Existence of nontrivial solutions for p -Kirchhoff type equations
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Existence of nontrivial solutions for p-Kirchhoff type equations Chunhan Liu* , Jianguo Wang and Qingling Gao * Correspondence: [email protected] Department of Mathematics, Qilu Normal University, Jinan, 250013, P.R. China
Abstract In this paper, the linking theorem and the mountain pass theorem are used to show the existence of nontrivial solutions for the p-Kirchhoff equations without assuming Ambrosetti-Rabinowitz type growth conditions, nontrivial solutions are obtained. MSC: 35J60; 35J25 Keywords: linking theorem; mountain pass theorem; nontrivial solutions
1 Introduction In this paper, we consider the nonlocal elliptic problem of the p-Kirchhoff type given by [M( |∇u|p dx)]p– (–p u) = f (x, u), in , () u = , on ∂, where ⊂ RN is a bounded domain, and p u = div(|∇u|p– ∇u) is the p-Laplacian with < p < N. Recently, the equation –(a + b |∇u| dx)u = f (x, u), in , () u = , on ∂, began to attract the attention of several researchers only after Lion [] had proposed an abstract framework for this problem. Perera and Zhang [] obtained a nontrivial solution of () by using the Yang index and critical group. They revisited () via invariant sets of decent flow and obtained the existence of a positive solution, a negative, and a sign-changing solutions in []. The study of Kirchhoff-type equations has been extended to the following case involving the p-Laplacian: –M( p |∇u|p dx) div(|∇u|p– ∇u) = f (x, u), in , u = ,
on ∂,
for details see [–]. One of the authors has done some related work on this field. Liu [] gave infinite solutions to the following equation via the fountain theorem and the dual fountain theorem: [M( (|∇u|p + λ(x)|u|p ) dx)]p– (–p u + λ(x)|u|p– u) = f (x, u), in , |∇u|p– ∂u = η|u|p– , on ∂. ∂ν ©2013 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu et al. Boundary Value Problems 2013, 2013:279 http://www.boundaryvalueproblems.com/content/2013/1/279
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However, to the best of our knowledge, there have been few papers dealing with equation () using the linking theorem and the mountain pass theorem. This paper will make some contribution to this research field. It is well known (see []) that the eigenvalue problem
–p u = λ|u|p– u u=
in , on ∂,
has the first eigenvalue λ > , which is simple, and has an associated eigenfunction φ > . It is also known that λ is an isolated point of σ (–p ), the spectrum of –p , which contains at least an eigenvalue sequence {λn } and < λ < λ ≤ λ ≤ · · · ≤ λn → ∞. Let ,p W () =
p p u ∈ L () : |∇u| dx < ∞ and u|∂ =
,p be a Banach space with the norm u = u p = ( |∇u|p dx) p for u ∈ W (). W = span{φ } be the one-dimensional eigenspace associated with λ , where φ = . Let ,p ,p p–
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