Qualitative properties of positive solutions to mixed local and nonlocal critical problems in $\mathbb{R}^n$

TL;DR

This paper studies positive solutions to a mixed local and nonlocal critical elliptic equation in Rn, establishing existence, regularity, decay, and symmetry properties of solutions, and clarifying the solution regularity depending on the parameter p.

Contribution

It proves the existence, regularity, decay estimates, and symmetry of positive solutions to a mixed local and nonlocal critical elliptic equation, extending previous weak solution results.

Findings

Viscosity solutions are regular and belong to specific Hölder and Sobolev spaces.

Solutions exhibit decay at infinity and are radially symmetric.

Existence of positive solutions with qualitative properties is established.

Abstract

We consider the following mixed local and non-local critical elliptic equation: \begin{equation*}\label{0.1} \left\{ \begin{array}{lll} -\Delta u+(-\Delta)^su=\lambda h u^{p}+u^{2^*-1}, &\text{in}\,\, \mathbb{R}^n, u>0, &\text {in} \,\, \mathbb{R}^n, \lim\limits_{|x|\to\infty} u(x) = 0, \end{array} \right. \end{equation*} where $n\geqslant4, \,\, p\in (0,2^*-1),\,\, 2^*:=\frac{2n}{n-2}$ and $h$ is a positive function. We first show the existence and regularity results of viscosity solutions to the above critical elliptic equation. More precisely, from \cite{Su-Xu} weak solutions are obtained and we prove they are indeed viscosity solutions and their regularity is: \( u \in C^{\alpha}(\mathbb{R}^n) \) for $p\in(0,1);$ \( u \in C^{2,\beta}(\mathbb{R}^n) \) for $p\in [1, 2^*-1).$ Moreover, for $p\in [1, 2^*-1)$, these viscosity solutions are indeed classical ones and we then prove the existence of positive solutions with the qualitative properties such as the decay estimates and the radial symmetry.