On eigenvalues problems for the $p(x)$-Laplacian

TL;DR

This paper investigates the spectrum of nonlinear eigenvalue problems involving the variable exponent $p(x)$-Laplacian, revealing a continuous spectrum and characterizing the first eigenvalue using variational methods.

Contribution

It demonstrates that the spectrum includes a continuous set of eigenvalues and identifies the first eigenvalue via Lagrange multipliers, extending understanding of $p(x)$-Laplacian eigenproblems.

Findings

Spectrum contains a continuous set of eigenvalues.

The smallest eigenvalue matches the first in the Ljusternik-Schnirelman sequence.

Variational methods effectively analyze the eigenvalues.

Abstract

This paper studies nonlinear eigenvalues problems with a double non homogeneity governed by the $p(x)$-Laplacian operator, under the Dirichlet boundary condition on a bounded domain of $\mathbb{R}^N(N\geq2)$. According to the type of the nonlinear part (sublinear, superlinear) we use the Lagrange multiplier's method, the Ekeland's variational principle and the Mountain-Pass theorem to show that the spectrum includes a continuous set of eigenvalues, which can in some contexts be all the set $\mathbb{R_+^{*}}$. Moreover, we show that the smallest eigenvalue obtained from the Lagrange multipliers is exactly the first eigenvalue in the Ljusternik-Schnirelman eigenvalues sequence. Key words: Nonlinear eigenvalue problems, $p(x)$-Laplacian, Lagrange multipliers, Ekeland variational principle, Ljusternik-Schnirelman principle, Mountain-Pass theorem.