Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities

Tacksun Jung,Q-Heung Choi

doi:10.1186/s13661-019-1165-5

Tacksun Jung, Q-Heung Choi

Open Access

https://doi.org/10.1186/s13661-019-1165-5

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Abstract

We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when pge 2). We obtain the third result by Leray–Schauder degree theory.

Highlights

The p-Laplacian boundary value problems with p-growth conditions arise in applications of nonlinear elasticity theory, electro-rheological fluids, and in non-Newtonian fluid theory of a porous medium
When p = p(x), the p(x)-Laplacian problems are inhomogeneous, so they may have singular phenomena like inf Λ = 0, where Λ is the set of the eigenvalues of the p(x)-Laplacian eigenvalue problem
The eigenvalue problem when α = 0, p(x) = p constant and 1 < p < ∞ has no singular phenomena, i.e., inf Λ > 0. It was proved in [9] that the eigenvalue problem when α = 0, p(x) = p constant and 1 < p < ∞ has a nondecreasing sequence of nonnegative eigenvalues λj, obtained by the Ljusternik–Schnirelman principle, tending to ∞ as j → ∞, and the corresponding orthonomalized eigenfunctions φj, j = 1, 2, . . . , where the first eigenvalue λ1 is positive and simple and only eigenfunctions associated with λ1 do not change sign, the set of eigenvalues is closed, the first eigenvalue λ1 is isolated

Summary

Introduction

The p-Laplacian boundary value problems with p-growth conditions arise in applications of nonlinear elasticity theory, electro-rheological fluids, and in non-Newtonian fluid theory of a porous medium (cf. [6, 7, 13]). Let us set τ can be rewritten as u = (–Lp – τ gp)–1 (b – τ )|u|p–2u+ – (a – τ )|u|p–2u– + sφ1p–1 .

Results

Conclusion