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

TL;DR

This work investigates a mixed local–nonlocal elliptic equation with Hardy potential, combining $- abla u$ and $(-\Delta)^s u$ and a singular weight $|x|^{-2}$ in a bounded domain. A key idea is a ground-state–type transformation $u(x)=|x|^{-d}w(x)$ with $d=\sqrt{\bar{\mu}}-\sqrt{\bar{\mu}-\mu}$ that yields a weighted, radial framework and enables a Harnack inequality, uniform bounds, and precise asymptotics $u(x) \sim C|x|^{-d}$ near the origin. The paper then develops a comprehensive existence theory across three regimes: linear ($p=2$) with a threshold phenomenon, superlinear ($p\in(2,2^*)$) via Mountain Pass, and sublinear ($p\in(1,2)$) with multiplicity and minimal-solution analysis, including a sharp blow-up analysis for minimizers. These results extend Brezis–Nirenberg-type theory to mixed local–nonlocal operators with Hardy potential and provide detailed asymptotics, variational thresholds, and multiplicity results relevant to models with diffusion and jumps.

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 $ Ω\subset \mathbb{R}^N $ is a bounded domain with smooth boundary, $ 0 < s < 1 $, $ 1 < p < 2^* $, with $ 2^* = \frac{2N}{N-2} $, $ λ> 0 $, and $ μ\in (0, \barμ) $ where $\bar μ= \left( \frac{N-2}{2} \right)^2$. 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.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.1: Lemma $3.5$ of stein_weiss1957fractional
• Lemma 2.2
• Lemma 2.3