Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

TL;DR

This paper investigates multiple solutions for a class of quasilinear elliptic equations with mixed local and nonlocal operators, singular nonlinearities, and superlinear growth, using variational methods and Nehari manifold analysis.

Contribution

It introduces a novel approach to analyze the Nehari manifold structure for mixed local-nonlocal problems with singular and superlinear terms, proving the existence of multiple solutions.

Findings

Existence of at least two solutions established.

Solutions are bounded in the $L^ Infty$ norm.

Analysis of the Nehari manifold decomposition is key to results.

Abstract

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a parameter $\lambda$. We study the existence of multiple solutions with the help of variational methods by restricting the associated energy functional on appropriate subsets of the Nehari manifold. Using the topological index and the structure of the fibering maps, we analyse a key splitting property of the associated Nehari manifold. This decomposition allows us to establish the existence of two distinct solutions. Additionally, we establish the $L^\infty$-bound for the solutions.