Superlinear elliptic equations with unbalanced growth and nonlinear boundary condition

TL;DR

This paper introduces a new norm in Musielak-Orlicz Sobolev spaces and proves the existence of multiple solutions for nonlinear elliptic equations with unbalanced growth and nonlinear boundary conditions.

Contribution

It presents a novel norm in Musielak-Orlicz Sobolev spaces and establishes multiple solution results for complex nonlinear elliptic problems with boundary conditions.

Findings

Boundedness of solutions for nonlinear Neumann problems

Existence of multiple solutions for variable exponent double phase problems

Use of mountain-pass and Nehari manifold methods

Abstract

In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma.