Abstract
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 obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.
Highlights
IntroductionWe consider the following double phase problem with nonlinear boundary condition and convection term given by
Let Ω ⊂ RN, N >, be a bounded domain with Lipschitz boundary ∂Ω
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence
Summary
We consider the following double phase problem with nonlinear boundary condition and convection term given by. Farkas-Fiscella-Winkert [13] studied singular Finsler double phase problems with nonlinear boundary condition and critical growth of the form. The proof is based on the surjectivity result for pseudomonotone operators and on the properties of the eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian. In the last section, we skip the convection term and use variational tools in order to prove the existence of two constant sign solutions for superlinear problems. It can be found, for example, in Papageorgiou-Winkert [30, Theorem 6.1.57].
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