Nehari manifold approach for superlinear double phase problems with variable exponents

TL;DR

This paper develops a Nehari manifold method to establish multiple solutions, including sign-changing ones, for superlinear double phase elliptic equations with variable exponents, broadening the understanding of such complex nonlinear problems.

Contribution

It introduces a novel application of the Nehari manifold approach to variable exponent double phase problems, providing multiplicity results and insights into nodal domains.

Findings

Existence of positive, negative, and sign-changing solutions.

Application of Nehari manifold method to complex nonlinear PDEs.

Analysis of nodal domains for sign-changing solutions.

Abstract

In this paper we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity, we prove a multiplicity result for such problems whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains.