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Abstract
We consider a perturbed version of the Robin eigenvalue problem for the p-Laplacian. The perturbation is (p - 1)-superlinear. Using the Nehari manifold method, we show that for all parameters lambda < {hat{lambda }}_1 (= the principal eigenvalue of the differential operator), there exists a ground-state nodal solution of the problem.
Highlights
The analysis developed in this paper includes the borderline case p = N
Using the Nehari manifold method, we show that if λ < λ1 (here λ1 is the principal eigenvalue of the differential operator u → −Δpu + ξ(z)|u|p−2u with Robin boundary condition), problem (Pλ) has a ground-state nodal solution
We say that λ ∈ R is an eigenvalue of the operator u → −Δpu+ξ(z)|u|p−2u with Robin boundary condition, if problem (2) admits a nontrivial solution u ∈ W 1,p(Ω), known as an eigenfunction corresponding to the eigenvalue λ
Summary
Suppose that Ω ⊆ RN is a bounded domain with a C2-boundary ∂Ω. In this paper, we study the following nonlinear parametric Robin problem:. The analysis developed in this paper includes the borderline case p = N. In this situation, the Dirichlet energy |Du|N dx is conformally invariant. Problem (Pλ) contains the perturbation u → ξ(z)|u|p−2u with the potential function ξ ∈ L∞(Ω), ξ(z) 0 for a.a. z ∈ Ω. We can view problem (Pλ) as a superlinear perturbation of the Robin eigenvalue problem for the operator u → −Δpu + ξ(z)|u|p−2u. In the boundary condition, ∂∂nup denotes the conormal derivative of u corresponding to the p-Laplace differential operator. This directional derivative is interpreted using the
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