Abstract
This paper is concerned with the following Kirchhoff-type problems with a nonsmooth potential: \(- (a+b\int_{\Omega} \vert \nabla u\vert ^{2}\,\mathrm {d}x )\Delta u\in\partial j(x,u) \) for a.a. \(x\in\Omega\), \(u=0 \) on ∂Ω. Using the nonsmooth mountain pass theorem, the nonsmooth local linking theorem, and the nonsmooth fountain theorem, we establish the existence and multiplicity of solutions for the problem. All this is based on the nonsmooth critical point. Some recent results in the literature are generalized and improved.
Highlights
In recent years, various Kirchhoff-type problems have been widely discussed by lots of authors
Liang et al in [ ] firstly studied the bifurcation phenomena of problem ( . ) with the right-hand side of the first equation replaced by νf (x, u) by using the topological degree and variational methods
We introduce a nonsmooth version fountain theorem which was proved by Dai [ ]
Summary
Various Kirchhoff-type problems have been widely discussed by lots of authors. (i) It is not difficult to see that there exist many functions, which, respectively, satisfy Theorem . There exist functions j(x, u), which satisfy all hypotheses of Theorem .
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