Mixed local-nonlocal quasilinear problems with critical nonlinearities

Abstract

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -Δ_p u+(-Δ_p)^{s} u & = & λ|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }Ω,\\ u & = & 0 & \mbox{ on } \mathbb{R}^{N} \setminus Ω, \end{array} \right. $$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded open set with smooth boundary, dimension $N\ge 2$, 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.