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arXiv · 2308.00205

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

Aboubacar Marcos·Janvier Soninhekpon
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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.

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Aboubacar Marcos, Janvier Soninhekpon. 2023-08-25. On eigenvalues problems for the $p(x)$-Laplacian. https://doi.org/10.1016/j.jmaa.2024.128143

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