Abstract
The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.
Highlights
Let Ω ⊂ RN be a bounded domain with a C2-boundary ∂Ω, and let V W1,p Ω and V0 W01,p Ω, 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively
We are going to study the existence of multiple solutions of 1.1 for two different classes of j which are in some sense complementary
If g : Ω × R → R is a Caratheodory function, that is, x → g x, s is measurable in Ω for all s ∈ R and s → g x, s is continuous in R for a.a. x ∈ Ω, ∂G x, s {g x, s } is single-valued, and problem 1.7 reduces to the following nonlinear elliptic problem depending on parameters a and b: find u ∈ V0 \ {0} and constants a ∈ R, b ∈ R such that
Summary
Let Ω ⊂ RN be a bounded domain with a C2-boundary ∂Ω, and let V W1,p Ω and V0 W01,p Ω , 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively. If g : Ω × R → R is a Caratheodory function, that is, x → g x, s is measurable in Ω for all s ∈ R and s → g x, s is continuous in R for a.a. x ∈ Ω, ∂G x, s {g x, s } is single-valued, and problem 1.7 reduces to the following nonlinear elliptic problem depending on parameters a and b: find u ∈ V0 \ {0} and constants a ∈ R, b ∈ R such that. The aim of this section is to provide an existence result of multiple solutions for all values of the parameter λ in an interval 0, λ0 , with λ0 > 0, guaranteeing that for any such λ there exist at least three nontrivial solutions of problem 1.4 , two of them having opposite constant sign and the third one being sign-changing or nodal.
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