Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential

Abstract

We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -Δu + (-Δ)^s u - μ\frac{u}{|x|^2} &= λ|u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } Ω\subset \mathbb{R}^N, u &= 0 \quad \text{in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary, $0<s<1$, $1<p<2^{\ast }$, with $2^{\ast }=\frac{2N}{N-2}$, $\lambda >0$, and $\mu \in (0,\bar{\mu })$ where $\bar{\mu }={\left(\frac{N-2}{2}\right)}^2$. The aim of this paper is twofold. First, we establish uniform asymptotic estimates for solutions of the problem by means of a suitable transformation. Then, according to the value of the exponent $p$, we analyze three distinct cases and prove the existence of a positive solution. Moreover, in the sublinear regime $1<p<2$, we demonstrate the existence of multiple positive solutions for small perturbations of the fractional Laplacian.