Existence and Multiplicity Results for an Elliptic Problem Involving Mixed Local and Nonlocal Operator

Abstract

In this work, we consider the following mixed local-nonlocal quasilinear ellipic problem
$$
(P_\lambda)\left\{
\begin{array}{rcll}
       -\Delta_p u+(-\Delta)_p^s u &= & \lambda f(u) & \text{in}\Omega,\\
       u &> & 0 & \text{in }\Omega,\\
       u & = & 0 & \text{in }\mathbb{R}^N \setminus\Omega,
\end{array}
\right.
$$
where Ω⊂R^N is a bounded regular domain in R^N with 0<s<1<p<N and f: R→R is a continuous function, that have a finite number of zeroes, changing sign between them. The main goal of this paper is to prove the existence and multiplicity of positive solutions for such problems by using variational methods.