Uniqueness Results for Mixed Local and Nonlocal Equations with Singular Nonlinearities and Source Terms
Abdelhamid Gouasmia
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} -Δ_{p} u + (-Δ)^{s}_{q} u = f(x) u^{-α} + g(x) u^β, \quad u > 0 \quad \text{in } Ω; \quad u = 0, \quad \text{in } \mathbb{R}^{N} \setminus Ω, \end{equation} where \( Ω\subset \mathbb{R}^N \) is an open bounded domain with a \( C^{2} \) boundary \( \partial Ω\), 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 \( 0 < β< q - 1 \) and \( α> 0 \). The function \( f \) is non-zero and belongs to a suitable Lebesgue space \( L^{r}(Ω) \) for some \( r \in [1, \infty] \), or satisfies a growth condition involving negative powers of the distance function \( d(\cdot) \) near the boundary \( \partial Ω\). 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 \( f(x) \sim d^{-δ}(x) \) and \( x \) is near the boundary, under the condition \( δ\geq p \). Our approach leverages the Picone identities on one hand and the interaction between the local and non-local terms on the other hand.