Multiple solutions of double phase variational problems with variable exponent

TL;DR

This paper establishes the existence of multiple solutions for a class of double phase variable exponent elliptic equations using variational methods in generalized Orlicz-Sobolev spaces, extending previous constant exponent results.

Contribution

It extends prior work by handling variable exponents in double phase problems and weakens hypotheses using a weighting method to address compactness issues.

Findings

Proves multiple solutions exist for variable exponent double phase problems.

Extends previous constant exponent results to variable exponents.

Employs critical points theory in generalized Orlicz-Sobolev spaces.

Abstract

This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{\alpha (x)-2}u=f(x,u)$ in $ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic operator $\mathbf{ A}$ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $ | \xi | ^{q(x)-2}\xi $ for small $| \xi | $ and like $| \xi | ^{p(x)-2}\xi $ for large $ | \xi | $, where $1<\alpha (\cdot )\leq p(\cdot )<q(\cdot )<N$. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Our results extend the previous works Azzollini, d'Avenia, and Pomponio (2014) and Chorfi and R\u{a}dulescu (2016), from the case when exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and $% q(\cdot )$ are functions. We also substantially weaken some of their hypotheses overcome the lack of compactness by using the weighting method.