Multiple Solutions with Sign Information for a Class of Parametric Superlinear (p, 2)-Equations

Abstract

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian (with p > 2) and a Laplacian (a two phase equation). The reaction consists of a parametric (p − 1)-superlinear term and a (p − 1)-sublinear perturbation. We show that for all λ > 0 big, the problem has at least three nontrivial smooth solutions, all with sign information. Also we determine their asymptotic behaviour as the parameter λ → ∞. When we strengthen the regularity of the perturbation term, we produce a second nodal solution, for a total of four solutions, all with sign information.

Multiplicity results for some quasilinear elliptic problems with concave nonlinearities

“…In the case of p > 2, λ > 0 and µ > 0, in [23] the authors considered problem (1.1) with nonlinearity f (x, u) being resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian and obtained six nontrivial smooth solutions under λ is small enough by using variational methods together with truncation and comparison techniques and Morse theory (critical groups). The authors in [12,17] did some similar work.…”

“…When p < N , there have been essential many works (such as [4,5,11,12,13,17,21,23,24,29,30]) to deal with the existence of nontrivial solutions or the existence of nodal solutions for problem (1.1). Moreover, almost all of the works involve the nonlinearity f (x, u) with standard subcritical polynomial growth, say, (SCP) : There exist positive constants c 0 and q…”