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

Existence of sign-changing solutions to quasilinear elliptic equations of the form -Sigma(N)(i,j=1) D-j(a(ij)(x,u)D(i)u) + 1/2 Sigma(N)(i,j=1) D(s)a(ij)(x,u)D(i)uD(j)u = lambda f(x, u) Omega under the Dirichlet boundary condition, where Omega subset of R-N (N >= 2) is a bounded domain with smooth boundary and lambda > 0 is a parameter, is studied. In particular, we examine how the number of sign-changing solutions depends on the parameter lambda. In the case considered here, there exists no nontrivial solution for lambda sufficiently small. We prove that, as lambda 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.