Existence results for double phase problems depending on Robin and Steklov eigenvalues for the $p$-Laplacian

TL;DR

This paper establishes existence results for double phase problems involving the p-Laplacian, with nonlinear boundary conditions, where solutions depend on Robin and Steklov eigenvalues, using variational and comparison methods.

Contribution

It provides new existence results for double phase problems with boundary conditions influenced by Robin and Steklov eigenvalues, under general assumptions.

Findings

Existence of solutions depends on Robin and Steklov eigenvalues.

Use of variational, truncation, and comparison techniques.

Results apply to problems with gradient dependence and nonlinear boundary conditions.

Abstract

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.