Nodal solutions for the nonlinear Robin problem in Orlicz spaces

Abstract

In this paper we consider a non-linear Robin problem driven by the Orlicz $g$-Laplacian operator. Using variational technique combined with a suitable truncation and Morse theory (critical groups), we prove two multiplicity theorems with sign information for all the solutions. In the first theorem, we establish the existence of at least two non-trivial solutions with fixed sign. In the second, we prove the existence of at least three non-trivial solutions with sign information (one positive, one negative, and the other change sign) and order. The result of the nodal solution is new for the non-linear $g$-Laplacian problems with the Robin boundary condition.