Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition

Abstract

In this paper, we study and discuss nonlinear elliptic equations with Neumann boundary condition for oscillation problem. We obtain infinitely many positive and negative solutions of (1.1) which are all nonconstant, and get at least two nonconstant solutions in every order interval under resonance case. Moreover, we yield infinitely many sign-changing solutions of (1.1) under some assumptions. Furthermore, we give a precise description of critical groups of some kinds of critical points. We draw the conclusions by using sub-sup solution method, mountain pass theorem in order intervals, Leray–Schauder degree theory and Morse theory.