{R}^n\), with positive constants \(C_1, C_2\).
The operator \(\mathcal{L}\) is the nonlocal \(p\)-Laplace operator defined by \[\label{L} \mathcal{L}(u)(x) = \mathrm{P.V.} \int_{\mathbb{R}^n} |u(x) - u(y)|^{p-2} (u(x) - u(y)) K(x,y) \, dy,\tag{3}\] where P.V. stands for the Cauchy principal value, and the kernel \(K\) is measurable, symmetric in \(x\) and \(y\), and satisfies for some \(\Lambda \geq 1\) \[\label{K} \frac{\Lambda^{-1}}{|x-y|^{n + ps}} \leq K(x,y) \leq \frac{\Lambda}{|x-y|^{n + ps}} \quad \text{for almost every } x,y \in \mathbb{R}^n.\tag{4}\]
When \(\mathcal{A}(x,\xi) = |\xi|^{p-2}\xi\), the operator \(\mathrm{div}(\mathcal{A}(x, \nabla u))\) reduces to the \(p\)-Laplace operator \(\Delta_p u\). Similarly, if \(K(x,y) = |x-y|^{-n - ps}\), then \(\mathcal{L}\) coincides with the fractional \(p\)-Laplace operator \((-\Delta_p)^s\). Consequently, equation 1 generalizes the mixed local and nonlocal \(p\)-Laplace equation \[\label{meqn1} -\Delta_p u + (-\Delta_p)^s u = f.\tag{5}\]
Mixed local and nonlocal PDEs have recently garnered considerable attention due to their applications in diverse physical models (see [1] and references therein).
When \(f \equiv 0\), various regularity results have been established for the equation 5 . For the linear case \(p=2\), Harnack inequalities for globally nonnegative solutions have been obtained via probabilistic methods (see [2], [3]). In the quasilinear setting, several regularity results including local boundedness and Harnack inequalities have been proven using purely analytic techniques in [4], [5].
For nontrivial right-hand sides \(f\), recent advances include results for \(p=2\) where \(f=gu\) with \(g\) belonging to the Kato class, establishing Harnack inequalities for globally nonnegative solutions of 5 via probabilistic arguments [6]. Additionally boundedness and Hölder regularity results appear in [7]–[9]. Weak Harnack inequalities for sign-changing supersolutions with bounded, nonpositive \(g\) are shown in [10].
When \(p \neq 2\), assuming various conditions on \(f\), boundedness, higher Hölder and \(C^{1,\alpha}\) regularity have been obtained in [11]–[14]. Lipschitz potential estimates for \(f \in L^1_{\mathrm{loc}}(\Omega)\) are found in [15], while global gradient estimates under divergence-form \(f\) appear in [16]. For measure data, potential and gradient estimates are available in [17], [18]. Moreover, weak Harnack inequalities for nonlinear source terms of the form \(f = g(x)|u|^{p-2} u\) with bounded, nonpositive \(g\) are presented in [19].
In this work, we provide sufficient conditions on the nonhomogeneity \(f\) that guarantees weak Harnack inequality, local boundedness and Harnack inequality for sign-changing solutions of 1 . Our main results in this article reads as follows:
First, we have the following weak Harnack inequality, which is known for the equation 5 with \(f\equiv 0\) in [5].
Theorem 1 (Weak Harnack Inequality). Let \(x_0 \in \mathbb{R}^n\) and \(0 < r \leq 1\) satisfy \(r < \frac{R}{2}\). Assume \(0 < s < 1 < p < n\) and \(f \in L^{\frac{q}{p}}(B_R(x_0))\) with \(q > n\). Suppose \(u\) is a weak supersolution of 1 with \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Then, for every \(0 < \eta < \kappa(p-1)\), there exists a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda,\eta) > 0\) such that \[\label{mthmest} \left( \fint_{B_{\frac{r}{2}}(x_0)} u^\eta \right)^{\frac{1}{\eta}} \leq c \left( \inf_{B_{\frac{r}{2}}(x_0)} u + r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} \right),\qquad{(1)}\] where the tail term \(\mathrm{Tail}(u_-; x_0, R)\) is defined in 6 .
For the homogeneous case \(f \equiv 0\), the classical proof of Theorem 1 relies heavily on the Krylov–Safonov covering lemma and the expansion of positivity technique. Here, we provide two different proofs that circumvent these classical tools and the De Giorgi framework. Moreover, as far as we are aware, these proofs are new even for \(p=2\) with \(f\equiv 0\).
The first approach adapts Moser’s iteration scheme [20] to the nonlinear nonlocal setting, using the John–Nirenberg inequality (Theorem 10). This method, previously employed in the purely local [20] and purely nonlocal contexts [21], centers on a logarithmic estimate (Lemma 14) connecting integral averages with positive and negative powers of supersolutions (see 40 ).
A key ingredient is controlling averages of negative powers through the infimum of the supersolution (Lemma 17), leading to a preliminary weak Harnack inequality (estimate 42 ). This is then refined using a reverse Hölder inequality (Lemma 18) to complete the proof of Theorem 1.
The second approach leverages a modified Bombieri–Giusti lemma (Lemma 11) originally developed in [22], which bypasses both the John–Nirenberg and Krylov–Safonov lemmas. This powerful tool has been applied in various local and nonlocal problems [23]–[26]. The proof integrates three key components: a supremum estimate for inverse of supersolutions (Lemma 17), a reverse Hölder inequality (Lemma 18), and a tailored logarithmic estimate (Lemma 20).
Both approaches rest on precise energy estimates (see Lemma 13, equations ?? and ?? ), obtained via two crucial algebraic inequalities ?? and ?? respectively, which handles the nonlinear structure and coupling between local and nonlocal terms. Such energy estimates combined with the Sobolev embeddings and Moser’s iteration technique proved Lemma 17 and Lemma 18. Additionally, a weighted Poincaré inequality (Lemma 9) is instrumental in deriving the final logarithmic estimates in Lemma 20. Notably, the proof of the reverse Hölder inequality in Lemma 18 differs from that in [5].
Our second main result establishes local boundedness, and its proof follows the approach of [5], originally carried out for the case \(f \equiv 0\) in equation 5 . Further, local boundedness result for nonzero \(f\) can be found in [13] when \(p\geq 2\).
Theorem 2 (Local boundedness). Let \(x_0 \in \mathbb{R}^n\) and \(0 < r \leq 1\) be such that \(B_R(x_0)\subset\Omega\) with \(r\leq R\). Assume that \(0 < s < 1 < p < n\), \(\kappa = \frac{n}{n-p}\), and \(f \in L^{\frac{q}{p}}(B_R(x_0))\) with \(q > n\). Suppose that \(u\) is a weak subsolution of 1 . Then, there exists a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\label{mthmbddest} \sup_{B_{\frac{r}{2}}(x_0)} u \leq \delta \, \mathrm{Tail}\Big(u_+; x_0, \frac{r}{2}\Big) + c \, \delta^{-\frac{(p-1)\kappa}{p(\kappa-1)}} \left( \fint_{B_r(x_0)} u_+^p \, dx \right)^{\frac{1}{p}} + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}},\qquad{(2)}\] where the tail term \(\mathrm{Tail}\big(u_+; x_0, \frac{r}{2}\big)\) is defined in 6 .
Finally, we have the following Harnack inequality, which is known for the equation 5 with \(f\equiv 0\) in [5]. The proof is done by combining the above local boundedness result along with a tail estimate and a preliminary weak Harnack inequality.
Theorem 3 (Harnack Inequality). Let \(x_0 \in \mathbb{R}^n\) and \(0 < r \leq 1\) satisfy \(r < \frac{R}{2}\). Assume that \(0 < s < 1 < p < n\) and \(f \in L^{\frac{q}{p}}(B_R(x_0))\) with \(q > n\). Suppose that \(u\) is a weak solution of 1 with \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Then, there exists a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\label{mthm2est} \sup_{B_{\frac{r}{2}}(x_0)} u \leq c \left( \inf_{B_r(x_0)} u + r^{p'} R^{-p'} \, \mathrm{Tail}(u_-; x_0, R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}}\|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} \right),\qquad{(3)}\] where the tail term \(\mathrm{Tail}(u_-; x_0, R)\) is defined in 6 .
Throughout the rest of the paper, we shall use the following conventions unless stated otherwise:
We fix \(0 < s < 1 < p < n\) and define \(\kappa = \frac{n}{n-p}\).
The domain \(\Omega\) denotes a bounded open subset of \(\mathbb{R}^n\).
The source function satisfies \(f \in L_{\mathrm{loc}}^{\frac{q}{p}}(\Omega)\) with \(q > n\).
For \(k \in \mathbb{R}\), we denote \(k_\pm = \max\{0, \pm k\}\).
The conjugate exponent of \(l>1\) is denoted by \(l'\) and defined by \(l'=\frac{l}{l-1}\).
For a point \(x_0 \in \mathbb{R}^n\) and radius \(r > 0\), we denote the ball \[B_r(x_0) = \{ y \in \mathbb{R}^n : |y - x_0| < r \}.\]
The underlying space of a weak subsolution or supersolution or solution of 1 in \(\Omega\) will be \(W_{\mathrm{loc}}^{1,p}(\Omega) \cap L^{p-1}_{ps}(\mathbb{R}^n)\).
We denote by \(d\mu = K(x,y) \, dx dy\) is defined using the kernel \(K\) from 4 .
The Lebesgue measure of a measurable set \(S\) is denoted by \(|S|\).
The barred integral symbol denotes the corresponding integral average.
For a measurable function \(u\) and weight \(\phi\), we write \[(u)_{\phi, B_r(x_0)} = \frac{\int_{B_r(x_0)} u \phi \, dx}{\int_{B_r(x_0)} \phi \, dx}.\]
The function \(J_p : \mathbb{R} \to \mathbb{R}\) is defined by \[J_p(t) = |t|^{p-2} t.\]
The constants \(C_1,C_2\) and \(\Lambda\) will be the same constants as found in 2 and 4 respectively.
If a constant \(C\) depends on parameters \(r_1, r_2, \dots, r_k\), we write \[C = C(r_1, r_2, \dots, r_k).\]
The paper is organized as follows. Section 2 introduces the functional framework and recalls auxiliary results. In Section 3, we establish the required energy estimates. Section 4 is devoted to proving supersolution and tail estimates. Finally, Section 5 contains the proofs of the main results.
Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set and let \(0 < s < 1 < p < \infty\). We consider the space \(W^{1,p}(\Omega)\) as the standard Sobolev space [27]. In order to study mixed local and nonlocal problems, it is convenient to consider the space \(W_0^{1,p}(\Omega)\) (see [11] and the references therein) defined as the closure of \(C_c^\infty(\Omega)\) with respect to the global norm \(\Big(\int_{\mathbb{R}^n}|\nabla u|^p\,dx\Big)^\frac{1}{p}\). In addition, we make use of the fractional Sobolev space \(W^{s,p}(\Omega)\), defined by (see [28]) \[W^{s,p}(\Omega) = \left\{ u \in L^p(\Omega) : \int_{\Omega} \int_{\Omega} \frac{|u(x) - u(y)|^p}{|x - y|^{n + ps}} \, dx \, dy < \infty \right\},\] and equipped with the norm \[\|u\|_{W^{s,p}(\Omega)} = \left( \int_{\Omega} |u(x)|^p \, dx + \int_{\Omega} \int_{\Omega} \frac{|u(x) - u(y)|^p}{|x - y|^{n + ps}} \, dx \, dy \right)^{\frac{1}{p}}.\]
The fractional Sobolev space with zero boundary values is given by \[W_0^{s,p}(\Omega) = \left\{ u \in W^{s,p}(\mathbb{R}^n) : u = 0 \text{ in } \mathbb{R}^n \setminus \Omega \right\}.\]
For \(0 < s \leq 1\), the local Sobolev space \(W^{s,p}_{\mathrm{loc}}(\Omega)\) consists of all functions \(u\) such that \(u \in W^{s,p}(\Omega')\) for every open subset \(\Omega' \Subset \Omega\), where \(\Omega' \Subset \Omega\) means that the closure \(\overline{\Omega'}\) is compactly contained in \(\Omega\).
The next lemma asserts that, under appropriate smoothness conditions on \(\Omega\), the classical Sobolev space embeds continuously into the fractional Sobolev space (cf. [28]).
Lemma 4. Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^n\), with \(1 < p < \infty\) and \(0 < s < 1\). There exists a positive constant \(C = C(n,p,s)\) such that \[\|u\|_{W^{s,p}(\Omega)} \leq C \|u\|_{W^{1,p}(\Omega)}\] for every \(u \in W^{1,p}(\Omega)\).
A related result for fractional Sobolev spaces with zero boundary conditions is given next (see [29]). Unlike Lemma 4, this result holds for any bounded domain since functions in \(W_0^{1,p}(\Omega)\) admit a zero extension outside \(\Omega\).
Lemma 5. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), with \(1 < p < \infty\) and \(0 < s < 1\). There exists a positive constant \(C = C(n,p,s,\Omega)\) such that \[\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|u(x) - u(y)|^p}{|x - y|^{n + ps}} \, dx \, dy \leq C \int_{\Omega} |\nabla u|^p \, dx\] for every \(u \in W_0^{1,p}(\Omega)\).
For \(0 < s < 1 < p < \infty\), we define the tail space \[L^{p-1}_{ps}(\mathbb{R}^n) = \left\{ v \in L^{p-1}_{\mathrm{loc}}(\mathbb{R}^n) : \int_{\mathbb{R}^n} \frac{|v(y)|^{p-1}}{(1 + |y|)^{n + ps}} \, dy < \infty \right\}.\] Furthermore, for \(x_0 \in \mathbb{R}^n\) and \(\rho > 0\), the mixed local and nonlocal tail quantity, which appears in our main results, is defined by \[\label{tail} \mathrm{Tail}(v; x_0, \rho) = \left( \rho^p \int_{\mathbb{R}^n \setminus B_\rho(x_0)} \frac{|v(y)|^{p-1}}{|y - x_0|^{n + ps}} \, dy \right)^{\frac{1}{p-1}}.\tag{6}\] By construction, for any \(v \in L^{p-1}_{ps}(\mathbb{R}^n)\), the tail quantity \(\mathrm{Tail}(v; x_0, \rho)\) is finite for every \(x_0 \in \mathbb{R}^n\) and \(\rho > 0\).
Next, we define the notion of weak subsolutions or supersolutions for equation 1 .
Definition 6. A function \(u \in W_{\mathrm{loc}}^{1,p}(\Omega) \cap L^{p-1}_{ps}(\mathbb{R}^n)\) is a weak subsolution (or supersolution) of 1 if for every \(\Omega' \Subset \Omega\) and every nonnegative test function \(\phi \in W_0^{1,p}(\Omega')\), we have \[\label{weaksubsupsoln} \int_{\Omega'} \mathcal{A}(x, \nabla u) \cdot \nabla \phi \, dx + \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} J_p(u(x) - u(y)) (\phi(x) - \phi(y)) \, d\mu \leq (\text{ or }\geq) \int_{\Omega'} f \phi \, dx,\qquad{(4)}\] where \(\mathcal{A}\) is as defined in 2 with \[J_p(u(x) - u(y)) = |u(x) - u(y)|^{p-2} (u(x) - u(y)) \quad \text{and} \quad d\mu = K(x,y) \, dx \, dy,\] where \(K\) is as defined in 4 . Moreover, we say that \(u \in W_{\mathrm{loc}}^{1,p}(\Omega) \cap L^{p-1}_{ps}(\mathbb{R}^n)\) is a weak solution of 1 if the equality in ?? holds for every \(\phi\in W_0^{1,p}(\Omega')\) without any sign restriction.
It immediately follows from Definition 6 that \(u\) is a weak subsolution or supersolution of 1 if and only if, for any constant \(c \in \mathbb{R}\), the translated function \(u + c\) is also a weak subsolution or supersolution of 1 .
The following iteration lemma from [30] will be useful for us.
Lemma 7. Let \(0 \leq T_0 \leq t \leq T_1\) and assume that \(f : [T_1, T_2] \to [0, \infty)\) is a nonnegative bounded function. Suppose that for \(T_0 \leq t < s \leq T_1\), we have \[f(t) \leq A (t - s)^{-\alpha} + B + \theta f(s),\] where \(A, B, \alpha, \theta\) are nonnegative constants and \(\theta < 1\). Then, there exists a constant \(c = c(\alpha, \theta)\) such that for every \(\rho, R\) with \(T_0 \leq \rho < R \leq T_1\), we have \[f(\rho) \leq c \big( A (R - \rho)^{-\alpha} + B \big).\]
The following version of the Gagliardo-Nirenberg-Sobolev inequality will be useful; see [31].
Lemma 8. Let \(\Omega \subset \mathbb{R}^n\) be an open set with finite measure \(|\Omega| < \infty\), and set \(\kappa = \frac{n}{n - p}\) for \(1 < p < n\). There exists a positive constant \(C = C(n,p)\) such that \[\label{e46friedrich} \left( \int_\Omega |u|^{p \kappa} \, dx \right)^{\frac{1}{p \kappa}} \leq C \left( \int_\Omega |\nabla u|^p \, dx \right)^{\frac{1}{p}}\qquad{(5)}\] holds for every \(u \in W_0^{1,p}(\Omega)\).
A weighted Poincaré inequality, which follows from a careful inspection of the proof in [32] and a change of variables, is stated below.
Lemma 9. Let \(r > 0\) and let \(\phi : B_r(x_0) \to [0,\infty)\) be a radially decreasing function with \(\phi \equiv 1\) in \(B_{\frac{r}{2}}(x_0)\). Then there exists a positive constant \(C = C(n,p)\) such that \[\label{wgtine} \int_{B_r(x_0)} |u - (u)_{\phi, B_r(x_0)}|^p \, \phi \, dx \leq C r^p \int_{B_r(x_0)} |\nabla u|^p \, \phi \, dx\qquad{(6)}\] holds for every \(u \in W^{1,p}(B_r(x_0))\).
The following theorem is a classical John-Nirenberg Lemma (see [33]).
Theorem 10. Let \(w \in L_{\mathrm{loc}}^1(\Omega)\) and suppose there exists a constant \(K\) such that \[\fint_{B_r(x_0)} |w(x) - (w)_{1, B_r(x_0)}| \, dx \leq K\] whenever \(B_{2r}(x_0) \subset \Omega\). Then there exists a constant \(\nu = \nu(n) > 0\) such that \[\fint_{B_r(x_0)} e^{\nu |w(x) - (w)_{1, B_r(x_0)}|} \, dx \leq 2,\] whenever \(B_{2r}(x_0) \subset \Omega\).
We also employ a variant of an abstract lemma originally due to Bombieri and Giusti [22], adapted in [34].
Lemma 11. Let \(\nu\) be a Borel measure and let \(\theta\), \(A\), and \(\gamma\) be positive constants, with \(0 < \delta < 1\) and \(0 < q \leq \infty\). Suppose \(U_\sigma\) are bounded measurable sets with \(U_{\sigma'} \subseteq U_\sigma\) for \(0 < \delta \leq \sigma' < \sigma \leq 1\). If \(q < \infty\), assume the doubling condition \[\nu(U_1) \leq A \nu(U_\delta).\] Let \(f\) be a positive measurable function on \(U_1\) satisfying the reverse Hölder inequality \[\left( \fint_{U_{\sigma'}} f^q \, d\nu \right)^{\frac{1}{q}} \leq \left( \frac{A}{(\sigma - \sigma')^\theta} \fint_{U_\sigma} f^s \, d\nu \right)^{\frac{1}{s}}\] for \(0 < s < q\), and \[\nu\left( \{ x \in U_1 : \log f > \lambda \} \right) \leq \frac{A \nu(U_\delta)}{\lambda^\gamma}\] for all \(\lambda > 0\). Then \[\left( \int_{U_\delta} f^q \, d\nu \right)^{\frac{1}{q}} \leq C,\] where \(C\) depends only on \(\theta\), \(\delta\), \(\gamma\), \(q\), and \(A\).
Finally, we state some algebraic inequalities that are crucial to obtain energy estimates for weak supersolutions of 1 . The inequalities ?? and ?? below are taken from [21] and [35] respectively.
Lemma 12. Let \(a,b > 0\) and \(p \in (1, \infty)\).
For any \(\tau_1, \tau_2 \in [0,1]\) and any \(\epsilon > p-1\), there exist constants \(c_i = c_i(p, \epsilon) > 0\), \(i=1,2\), bounded when \(\epsilon\) is bounded away from \(p-1\), such that \[\label{alg2} J_p(b - a) \big(\tau_1^p a^{-\epsilon} - \tau_2^p b^{-\epsilon} \big) \geq c_1 \left| \tau_2 b^{\frac{p-1-\epsilon}{p}} - \tau_1 a^{\frac{p-1-\epsilon}{p}} \right|^p - c_2 |\tau_2 - \tau_1|^p \left( b^{p-1-\epsilon} + a^{p-1-\epsilon} \right).\qquad{(7)}\]
For any \(\tau_1, \tau_2 \geq 0\) and any \(\epsilon \in (0, p-1)\), there exists a constant \(C = C(p) > 1\) such that \[\label{alg1} J_p(b - a) \big(\tau_1^p a^{-\epsilon} - \tau_2^p b^{-\epsilon} \big) \geq \frac{\zeta(\epsilon)}{C} \left| \tau_2 b^{\frac{p-1-\epsilon}{p}} - \tau_1 a^{\frac{p-1-\epsilon}{p}} \right|^p - \left( \zeta(\epsilon) + 1 + \epsilon^{1-p} \right) |\tau_2 - \tau_1|^p \left( b^{p-1-\epsilon} + a^{p-1-\epsilon} \right),\qquad{(8)}\] where \(\zeta(\epsilon) = \epsilon \left(\frac{p}{p-1-\epsilon}\right)^p\). Moreover, if \(0 < p-1-\epsilon < 1\), one may choose \(\zeta(\epsilon) = \frac{\epsilon p^p}{p-1-\epsilon}\).
In this section, we establish energy estimates for weak supersolutions of 1 . The first one provides an energy estimate for negative exponents and the second one for positive exponents, respectively.
Lemma 13. Suppose that \(0 < r \leq 1\) with \(r < R\). Let \(u\) be a weak supersolution of 1 such that \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Assume \(v = u + d\), where \(d > 0\), and let \(\psi \in C_c^\infty(B_r(x_0))\) be nonnegative in \(B_r(x_0)\).
Then for any \(\epsilon > p - 1\) and every \(\delta > 0\), there exists a positive constant \(C = C(p,C_1, C_2, \Lambda, \epsilon)\) such that \[\label{negeng1} \begin{align} &\int_{B_r(x_0)} \psi^p \Big| \nabla \Big( v^{-\frac{\alpha}{p}} \Big) \Big|^p \, dx \\ &\leq C \frac{\alpha^p}{\epsilon} \Bigg\{ \epsilon^{1-p} \int_{B_r(x_0)} v^{-\alpha} |\nabla \psi|^p \, dx + \int_{B_r(x_0)} \int_{B_r(x_0)} \big( v^{-\alpha}(x) + v^{-\alpha}(y) \big) |\psi(x) - \psi(y)|^p \, d\mu \\ &\quad + \Bigg( \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{dy}{|x - y|^{n + ps}} +d^{1-p} \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|x-y|^{n + ps}} \, dy \Bigg) \int_{B_r(x_0)} v^{-\alpha} \psi^p \, dx \\ &\quad + d^{1-p} \|f\|_{L^{\frac{q}{p}}(B_r(x_0))} \Bigg( \delta \big\| \psi v^{-\frac{\alpha}{p}} \big\|_{L^{p \kappa}(B_r(x_0))}^p + \delta^{-\frac{n}{q-n}} \big\| \psi v^{-\frac{\alpha}{p}} \big\|_{L^p(B_r(x_0))}^p \Bigg) \Bigg\}, \end{align}\qquad{(9)}\] where \(\alpha = \epsilon - p + 1\). Moreover, the constant \(C\) in ?? is bounded as long as \(\epsilon\) is bounded away from \(p-1\).
Then for any \(\epsilon \in (0, p - 1)\) and every \(\delta > 0\), there exists a positive constant \(C = C(C_1, C_2, \Lambda, p)\) such that \[\label{negeng2} \begin{align} &\int_{B_r(x_0)} \psi^p \Big| \nabla \Big( v^{\frac{\beta}{p}} \Big) \Big|^p \, dx \\ &\leq C \frac{\beta^p}{\epsilon} \Bigg\{ \epsilon^{1-p} \int_{B_r(x_0)} v^{\beta} |\nabla \psi|^p \, dx \\ &\quad + \Big( \zeta(\epsilon) + 1 + \epsilon^{1-p} \Big) \int_{B_r(x_0)} \int_{B_r(x_0)} \big( v^{\beta}(x) + v^{\beta}(y) \big) |\psi(x) - \psi(y)|^p \, d\mu \\ &\quad + \Bigg( \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{dy}{|x - y|^{n + ps}} + d^{1-p} \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|x-y|^{n + ps}} \, dy \Bigg) \int_{B_r(x_0)} v^{\beta} \psi^p \, dx \\ &\quad + d^{1-p} \|f\|_{L^{\frac{q}{p}}(B_r(x_0))} \Bigg( \delta \big\| \psi v^{\frac{\beta}{p}} \big\|_{L^{p \kappa}(B_r(x_0))}^p + \delta^{-\frac{n}{q-n}} \big\| \psi v^{\frac{\beta}{p}} \big\|_{L^p(B_r(x_0))}^p \Bigg) \Bigg\}, \end{align}\qquad{(10)}\] where \(\beta = p - 1 - \epsilon\). Here, \(\zeta(\epsilon) = \frac{\epsilon p^p}{\beta^p}\). Moreover, if \(0 < \beta < 1\), then we may choose \(\zeta(\epsilon) = \frac{\epsilon p^p}{\beta}\).
Proof. We prove both inequalities simultaneously. To this end, let \(\epsilon > 0\) be such that \(\epsilon \neq p - 1\), and define \(\gamma = \epsilon - p + 1\). By choosing \(\phi = v^{-\epsilon} \psi^{p}\) as a test function in ?? , we obtain \[\label{IJK} I + J \leq K,\tag{7}\] where \[I = \int_{\Omega} \mathcal{A}(x, \nabla v) \nabla \left(-v^{-\epsilon} \psi^{p}\right) \, dx,\] \[J = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} J_p(v(x) - v(y)) \bigl(v^{-\epsilon}(y) \psi^{p}(y) - v^{-\epsilon}(x) \psi^{p}(x)\bigr) \, d\mu, \quad \text{and} \quad K = - \int_{\Omega} f v^{-\epsilon} \psi^{p} \, dx.\]
Estimate of \(I\): To estimate \(I\), we use the hypothesis 2 on \(\mathcal{A}\) along with Young’s inequality, yielding \[p \psi^{p-1} \nabla \psi \cdot \mathcal{A}(x, \nabla v) v^{-\epsilon} \leq \frac{C_1 \epsilon}{2} \psi^{p} v^{-\epsilon - 1} |\nabla v|^{p} + c(p, C_1, C_2) \epsilon^{1-p} v^{-\gamma} |\nabla \psi|^{p}.\] Further applying the hypothesis 2 on \(\mathcal{A}\) and the above estimate, we get \[\label{estI} I \geq c_1(p, C_1, C_2) \frac{\epsilon}{|\gamma|^{p}} \int_{B_r(x_0)} \psi^{p} \left| \nabla \left( v^{-\frac{\gamma}{p}} \right) \right|^{p} \, dx - c(p, C_1, C_2) \epsilon^{1-p} \int_{B_r(x_0)} v^{-\gamma} |\nabla \psi|^{p} \, dx.\tag{8}\]
Estimate of \(J\): We first write \[J = J_1 - 2 J_2,\] where \[J_1 = \int_{B_r(x_0)} \int_{B_r(x_0)} J_p(v(x) - v(y)) \bigl(v^{-\epsilon}(y) \psi^{p}(y) - v^{-\epsilon}(x) \psi^{p}(x)\bigr) \, d\mu,\] and \[J_2 = \int_{x \in B_r(x_0)} \int_{y \in \mathbb{R}^n \setminus B_r(x_0)} J_p(v(x) - v(y)) v^{-\epsilon}(x) \psi^{p}(x) \, d\mu.\]
Estimate of \(J_1\): For \(\epsilon > p - 1\), applying the inequality ?? from Lemma 12 yields \[\label{estJ1} \begin{align} J_1 &\geq c_1 \int_{B_r(x_0)} \int_{B_r(x_0)} \left| \psi(x) v^{-\frac{\gamma}{p}}(x) - \psi(y) v^{-\frac{\gamma}{p}}(y) \right|^{p} \, d\mu \\ &\quad - c_2 \int_{B_r(x_0)} \int_{B_r(x_0)} \bigl(v^{-\gamma}(x) + v^{-\gamma}(y)\bigr) |\psi(x) - \psi(y)|^{p} \, d\mu, \end{align}\tag{9}\] where the constants \(c_i = c_i(p, \epsilon) > 0\) for \(i=1,2\) remain bounded as long as \(\epsilon\) stays away from \(p - 1\).
For \(\epsilon \in (0, p - 1)\), by using the inequality ?? in Lemma 12, we obtain \[\label{estJ11} \begin{align} J_1 &\geq \frac{\zeta(\epsilon)}{C(p)} \int_{B_r(x_0)} \int_{B_r(x_0)} \left| \psi(x) v^{-\frac{\gamma}{p}}(x) - \psi(y) v^{-\frac{\gamma}{p}}(y) \right|^{p} \, d\mu \\ &\quad - \left(\zeta(\epsilon) + 1 + \epsilon^{1-p}\right) \int_{B_r(x_0)} \int_{B_r(x_0)} \bigl(v^{-\gamma}(x) + v^{-\gamma}(y)\bigr) |\psi(x) - \psi(y)|^{p} \, d\mu, \end{align}\tag{10}\] where \(\zeta(\epsilon) = \frac{\epsilon p^{p}}{(p - 1 - \epsilon)^{p}}\). Moreover, if \(0 < p - 1 - \epsilon < 1\), then \(\zeta(\epsilon)\) can be chosen as \(\frac{\epsilon p^{p}}{p - 1 - \epsilon}\).
Estimate of \(J_2\): For any \(x \in B_R(x_0)\) and \(y \in \mathbb{R}^n\), we note that \[J_p(v(x) - v(y)) \leq c(p) \bigl(v^{p-1}(x) + u_-^{p-1}(y)\bigr), \quad \text{and} \quad v^{-\epsilon}(x) \leq d^{1-p} v^{-\gamma}(x).\] Since \(u \geq 0\) in \(B_R(x_0)\), the above imply \[\label{estJ2} \begin{align} J_2 &\leq C(\Lambda, p) \Bigg( \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{dy}{|x - y|^{n + ps}} + d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-^{p-1}(y)}{|x-y|^{n + ps}} \, dy \Bigg) \\ &\quad \times \int_{B_r(x_0)} v^{-\gamma} \psi^{p} \, dx. \end{align}\tag{11}\]
Therefore, if \(\epsilon > p - 1\), using 9 and 11 together, we conclude \[\label{estJ} \begin{align} J &\geq c_1 \int_{B_r(x_0)} \int_{B_r(x_0)} \left| \psi(x) v^{-\frac{\gamma}{p}}(x) - \psi(y) v^{-\frac{\gamma}{p}}(y) \right|^{p} \, d\mu \\ &\quad - c_2 \int_{B_r(x_0)} \int_{B_r(x_0)} \bigl(v^{-\gamma}(x) + v^{-\gamma}(y)\bigr) |\psi(x) - \psi(y)|^{p} \, d\mu \\ &\quad - 2 C(\Lambda, p) \Bigg( \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{dy}{|x - y|^{n + ps}} + d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-^{p-1}(y)}{|x - y|^{n + ps}} \, dy \Bigg) \\ &\quad \times \int_{B_r(x_0)} v^{-\gamma} \psi^{p} \, dx, \end{align}\tag{12}\] where the constants \(c_i = c_i(p, \epsilon) > 0\) for \(i=1,2\) remain bounded as long as \(\epsilon\) stays away from \(p - 1\). Further, if \(\epsilon \in (0, p-1)\), then using 10 and 11 , we obtain \[\label{estJnew} \begin{align} J &\geq \frac{\zeta(\epsilon)}{C(p)} \int_{B_r(x_0)} \int_{B_r(x_0)} \left| \psi(x) v^{-\frac{\gamma}{p}}(x) - \psi(y) v^{-\frac{\gamma}{p}}(y) \right|^p \, d\mu \\ &\quad - \Big(\zeta(\epsilon) + 1 + \epsilon^{1-p} \Big) \int_{B_r(x_0)} \int_{B_r(x_0)} \bigl( v^{-\gamma}(x) + v^{-\gamma}(y) \bigr) |\psi(x) - \psi(y)|^p \, d\mu \\ &\quad - 2 C(\Lambda, p) \Bigg( \sup_{x \in \mathrm{supp} \, \psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{dy}{|x - y|^{n + ps}} + d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|x - y|^{n + ps}} \, dy \Bigg) \\ &\quad \times \int_{B_r(x_0)} v^{-\gamma} \psi^{p} \, dx. \end{align}\tag{13}\]
Estimate of \(K\): Using Hölder’s inequality, for every \(\delta > 0\), we have \[\label{estK} \begin{align} |K| &\leq d^{1-p} \int_{B_r(x_0)} |f| v^{-\gamma} \psi^{p} \, dx \leq d^{1-p} \| f \|_{L^{\frac{q}{p}}(B_r(x_0))} \Big\| \psi v^{-\frac{\gamma}{p}} \Big\|^{p}_{L^{\frac{p q}{q - p}}(B_r(x_0))} \\ &\leq d^{1-p} \| f \|_{L^{\frac{q}{p}}(B_r(x_0))} \Big\| \psi v^{-\frac{\gamma}{p}} \Big\|^{\frac{p(q - n)}{q}}_{L^{p}(B_r(x_0))} \Big\| \psi v^{-\frac{\gamma}{p}} \Big\|^{\frac{n p}{q}}_{L^{p \kappa}(B_r(x_0))} \\ &\leq d^{1-p} \| f \|_{L^{\frac{q}{p}}(B_r(x_0))} \left( \frac{n \delta}{q} \Big\| \psi v^{-\frac{\gamma}{p}} \Big\|^{p}_{L^{p \kappa}(B_r(x_0))} + \frac{q - n}{q} \delta^{-\frac{n}{q - n}} \Big\| \psi v^{-\frac{\gamma}{p}} \Big\|^{p}_{L^{p}(B_r(x_0))} \right), \end{align}\tag{14}\] where the second inequality follows from Hölder’s inequality. Moreover, since \(q > n\), the third and fourth inequalities follow by interpolation and Young’s inequality, respectively.
Therefore, if \(\epsilon > p - 1\), then combining the estimates 8 , 12 and 14 into 7 , the desired inequality ?? follows. Further, if \(\epsilon \in (0, p - 1)\), then combining the estimates 8 , 13 and 14 into 7 , the desired inequality ?? follows. This completes the proof. ◻
Next, we establish the following logarithmic estimate for weak supersolutions of 1 .
Lemma 14. (Logarithmic estimate) Suppose that \(0 < r \leq 1\) with \(r < \frac{R}{2}\). Let \(u\) be a weak supersolution of 1 such that \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). For \(d > 0\), set \(v = u + d\). Then there exists a positive constant \(c = c(n,p,s,C_1,C_2,\Lambda)\) such that \[\label{loglemeqn} \begin{align} \int_{B_{\frac{3r}{2}}(x_0)} |\nabla \log v|^p \psi^p \, dx &\leq c r^n \bigl(r^{-p} + d^{1-p} R^{-p} \mathrm{Tail}(u_-; x_0, R)^{p-1} \bigr) \\ &\quad + c d^{1-p} R^{\frac{n(q-p)}{q}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}, \end{align}\qquad{(11)}\] for every \(\psi \in C_c^\infty(B_{\frac{3r}{2}}(x_0))\) satisfying \[0 \leq \psi \leq 1 \quad \text{in } B_{\frac{3r}{2}}(x_0) \quad \text{and} \quad |\nabla \psi| \leq \frac{c}{r} \quad \text{in } B_{\frac{3r}{2}}(x_0),\] for some positive constant \(c = c(n,p)\).
Proof. Choosing the test function \(\phi = v^{1-p} \psi^p\) in ?? , we obtain \[\label{logesttest} 0 \leq I + J + K,\tag{15}\] where \[I = \int_{B_{2r}(x_0)} \mathcal{A}(x, \nabla v) \nabla \bigl( v^{1-p} \psi^p \bigr) \, dx,\] \[J = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} J_p(v(x) - v(y)) \bigl( v^{1-p}(x) \psi^p(x) - v^{1-p}(y) \psi^p(y) \bigr) \, d\mu,\] and \[K = - \int_{B_{2r}(x_0)} f v^{1-p} \psi^p \, dx.\]
Estimate of \(I\):
Using the property 2 of \(\mathcal{A}\) along with Young’s inequality, there exist constants \(c_i = c_i(n,p,C_1,C_2) > 0\), \(i=1,2\), such that \[\label{logI3}
I \leq - c_1 \int_{B_{\frac{3r}{2}}(x_0)} |\nabla \log v|^p \psi^p \, dx + c_2 r^{n-p}.\tag{16}\]
Estimate of \(J\):
By adapting the arguments from [36] and using the properties of \(\psi\) together with \(0 < r
\leq 1\) and \(2r < R\), there exists a constant \(c = c(n,p,s,\Lambda) > 0\) such that \[\label{logI1}
\begin{align}
J &\leq - \frac{1}{c} \int_{B_{2r}(x_0)} \int_{B_{2r}(x_0)} \left| \log\left(\frac{v(x)}{v(y)}\right) \right|^p \psi(y)^p \, d\mu \\
&\quad + c d^{1-p} r^n R^{-p} \mathrm{Tail}(u_-; x_0, R)^{p-1} + c r^{n-p}.
\end{align}\tag{17}\]
Estimate of \(K\):
Using the fact that \(2r < R\) and Hölder’s inequality, we have \[\label{I}
|K| = \left| \int_{B_{2r}(x_0)} f (u + d)^{1-p} \psi^p \, dx \right| \leq c(n,p) d^{1-p} R^{\frac{n(q-p)}{q}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}.\tag{18}\]
Combining 16 , 17 and 18 into 15 , we obtain the desired estimate ?? . ◻
Corollary 15. If \(d\) satisfies ?? , and in addition \(\psi \equiv 1\) in \(B_r(x_0)\) in Lemma 14, then from ?? we deduce the estimate \[\label{logappeqn} \int_{B_r(x_0)} |\nabla \log(u + d)|^p \, dx \leq C r^{n - p},\qquad{(12)}\] for some constant \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\).
Next, we establish the following energy estimate for weak subsolutions of 1 .
Lemma 16. Let \(u\) be a weak subsolution of 1 and denote \(w = (u-k)_+\) with \(k\in\mathbb{R}\). Then, for every \(\delta > 0\), there exists a positive constant \(c = c(p,C_1,C_2,\Lambda)\) such that \[\label{subengest} \begin{align} &\int_{B_r(x_0)} \psi^p \, |\nabla w|^p \, dx \\ &\leq c \Bigg\{ \int_{B_r(x_0)} w^p \, |\nabla \psi|^p \, dx + \int_{B_r(x_0)} \int_{B_r(x_0)} \max\{ w(x), w(y) \}^p \, |\psi(x)-\psi(y)|^p \, d\mu \\ &\quad + \sup_{x \in \mathrm{supp}\,\psi} \int_{\mathbb{R}^n \setminus B_r(x_0)} \frac{w^{p-1}(y)}{|x-y|^{n+ps}} \, dy \times \int_{B_r(x_0)} w\,\psi^p \, dx + {\int_{B_r(x_0)}|f|\,w\,\psi^p\,dx} \Bigg\}, \end{align}\qquad{(13)}\] whenever \(B_r(x_0) \subset \Omega\) and \(\psi \in C_c^\infty(B_r(x_0))\) is a nonnegative function.
Proof. The proof follows by arguing similarly as in the proof of [5], by choosing \(\varphi = w \psi^p\) as a test function in ?? . ◻
Lemma 17. Suppose \(0 < r \leq 1\) such that \(r < R\). Let \(u\) be a weak supersolution of 1 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Define \(v = u + d\), where \[\label{d} d > r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}}.\qquad{(14)}\] Let \(0 < \gamma < 1\). Then there exist constants \(C = C(n, p, q, s, \gamma, C_1, C_2, \Lambda) > 0\) and \(\theta = \theta(n, p, q, s) > 0\) such that \[\label{invest} \sup_{B_{\sigma' r}(x_0)} (u + d)^{-1} \leq \left( \frac{C}{(\sigma - \sigma')^\theta} \right)^{\frac{1}{t}} \left( \fint_{B_{\sigma r}(x_0)} (u + d)^{-t} \, dx \right)^{\frac{1}{t}},\qquad{(15)}\] holds for all \(0 < \gamma \leq \sigma' < \sigma \leq 1\) and for all \(t > 0\).
Proof. Let \(0 < \gamma < \sigma' < \sigma \leq 1\). Recall \(\kappa = \frac{n}{n-p}\), where \(1 < p < n\). For \(j = 0, 1, \ldots,\) define \[\sigma_j = \sigma - (\sigma - \sigma') \big(1 - \kappa^{-j}\big), \quad r_j = \sigma_j r, \quad \text{and} \quad B_j = B_{r_j}(x_0).\] Let \(\psi_j \in C_c^\infty(B_j)\) satisfy \[\label{psi} \begin{cases} 0 \leq \psi_j \leq 1 \text{ in } B_j, \\[6pt] |\nabla \psi_j| \leq \dfrac{c \kappa^j}{(\sigma - \sigma') r} \text{ in } B_j, \\[6pt] \psi_j \equiv 1 \text{ in } B_{j+1}, \text{ and } \mathrm{dist}(\mathrm{supp} \, \psi_j, \mathbb{R}^n \setminus B_j) \geq 2^{-j-1} r, \end{cases}\tag{19}\] for some constant \(c=c(n,p)>0\). Let \(\epsilon > p - 1\). For \(\alpha = \epsilon - p + 1 > 0\), we observe that \[\label{11} \fint_{B_{j+1}} v^{-\alpha \kappa} \, dx \leq \frac{1}{|B_{j+1}|} \int_{B_j} \left| v^{-\frac{\alpha}{p}} \psi_j \right|^{p \kappa} \, dx.\tag{20}\] By Lemma 8, we have \[\label{22} \begin{align} \int_{B_j} \left| v^{-\frac{\alpha}{p}} \psi_j \right|^{p \kappa} \, dx &\leq C \left( \int_{B_j} \left| \nabla \big( v^{-\frac{\alpha}{p}} \psi_j \big) \right|^p \, dx \right)^\kappa \\ &\leq C \left( \int_{B_j} \left| \nabla \big( v^{-\frac{\alpha}{p}} \big) \right|^p \psi_j^p \, dx \right)^\kappa + C \left( \int_{B_j} |\nabla \psi_j|^p v^{-\alpha} \, dx \right)^\kappa := (I + J), \end{align}\tag{21}\] where \(C = C(n,p) > 0\) is a constant.
Estimate of \(I\): Using the estimate ?? from Lemma 13, for every \(\delta>0\), we get \[\label{estIInew} \begin{align} & \int_{B_j} \psi_j^p \left| \nabla \left( v^{-\frac{\alpha}{p}} \right) \right|^p \, dx \\ &\leq C \alpha^p \Bigg\{ \int_{B_j} v^{-\alpha} |\nabla \psi_j|^p \, dx + \int_{B_j} \int_{B_j} \big( v^{-\alpha}(x) + v^{-\alpha}(y) \big) |\psi_j(x) - \psi_j(y)|^p \, d\mu \\ &\quad + \left( \sup_{x \in \mathrm{supp} \, \psi_j} \int_{\mathbb{R}^n \setminus B_j} \frac{dy}{|x - y|^{n + p s}} + d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|y - x_0|^{n + p s}} \, dy \right) \int_{B_j} v^{-\alpha} \psi_j^p \, dx \\ &\quad + d^{1-p} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))} \Big( \delta \|\psi_j v^{-\frac{\alpha}{p}}\|_{L^{p \kappa}(B_j)}^p + \delta^{-\frac{n}{q - n}} \|\psi_j v^{-\frac{\alpha}{p}}\|_{L^p(B_j)}^p \Big) \Bigg\} \\ &:= C \alpha^p (I_1 + I_2 + I_3 + I_4), \end{align}\tag{22}\] where \(C = C(C_1, C_2, \Lambda, p, \epsilon) > 0\) is a constant bounded as long as \(\epsilon\) stays away from \(p - 1\).
Estimate of \(I_1\): Using condition 19 on \(\psi_j\), we have \[\label{estI1} I_1 \leq C \left( \frac{\kappa^j}{(\sigma - \sigma') r_j} \right)^p \int_{B_j} v^{-\alpha} \, dx.\tag{23}\]
Estimate of \(I_2\): Again, by condition 19 on \(\psi_j\) and since \(0 < r \leq 1\), it follows that \[\label{estI2} I_2 \leq C \left( \frac{\kappa^j}{(\sigma - \sigma') r_j} \right)^p \int_{B_j} v^{-\alpha} \, dx.\tag{24}\]
Estimate of \(I_3\): For any \(x \in \mathrm{supp} \, \psi_j\) and \(y \in \mathbb{R}^n \setminus B_j\), using the property 19 , we have \[\frac{1}{|x - y|} = \frac{1}{|y - x_0|} \frac{|y - x_0|}{|x - y|} \leq \frac{1}{|y - x_0|} \left( 1 + \frac{|x - x_0|}{|x - y|} \right) \leq \frac{1}{|y - x_0|} \left(1 + \frac{r}{2^{-j-1} r} \right) \leq \frac{2^{j+2}}{|y - x_0|}.\] Using \(0 < r \leq 1\), this implies \[\sup_{x \in \mathrm{supp} \, \psi_j} \int_{\mathbb{R}^n \setminus B_j} \frac{dy}{|x - y|^{n + p s}} \leq C \frac{2^{j(n + p s)}}{(\sigma - \sigma')^p r_j^p}.\] Again, using \(0 < r \leq 1\) along with the assumption ?? on \(d\), we get \[d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|x-y|^{n + p s}} \, dy \leq \leq C \frac{2^{j(n + p s)}}{(\sigma - \sigma')^p r_j^p}.\] Hence, \[\label{estI3} I_3 \leq C \frac{2^{j(n + p s)}}{(\sigma - \sigma')^p r_j^p} \int_{B_j} v^{-\alpha} \, dx.\tag{25}\]
Estimate of \(I_4\): Set \(\delta = \delta_0 r^{q-n} R^{-\frac{(q-p)(q-n)}{q}}\) where \(\delta_0 > 0\) is a constant to be chosen later. Using the assumption ?? on \(d\) and \(r < R\), we have \[\label{estI4} I_4 \leq \delta_0 \|\psi_j v^{-\frac{\alpha}{p}}\|_{L^{p \kappa}(B_j)}^p + \delta_0^{-\frac{n}{q - n}} r^{-p} \|\psi_j v^{-\frac{\alpha}{p}}\|_{L^p(B_j)}^p.\tag{26}\] Combining the estimates 23 , 24 , 25 , and 26 into 22 , we obtain \[\label{estIII} \begin{align} I &\leq C \alpha^{p\kappa} \sum_{i=1}^4 I_i^\kappa \\ &= C \alpha^{p\kappa} \left( \frac{(2\kappa)^{j(n+ps+p)}}{(\sigma-\sigma')^{p} r_j^{p}} \int_{B_j} v^{-\alpha} \, dx \right)^\kappa + C \alpha^{p\kappa} \left( \delta_0^{\kappa} \big\| \psi_j v^{-\frac{\alpha}{p}} \big\|_{L^{p\kappa}(B_j)}^{p\kappa} + \delta_0^{-\frac{n\kappa}{q-n}} r^{-p\kappa} \big\| \psi_j v^{-\frac{\alpha}{p}} \big\|_{L^p(B_j)}^{p\kappa} \right). \end{align}\tag{27}\] In the above estimates \(C = C(n,p,s,C_1,C_2,\Lambda,\epsilon,\gamma) > 0\) is a constant bounded when \(\epsilon\) is bounded away from \(p-1\).
Now, choosing \(\delta_0 > 0\) such that \(C \alpha^{p\kappa} \delta_0^{\kappa} = \frac{1}{2}\), the above inequality simplifies to \[\label{estIV} \begin{align} I &\leq C \alpha^{p\kappa} \left( \frac{(2\kappa)^{j(n+ps+p)}}{(\sigma-\sigma')^{p} r_j^{p}} \int_{B_j} v^{-\alpha} \, dx \right)^\kappa + \left( \frac{1}{2} \big\| \psi_j v^{-\frac{\alpha}{p}} \big\|_{L^{p\kappa}(B_j)}^{p\kappa} + 2^{\frac{n}{q-n}} C^{\frac{n}{q-n}} \alpha^{p\kappa \left(1 + \frac{n}{q-n}\right)} r^{-p\kappa} \big\| \psi_j v^{-\frac{\alpha}{p}} \big\|_{L^p(B_j)}^{p\kappa} \right) \\ &\leq C \max \left\{ \alpha^{p\kappa}, \alpha^{p\kappa \left(1 + \frac{n}{q-n}\right)} \right\} \left( \frac{(2\kappa)^{j(n+ps+p)}}{(\sigma-\sigma')^{p} r_j^{p}} \int_{B_j} v^{-\alpha} \, dx \right)^\kappa + \frac{1}{2} \big\| \psi_j v^{-\frac{\alpha}{p}} \big\|_{L^{p\kappa}(B_j)}^{p\kappa}, \end{align}\tag{28}\] where \(C = C(n,p,s,C_1,C_2,\Lambda,\epsilon,\gamma) > 0\) is a constant bounded as long as \(\epsilon\) is bounded away from \(p-1\).
Estimate of \(J\): Using 19 , we have \[\label{estJJ} J \leq C \left( \frac{(2\kappa)^{j(n+ps+p)}}{(\sigma-\sigma')^{p} r_j^{p}} \int_{B_j} v^{-\alpha} \, dx \right)^\kappa,\tag{29}\] where \(C = C(n,p,s,\gamma) > 0\) is a constant.
Hence, combining 28 and 29 in 21 , we deduce \[\label{3} \int_{B_j} \left| v^{-\frac{\alpha}{p}} \psi_j \right|^{p\kappa} dx \leq C \left( \max \left\{ \alpha^{p\kappa}, \alpha^{p\kappa \left(1 + \frac{n}{q-n} \right)} \right\} + 1 \right) \left( \frac{(2\kappa)^{j(n+ps+p)}}{(\sigma-\sigma')^{p} r_j^{p}} \int_{B_j} v^{-\alpha} dx \right)^\kappa,\tag{30}\] where \(C = C(n,p,s,C_1,C_2,\Lambda,\epsilon,\gamma) > 0\) is bounded as long as \(\epsilon\) is bounded away from \(p-1\).
Using 30 in 20 , we obtain the estimate (In the first line below, I have removed the typo \(r_j^p\) from the denominator of the averaged integral.) \[\label{4} \begin{align} \fint_{B_{j+1}} v^{-\alpha \kappa} dx &\leq C \frac{r_j^{-p\kappa} |B_j|^\kappa}{|B_{j+1}|} \left( \frac{\left( \max \left\{ \alpha^{p}, \alpha^{p \left(1 + \frac{n}{q-n} \right)} \right\} + 1 \right) (2\kappa)^{j(n+ps+p)}}{(\sigma - \sigma')^{p}} \fint_{B_j} v^{-\alpha} dx \right)^\kappa \\ &\leq C \left( \frac{\left( \max \left\{ \alpha^{p}, \alpha^{p \left(1 + \frac{n}{q-n} \right)} \right\} + 1 \right) (2\kappa)^{j(n+ps+p)}}{(\sigma - \sigma')^{p}} \fint_{B_j} v^{-\alpha} dx \right)^\kappa, \end{align}\tag{31}\] where we used that the quantity \(\frac{r_j^{-p\kappa} |B_j|^\kappa}{|B_{j+1}|}\) is bounded by a constant independent of \(j, r, \sigma\) and \(\sigma'\). Also, \(C = C(n,p,s,C_1,C_2,\Lambda,\epsilon,\gamma) > 0\) is bounded as long as \(\epsilon\) is bounded away from \(p-1\).
We now apply the Moser iteration technique to 31 . To this end, choose \(\alpha_j = p \kappa^j\) for \(j=0,1,\dots\). Noting that \(\alpha_j \geq p\), iterating inequality 31 yields \[\begin{align} \left( \fint_{B_0} v^{-p} dx \right)^{\frac{1}{p}} &\geq \left( \frac{\sigma - \sigma'}{C} \right)^{1 + \kappa^{-1} + \cdots + \kappa^{-m+1}} \prod_{j=0}^{m-1} (2\kappa)^{-\frac{(l + n + ps + p) j}{p \kappa^j}} \left( \fint_{B_m} v^{-p \kappa^m} dx \right)^{\frac{1}{p \kappa^m}}, \end{align}\] where \(l = \frac{p q}{q - n}\) and \(C = C(n,p,q,s,C_1,C_2,\Lambda,\gamma) > 0\) is a constant. Following the approach in [34], letting \(m \to \infty\) in the above inequality, we conclude the desired estimate ?? . ◻
Lemma 18. Let \(0 < r \leq 1\) with \(r < R\). Suppose \(u\) is a weak supersolution of 1 such that \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Define \(v = u + d\), where \(d\) satisfies ?? . For any \(0 < \gamma < 1\), there exist constants \[C = C(n,p,q,s,C_1,C_2,\Lambda,\lambda,\gamma) > 0 \quad \text{and} \quad \theta = \theta(n,p,s) > 0\] such that \[\label{revest} \begin{align} \Bigg( \fint_{B_{\sigma' r}(x_0)} v^{\lambda} \, dx \Bigg)^{\frac{1}{\lambda}} \;\leq\; \Bigg( \frac{C}{(\sigma - \sigma')^{\theta}} \Bigg)^{\frac{1}{\mu}} \Bigg( \fint_{B_{\sigma r}(x_0)} v^{\mu} \, dx \Bigg)^{\frac{1}{\mu}}, \end{align}\qquad{(16)}\] for all \(0 < \gamma \leq \sigma' < \sigma \leq 1\) and \(0 < \mu < \lambda < \lambda_0\), where \(\lambda_0 = (p-1)\kappa\).
Remark 19. It is important to note that Lemma 18 is derived here using inequality ?? , which differs from the one employed to establish the reverse Hölder inequality for the equation 5 in [5] when \(f \equiv 0\). Moreover, the above inequality ?? is new for nontrivial \(f\).
Proof. We take \(0 < \gamma < \sigma' < \sigma \leq 1\) and recall that \(\kappa = \frac{n}{n-p}\) with \(1 < p < n\). We partition the interval \((\sigma, \sigma')\) into \(m\) subintervals, where \(j = 0,1,\ldots,m\) (with \(m\) to be chosen later), and define \[\sigma_j = \sigma - (\sigma - \sigma') \frac{1 - \kappa^{-j}}{1 - \kappa^{-m}}, \quad r_j = \sigma_j r, \quad B_j = B_{r_j}(x_0).\] Let \(\psi_j \in C_c^\infty(B_j)\) be as defined in 19 . Let \(\epsilon\in(0,p-1)\), then for \(\beta = p-1 - \epsilon > 0\), we obtain \[\label{1} \fint_{B_{j+1}} v^{\beta \kappa} \, dx \;\leq\; \frac{1}{|B_{j+1}|} \int_{B_j} \big| v^{\frac{\beta}{p}} \psi_j \big|^{p\kappa} \, dx.\tag{32}\]
Applying Lemma 8 gives \[\label{2} \begin{align} \int_{B_j} \big| v^{\frac{\beta}{p}} \psi_j \big|^{p\kappa} \, dx &\leq C \Bigg( \int_{B_j} \big| \nabla( v^{\frac{\beta}{p}} \psi_j ) \big|^p \, dx \Bigg)^{\kappa} \\ &\leq C \Bigg( \int_{B_j} \big| \nabla(v^{\frac{\beta}{p}}) \big|^p \psi_j^p \, dx \Bigg)^{\kappa} + C \Bigg( \int_{B_j} |\nabla \psi_j|^p v^{\beta} \, dx \Bigg)^{\kappa} \\ &=: I + J, \end{align}\tag{33}\] where \(C = C(n,p) > 0\).
Estimate of \(I\). Using estimate ?? from Lemma 13, we obtain \[\label{estII} \begin{align} \int_{B_j} \psi_j^p \Big| \nabla \big( v^{\frac{\beta}{p}} \big) \Big|^p \, dx &\leq C \Bigg\{ \int_{B_j} v^{\beta} |\nabla \psi_j|^p \, dx + \int_{B_j}\int_{B_j} (v^{\beta}(x) + v^{\beta}(y)) |\psi_j(x)-\psi_j(y)|^p \, d\mu \\ &\quad + \Bigg( \sup_{x \in \mathrm{supp}\,\psi_j} \int_{\mathbb{R}^n \setminus B_j} \frac{dy}{|x-y|^{n+ps}} + d^{1-p} \int_{\mathbb{R}^n \setminus B_R(x_0)} \frac{u_-(y)^{p-1}}{|y-x_0|^{n+ps}} \, dy \Bigg) \int_{B_j} v^{\beta} \psi_j^p \, dx \\ &\quad + d^{1-p} \|f\|_{L^{q/p}(B_R(x_0))} \Big( \delta \| \psi v^{\frac{\beta}{p}} \|_{L^{p\kappa}(B_j)}^p + \delta^{-\frac{n}{q-n}} \| \psi_j v^{-\frac{\alpha}{p}} \|_{L^p(B_j)}^p \Big) \Bigg\}, \end{align}\tag{34}\] where \(C = C(n,p,s,C_1,C_2,\Lambda,\beta) > 0\), bounded provided \(\beta\) remains away from \(p-1\).
Proceeding exactly as in the proof of Lemma 17, we arrive at \[\label{5} \begin{align} \Bigg( \fint_{B_{j+1}} v^{\beta \kappa} \, dx \Bigg)^{\frac{1}{\beta \kappa}} &\leq \Bigg( \frac{C (2\kappa)^{j(n+ps+p)}}{(\sigma - \sigma')^p} \fint_{B_j} v^{\beta} \, dx \Bigg)^{\frac{1}{\beta}}, \end{align}\tag{35}\] using the fact that \(\frac{r_j^{-p\kappa} |B_j|^{\kappa}}{|B_{j+1}|}\) is bounded independently of \(j, r, \sigma, \sigma'\). Here \(C = C(n,p,s,C_1,C_2,\Lambda,\beta) > 1\) is bounded if \(\beta\) stays away from \(p-1\).
To apply Moser iteration to 35 , we fix \(\lambda > \mu\) and choose \(m\) such that \(\mu \kappa^{m-1} \leq \lambda \leq \mu \kappa^m\). Let \(\rho_0 \leq \mu\) with \(\lambda = \kappa^m \rho_0\), and set \(\rho_j = \kappa^j \rho_0\) for \(j=0,1,\ldots,m\). Then, iterating 35 , we obtain \[\label{6} \begin{align} \Bigg( \fint_{B_m} v^{\lambda} \, dx \Bigg)^{\frac{1}{\lambda}} &\leq \Bigg( \frac{C (2\kappa)^m}{\sigma - \sigma'} \Bigg)^{\frac{n+ps+p}{\rho_{m-1}}} \Bigg( \fint_{B_{m-1}} v^{\rho_{m-1}} \, dx \Bigg)^{\frac{1}{\rho_{m-1}}} \\ &\leq \vdots \\ &\leq \Bigg( \frac{C_{\mathrm{prod}}(m)}{(\sigma - \sigma')^{\kappa^*}} \fint_{B_{\sigma r}(x_0)} v^{\rho_0} \, dx \Bigg)^{\frac{1}{\rho_0}}, \end{align}\tag{36}\] where \[C_{\mathrm{prod}}(m) = C^{\kappa^*} \prod_{j=0}^{m-1} (2\kappa)^{\frac{(n+ps+p)(j+1)}{\kappa^j}}, \qquad \kappa^* = (n+ps+p) \sum_{j=0}^{m-1} \kappa^{-j} = \frac{(n+ps+p)\kappa (1 - \kappa^{-m})}{\kappa - 1}.\]
Note that the constant \(C\) depends on \(\lambda\) due to the singularity of the constant in Lemma 13 at \(\epsilon=0\). Since \(C_{\mathrm{prod}}(m)\) is uniformly bounded in \(m\), applying Hölder’s inequality to 36 yields \[\label{66} \begin{align} \Bigg( \fint_{B_{\sigma' r}(x_0)} v^{\lambda} \, dx \Bigg)^{\frac{1}{\lambda}} &\leq \Bigg( \frac{C}{(\sigma - \sigma')^{\kappa^*}} \Bigg)^{\frac{1}{\rho_0}} \Bigg( \fint_{B_{\sigma r}(x_0)} v^{\mu} \, dx \Bigg)^{\frac{1}{\mu}}. \end{align}\tag{37}\]
Finally, since \(\mu \kappa^{m-1} \leq \rho_0 \kappa^m\), we deduce \(\rho_0 \geq \frac{\mu}{\kappa}\), and hence the desired estimate ?? follows with \[\theta = \frac{(n+ps+p)\kappa^2}{\kappa - 1}.\] ◻
Lemma 20. Suppose \(0 < r \leq 1\) with \(r < \frac{R}{2}\). Let \(u\) be a weak supersolution of 1 such that \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Define \(v = u + d\), where \(d\) satisfies ?? . Let \(\psi \in C_c^\infty(B_{\frac{3r}{2}}(x_0))\) be such that \[0 \leq \psi \leq 1, \qquad |\nabla \psi| \leq \frac{c}{r} \;\text{in } B_{\frac{3r}{2}}(x_0), \qquad \psi \equiv 1 \;\text{in } B_r(x_0),\] for some constant \(c = c(n,p) > 0\). Then there exists a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\mu_1 := \big| \{x \in B_r(x_0) : \log v + b > a\} \big| \;\leq\; C \frac{|B_{\frac{r}{2}}(x_0)|}{a^p}, \quad \forall a > 0,\] and \[\mu_2 := \big| \{x \in B_r(x_0) : -\log v - b > a\} \big| \;\leq\; C \frac{|B_{\frac{r}{2}}(x_0)|}{a^p}, \quad \forall a > 0,\] where \[\label{b} b = - (\log v)_{\psi,B_{\frac{3r}{2}}(x_0)} = - \frac{\int_{B_{\frac{3r}{2}}(x_0)} \psi^p \log v \, dx}{\int_{B_{\frac{3r}{2}}(x_0)} \psi^p \, dx}.\qquad{(17)}\]
Proof. Let \(f = \log v + b\). By applying the weighted Poincaré inequality in Lemma 9 with \(\phi = \psi^p\), there exists a constant \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\label{7} \int_{B_r(x_0)} |f|^p \, dx \;\leq\; \int_{B_{\frac{3r}{2}}(x_0)} |f|^p \psi^p \, dx \;\leq\; C r^p \int_{B_{\frac{3r}{2}}(x_0)} |\nabla \log v|^p \psi^p \, dx \;\leq\; C r^n,\tag{38}\] where the last inequality follows from Lemma 14 together with the assumption on \(d\).
Using 38 and applying Chebyshev’s inequality, for any \(a > 0\) we obtain \[\mu_1 + \mu_2 \;\leq\; \frac{2}{a^p} \int_{B_r(x_0)} |f|^p \, dx \;\leq\; C \frac{|B_{\frac{r}{2}}(x_0)|}{a^p},\] for some constant \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\). This proves the lemma. ◻
Lemma 21. Let \(u\) be a weak solution of 1 such that \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Then there exists a positive constant \(c = c(n,p,q,s,C_1,C_2,\Lambda)\) such that \[\mathrm{Tail}(u_+; x_0, r) \leq c \sup_{B_r(x_0)} u + c r^{p'} R^{-p'} \, \mathrm{Tail}(u_-; x_0, R) + c r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \, \| f \|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}},\] whenever \(0 < r < R\) with \(r \in (0,1]\).
Proof. Let \(M = \sup_{B_r(x_0)} u\) and let \(\psi \in C_c^\infty(B_r(x_0))\) be such that \(0 \leq \psi \leq 1\) in \(B_r(x_0)\), with \(\psi \equiv 1\) in \(B_{\frac{r}{2}}(x_0)\) and \(|\nabla \psi| \leq \frac{8}{r}\) in \(B_r(x_0)\). Setting \(w = u - 2M\) and choosing \(\varphi = w \psi^p\) as a test function in ?? , we obtain \[\label{tailtst} I_1 + I_2 + I_3 - I_4 = 0,\tag{39}\] where \[I_1 = \int_{B_r(x_0)} \mathcal{A}(x,\nabla u) \nabla (w \psi^p) \, dx,\]
\[I_2 = \int_{B_r(x_0)} \int_{B_r(x_0)} J_p(u(x)-u(y)) \big(w(x)\psi^p(x) - w(y)\psi^p(y)\big) \, d\mu,\]
\[I_3 = 2 \int_{B_r(x_0)} \int_{\mathbb{R}^n \setminus B_r(x_0)} J_p(u(x)-u(y)) w(x) \psi^p(x) \, d\mu, \quad \text{and} \quad I_4 = \int_{B_r(x_0)} f w \psi^p \, dx.\]
Taking into account that \(r\in(0,1]\) and proceeding exactly as in the estimates of \(I_i\), \(i=1,2,3\), in [5], we obtain \[I_1 + I_2 + I_3 \geq c M r^{-p} \, \mathrm{Tail}(u_+; x_0, r)^{p-1} |B_r(x_0)| - c M R^{-p} \, \mathrm{Tail}(u_-; x_0, R)^{p-1} |B_r(x_0)| - c M^p r^{-p} |B_r(x_0)|,\] for some positive constant \(c = c(n,p,s,C_1,C_2,\Lambda)\). Moreover, we observe that \[I_4 \leq 3M \, {\| f \|_{L^{\frac{q}{p}}(B_R(x_0))}} \, |B_r(x_0)|^{\frac{q-p}{q}}.\]
Combining the above estimates in 39 , and using that \(0<r<R\), the result follows. ◻
We establish the result in three steps.
Step 1. We claim that there exist constants \(\epsilon = \epsilon(n,p,q,s,C_1,C_2,\Lambda) > 0\) and \(c = c(n,p,q,s,C_1,C_2,\Lambda,\epsilon) > 0\) such that \[\label{neg-poseqn} \Bigg( \fint_{B_r(x_0)} v^{\epsilon} \, dx \Bigg)^{\frac{1}{\epsilon}} \;\leq\; c \Bigg( \fint_{B_r(x_0)} v^{-\epsilon} \, dx \Bigg)^{-\frac{1}{\epsilon}},\tag{40}\] where \(v = u + d\) with \(d\) satisfying ?? .
To prove this, set \(w = \log v\). By the classical Poincaré inequality, \[\label{Poinapp1} \int_{B_r(x_0)} |w(x) - (w)_{1,B_r(x_0)}|^p \, dx \;\leq\; c_1 r^p \int_{B_r(x_0)} |\nabla w|^p \, dx,\tag{41}\] for some constant \(c_1 = c_1(n,p) > 0\). Using Hölder’s inequality together with ?? and 41 , we deduce \[\fint_{B_r(x_0)} |w(x) - (w)_{1,B_r(x_0)}| \, dx \;\leq\; K,\] where \(K = K(n,p,q,s,C_1,C_2,\Lambda) > 0\). By Theorem 10, there exists a constant \(\nu = \nu(n) > 0\) such that \[\fint_{B_r(x_0)} e^{\nu |w(x) - (w)_{1,B_r(x_0)}|} \, dx \;\leq\; 2.\] Hence, \[\fint_{B_r(x_0)} e^{\frac{\nu w}{K}} \, dx \cdot \fint_{B_r(x_0)} e^{-\frac{\nu w}{K}} \, dx \;\leq\; 4.\] This yields 40 with \(\epsilon = \frac{\nu}{K}\) and \(c = 4^{1/\epsilon}\).
Step 2. Choosing \(\sigma = 1\), \(\sigma' = \frac{1}{2}\), and \(t = \epsilon\) in Lemma 17, and combining with 40 , we obtain \[\label{pwh1} \Bigg( \fint_{B_r(x_0)} v^{\epsilon} \, dx \Bigg)^{\frac{1}{\epsilon}} \;\leq\; c \Big( \inf_{B_{\frac{r}{2}}(x_0)} u + d \Big),\tag{42}\] for some constant \(c = c(n,p,q,s,C_1,C_2,\Lambda,\epsilon) > 0\).
Step 3. Let \(\eta \in (0,(p-1)\kappa)\).
If \(\eta < \epsilon\), applying Hölder’s inequality to 42 gives \[\label{pwh2} \Bigg( \fint_{B_r(x_0)} u^{\eta} \, dx \Bigg)^{\frac{1}{\eta}} \;\leq\; c \Big( \inf_{B_{\frac{r}{2}}(x_0)} u + d \Big),\tag{43}\] where \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 0\).
If \(\eta > \epsilon\), then choosing \(\mu = \epsilon\), \(\lambda = \eta\), \(\sigma = 1\), \(\sigma' = \frac{1}{2}\) in Lemma 18, and combining with 42 , we again obtain \[\label{fwh} \Bigg( \fint_{B_r(x_0)} u^{\eta} \, dx \Bigg)^{\frac{1}{\eta}} \;\leq\; c \Big( \inf_{B_{\frac{r}{2}}(x_0)} u + d \Big),\tag{44}\] where \(c = c(n,p,q,s,C_1,C_2,\Lambda,\eta) > 0\).
Now, for any \(\delta > 0\), we choose \[d = r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0,R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} + \delta.\] Then from 43 and 44 , for any \(\delta > 0\), \[\label{pwh3} \Bigg( \fint_{B_{\frac{r}{2}}(x_0)} u^{\eta} \, dx \Bigg)^{\frac{1}{\eta}} \;\leq\; c \Big( \inf_{B_{\frac{r}{2}}(x_0)} u + r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0,R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} + \delta \Big),\tag{45}\] where \(c = c(n,p,q,s,C_1,C_2,\Lambda,\eta) > 0\). Letting \(\delta \to 0\) in 45 gives the desired estimate ?? . This completes the proof.
Let \(v = u + d\), where \(d\) satisfies ?? . Assume \(\frac{1}{2} \leq \sigma' < \sigma \leq 1\). Define \(U_m = B_{mr}(x_0)\) for every \(m \in [\frac{1}{2},1]\). Set \[w_1 = e^{-b} v^{-1}, \qquad w_2 = e^b v,\] where \[b = - \frac{\int_{B_{\frac{3r}{2}}(x_0)} \psi^p \log v \, dx}{\int_{B_{\frac{3r}{2}}(x_0)} \psi^p \, dx},\] as in ?? .
By Lemma 17, there exist constants \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\) and \(\theta = \theta(n,p,q,s) > 0\) such that \[\label{invap1} \sup_{U_{\sigma'}} w_1 \;\leq\; \Bigg( \frac{C}{(\sigma - \sigma')^{\theta}} \fint_{U_\sigma} w_1^t \, dx \Bigg)^{\frac{1}{t}}\tag{46}\] for every \(t > 0\). Moreover, by Lemma 20, there exists a constant \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\label{logestap1} \big| \{ x \in U_1 : \log w_1 > a \} \big| \;\leq\; C \frac{|U_{\frac{1}{2}}|}{a^p}, \quad \forall a > 0.\tag{47}\] Hence, by Lemma 11, \[\label{BGap1} \sup_{U_{\frac{1}{2}}} w_1 \;\leq\; C,\tag{48}\] for some \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\).
Similarly, by Lemma 18, there exist constants \(C = C(n,p,q,s,C_1,C_2,\Lambda,\eta) > 0\) and \(\theta = \theta(n,p,s) > 0\) such that \[\label{revHap1} \Bigg( \fint_{U_{\sigma'}} w_2^\eta \, dx \Bigg)^{\frac{1}{\eta}} \;\leq\; \Bigg( \frac{C}{(\sigma - \sigma')^{\theta}} \fint_{U_\sigma} w_2^\mu \, dx \Bigg)^{\frac{1}{\mu}},\tag{49}\] for every \(0 < \mu < \eta < (p-1)\kappa\). In addition, by Lemma 20, there exists a constant \(C = C(n,p,q,s,C_1,C_2,\Lambda) > 0\) such that \[\label{logestap2} \big| \{ x \in U_1 : \log w_2 > a \} \big| \;\leq\; C \frac{|U_{\frac{1}{2}}|}{a^p}, \quad \forall a > 0.\tag{50}\] Thus, by Lemma 11, \[\label{BGap2} \Bigg( \fint_{U_{\frac{1}{2}}} w_2^\eta \, dx \Bigg)^{\frac{1}{\eta}} \;\leq\; C,\tag{51}\] for some \(C = C(n,p,q,s,C_1,C_2,\Lambda,\eta) > 0\).
Now, for any \(\delta > 0\), we choose \[d = r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0,R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} + \delta.\] Multiplying 48 and 51 and then letting \(\delta \to 0\) yields the desired result ?? . This completes the proof.
Taking into account Lemma 16, the proof follows the lines of the proof of [5]. More precisely, in this way we obtain the estimate \[\sup_{B_{\frac{r}{2}}(x_0)} u \leq \bar{k},\] where \[\bar{k} = \delta \, \mathrm{Tail}\Big(u_+; x_0, \frac{r}{2}\Big) + c_0^{\frac{1}{\beta}} b^{\frac{1}{\beta^2}} \left( \fint_{B_r(x_0)} u_+^p \, dx \right)^{\frac{1}{p}} + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \| f \|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}},\] with \[c_0 = c(n,p,q,s,C_1,C_2,\Lambda) \, \delta^{\frac{(1-p)\kappa}{p}}, \quad b = 2^{\left(\frac{n+ps+p-1}{p} + \frac{\kappa-1}{\kappa}\right)\kappa}, \quad \beta = \kappa - 1, \quad \kappa = \frac{n}{n-p}.\]
Let \(0 < \rho < r\). Then, by Theorem 2 and Lemma 21, for every \(\delta \in (0,1]\), there exists a positive constant \(c = c(n,p,q,s,C_1,C_2,\Lambda)\) such that \[\label{mthm2-1} \begin{align} \sup_{B_{\frac{\rho}{2}}(x_0)} u &\leq \delta \, \mathrm{Tail}\Big(u_+; x_0, \frac{\rho}{2}\Big) + c \, \delta^{-\gamma} \left( \fint_{B_{\rho}(x_0)} u^p \, dx \right)^{\frac{1}{p}} + \rho^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)}\|f\|_{L^{\frac{q}{p}}(B_\rho(x_0))}^{\frac{1}{p-1}} \\ &\leq \delta \Big( c \sup_{B_\rho(x_0)} u + c \rho^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R)\Big) + c \, \delta^{-\gamma} \left( \fint_{B_\rho(x_0)} u^p \, dx \right)^{\frac{1}{p}} \\ &\qquad + c\rho^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}}, \end{align}\tag{52}\] where \(\gamma = \frac{(p-1)\kappa}{p(\kappa-1)}\) with \(\kappa = \frac{n}{n-p}\).
Let \(\frac{1}{2} \leq \sigma' < \sigma \leq 1\) and \(\rho = (\sigma - \sigma')r\). Using a covering argument, we obtain \[\begin{align} \sup_{B_{\sigma' r}(x_0)} u &\leq c \, \frac{\delta^{-\gamma}}{(\sigma - \sigma')^{\frac{n}{p}}} \left( \fint_{B_{\sigma r}(x_0)} u^p \, dx \right)^{\frac{1}{p}} + c \delta \sup_{B_{\sigma r}(x_0)} u + c \delta r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) \\ &\qquad+ c\frac{r^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)}}{(\sigma-\sigma')^\frac{n-p}{p-1}}\|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} \\ &\leq c \, \frac{\delta^{-\gamma}}{(\sigma - \sigma')^{\frac{n}{p}}} \big( \sup_{B_{\sigma r}(x_0)} u \big)^{\frac{p-t}{p}} \left( \fint_{B_{\sigma r}(x_0)} u^t \, dx \right)^{\frac{1}{t}} + c \delta \sup_{B_{\sigma r}(x_0)} u + c \delta r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) \\ &\qquad + c\frac{r^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)}}{(\sigma-\sigma')^\frac{n-p}{p-1}}\|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}}, \end{align}\] for every \(t \in (0,p)\) with a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 1\).
Applying Young’s inequality with exponents \(\frac{p}{t}\) and \(\frac{p}{p-t}\) to the above estimate and choosing \(\delta = \frac{1}{4c}\in(0,1]\), we obtain \[\label{heqn} \sup_{B_{\sigma' r}(x_0)} u \leq \frac{1}{2} \sup_{B_{\sigma r}(x_0)} u + \frac{c}{(\sigma - \sigma')^m} \Bigg\{\Big( \fint_{B_r(x_0)} u^t \, dx \Big)^{\frac{1}{t}} + r^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)}\|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}}\Bigg\} + c r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R),\tag{53}\] where \(m=\max\{\frac{n}{t},\frac{p-n}{p-1}\}\) for every \(t \in (0,p)\) with a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda) > 0\).
Using Lemma 7 in the above estimate, it follows that \[\sup_{B_{\frac{r}{2}}(x_0)} u \leq c \left( \fint_{B_r(x_0)} u^t \, dx \right)^{\frac{1}{t}} + c r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) + cr^\frac{p-n}{p-1}R^\frac{n(q-p)}{q(p-1)}\|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}},\] for every \(t \in (0,p)\) with a constant \(c = c(n,p,q,s,C_1,C_2,\Lambda,t) > 0\).
Let \(\epsilon>0\), then choosing \[d = r^{p'} R^{-p'} \mathrm{Tail}(u_-; x_0, R) + r^{\frac{p-n}{p-1}} R^{\frac{n(q-p)}{q(p-1)}} \|f\|_{L^{\frac{q}{p}}(B_R(x_0))}^{\frac{1}{p-1}} + \epsilon,\] in the estimate 42 and letting \(\epsilon \to 0\), and combining the resulting estimate with the above inequality 53 , the theorem follows.
The author thanks IISER Berhampur for the seed grant: IISERBPR/RD/OO/2024/15, Date: February 08, 2024.