Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations

Abstract

Existence of sign-changing solutions to quasilinear elliptic equations of the form [Formula: see text] under the Dirichlet boundary condition, where [Formula: see text] ([Formula: see text]) is a bounded domain with smooth boundary and [Formula: see text] is a parameter, is studied. In particular, we examine how the number of sign-changing solutions depends on the parameter [Formula: see text]. In the case considered here, there exists no nontrivial solution for [Formula: see text] sufficiently small. We prove that, as [Formula: see text] becomes large, there exist both arbitrarily many sign-changing solutions with negative energy and arbitrarily many sign-changing solutions with positive energy. The results are proved via a variational perturbation method. We construct new invariant sets of descending flow so that sign-changing solutions to the perturbed equations outside of these sets are obtained, and then we take limits to obtain sign-changing solutions to the original equation.