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.
Highlights
Let W ,p( ) = W ⊕ V , where V = {u ∈ W ,p( ) : uφ p– dx = }, there exists λ > λ such that
In this paper, we consider the nonlocal elliptic problem of the p-Kirchhoff type given by [M( |∇u|p dx)]p– (– pu) = f (x, u), in, ( )u =, on ∂, where ⊂ RN is a bounded domain, and < p < N.Recently, the equation pu = div(|∇u|p– ∇u) is the p-Laplacian with–(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
The study of Kirchhoff-type equations has been extended to the following case involving the p-Laplacian:
Summary
Let W ,p( ) = W ⊕ V , where V = {u ∈ W ,p( ) : uφ p– dx = }, there exists λ > λ such that Theorem [ ] (Linking theorem) Let X = Y ⊕ Z be a Banach space with dim Y < ∞. If satisfies the (PS) condition with c = inf max γ (u) , γ ∈ u∈M
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