Uniqueness Results for Mixed Local and Nonlocal Equations with Singular Nonlinearities and Source Terms (2411.01026v1)

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.