Uniqueness Results for Mixed Local and Nonlocal Equations with Singular Nonlinearities and Source Terms

TL;DR

This paper proves uniqueness and existence of solutions for a complex PDE involving local and nonlocal operators with singular nonlinearities, and establishes conditions for non-existence near the boundary.

Contribution

It introduces a novel comparison principle for infinite energy solutions and provides new existence, regularity, and non-existence results for mixed local and nonlocal equations with singular terms.

Findings

Uniqueness of infinite energy solutions established.

Existence of weak solutions demonstrated under various conditions.

Non-existence results near the boundary for certain singularities.

Abstract

This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x) u^{-\alpha} + g(x) u^{\beta}, \quad u > 0 \quad \text{in } \Omega; \quad u = 0, \quad \text{in } \mathbb{R}^{N} \setminus \Omega, \end{equation} where \( \Omega \subset \mathbb{R}^N \) is an open bounded domain with a \( C^{2} \) boundary \( \partial \Omega \), and \( N > p \). We assume that \( 0 < s < 1 \) and \( 1 < p, q < \infty \), with the conditions \( q = p \) or \( q < p \), corresponding to the homogeneous and non-homogeneous cases, respectively. The parameters satisfy \( 0 < \beta < q - 1 \) and \( \alpha > 0 \). The function \( f \) is non-zero and belongs to a suitable Lebesgue space \( L^{r}(\Omega) \) for some \( r \in [1, \infty] \), or satisfies a growth condition involving negative powers of the distance function \( d(\cdot) \) near the boundary \( \partial \Omega \). Additionally, \( g \) is a nonnegative function within appropriate Lebesgue spaces. The primary objectives of this paper are twofold. First, we establish the uniqueness of infinite energy solutions to problem \eqref{A} by introducing a novel comparison principle under certain conditions. Second, we derive several existence results for weak solutions in various senses, accompanied by regularity results for problem \eqref{A}. Furthermore, we present a non-existence result when the function \( f(x) \sim d^{-\delta}(x) \) and \( x \) is near the boundary, under the condition \( \delta \geq p \). Our approach leverages the Picone identities on one hand and the interaction between the local and non-local terms on the other hand.