Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations

TL;DR

This paper investigates the existence of solutions for $(p,q)$-Laplacian equations with nonlinear terms and non-homogeneous boundary conditions, providing spectral analysis and applying variational methods to establish solution existence under certain asymptotic conditions.

Contribution

It offers a complete spectral characterization of the $(p,q)$-Laplacian with weights and introduces new existence results for nonlinear problems based on spectral thresholds.

Findings

Spectral description of the eigenvalue problem with weights and boundary parameters.

Existence of weak solutions below the first eigenvalue of the $q$-Laplacian.

Existence results when nonlinearities are below the first Steklov-Neumann eigenvalue line.

Abstract

We study the solvability of $(p,q)$-Laplacian problems with nonlinear reaction terms and non-homogeneous Neumann boundary conditions. First, we provide a complete description of the spectrum of the eigenvalue problem involving the $(p,q)$-Laplacian with weights and a spectral parameter present in both the differential equation and on the boundary. Then, using variational methods and critical point theory, we prove the existence of weak solutions for the nonlinear problem when the nonlinearities involved remain asymptotically, in some sense, below the first eigenvalue of the $q$-Laplacian problem with weights and a spectral parameter present in both the differential equation and on the boundary. We also establish an existence result for the nonlinear problem when the nonlinearities involved remain asymptotically below the first Steklov-Neumann eigenvalue-line, which is a line connecting the first Steklov and first Neumann eigenvalues for $q$-Laplacian problems with weights and a spectral parameter present either in the differential equation or on the boundary.