Constant sign and nodal solutions for nonhomogeneous Robin boundary value problems with asymmetric reactions

TL;DR

This paper proves the existence of multiple solutions, including positive, negative, and sign-changing, for a nonlinear elliptic equation with Robin boundary conditions under certain conditions, using variational methods.

Contribution

It introduces new results on multiple solutions for nonhomogeneous elliptic problems with asymmetric reactions, employing advanced variational and topological techniques.

Findings

At least four nontrivial solutions exist for small parameters.

Two solutions are positive, one is negative, and one is sign-changing.

Solutions are obtained via critical point theory and Morse theory.

Abstract

We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques.