Mixed local-nonlocal quasilinear problems with critical nonlinearities

TL;DR

This paper investigates the existence and multiplicity of solutions for a mixed local-nonlocal quasilinear PDE involving critical nonlinearities, using variational and topological methods across different parameter regimes.

Contribution

It introduces a novel analysis of a combined p-Laplacian and fractional p-Laplacian operator with critical nonlinearities, exploring three scenarios based on the exponent q.

Findings

Established existence of solutions in various regimes of q.

Identified multiple solutions using topological methods.

Extended variational techniques to mixed local-nonlocal operators.

Abstract

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on } \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb{R}^N$ is a bounded open set with smooth boundary, dimension $N\geq 2$, parameter $\lambda>0$, exponents $0<s<1<p<N$, while $q\in(1,p^{\ast})$ with $p^{\ast}=\frac{Np}{N-p}$. The problem is driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian. We analyze three different scenarios depending on exponent $q$. For this, we combine variational methods with some topological techniques, such as the Krasnoselskii genus and the Lusternik-Schnirelman category theories.