The spectrum and an index formula for the Neumann p−p-p−Laplacian and multiple solutions for problems with a crossing nonlinearity (original) (raw)
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In this article, we study eigenvalue problems with the p-Laplacian operator: −(|y′|p−2y′)′ = (p− 1)(λρ(x)− q(x))|y|p−2y on (0, πp), where p > 1 and πp ≡ 2π/(p sin(π/p)). We show that if ρ ≡ 1 and q is singlewell with transition point a = πp/2, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same result also holds for p-Laplacian problem with single-barrier ρ and q ≡ 0. Applying these results, we extend and improve a result by [24] by using finitely many eigenvalues and by generalizing the string equation to p-Laplacian problem. Moreover, our results also extend a result of Huang [14] on the estimate of the first instability interval for Hill equation to single-well function q.
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Purpose The purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian. Design/methodology/approach The approach is based on variational methods and critical point theory. Findings Theorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4. Originality/value The paper is original and the authors think the results are new.