Abstract
By a variant version of mountain pass theorem, the existence and multiplicity of solutions are obtained for a class of superlinear -Laplacian equations: . In this paper, we suppose neither satisfies the superquadratic condition in Ambrosetti-Rabinowitz sense nor is nondecreasing with respect to .
Highlights
Introduction and main resultsIn this paper, we consider the following superlinear p-Laplacian equations with Dirichlet boundary value condition:− pu = f (x, u), x ∈ Ω, u = 0, x ∈ ∂Ω, (1.1)where pu is the p-Laplacian operator: pu = div(|∇u|p−2∇u) with p > 1, Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω, f ∈ C(Ω × R, R) is subcritical in t, that is, there is a q ∈ (p, np/(n − p)) when N > p; q ∈ (p, +∞) when N ≤ p such that f (x,t) lim t→∞ |t|q−1 =0 (1.2)uniformly in a.e. x ∈ Ω
Where pu is the p-Laplacian operator: pu = div(|∇u|p−2∇u) with p > 1, Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω, f ∈ C(Ω × R, R) is subcritical in t, that is, there is a q ∈ (p, np/(n − p)) when N > p; q ∈ (p, +∞) when N ≤ p such that f (x,t) uniformly in a.e. x ∈ Ω
We can see that we extend the results of [8, 18, 19] in superlinear case on two hands
Summary
We consider the following superlinear p-Laplacian equations with Dirichlet boundary value condition:. Li and Zhou extend the results to p > 1 in [8] (see [8, Remark 1.2]) In their discussion, they suppose (f1), Assumption 1.1, and the following condition hold: (f5) limt→0+ ( f (x, t)/tp−1) = p(x) and limt→+∞( f (x, t)/tp−1) = +∞ uniformly in a.e. x ∈ Ω, where p(x) ≡ l ∈ [0, λ1) (in [8, 18]) or p ∈ L∞(Ω) with p ∞ < λ1 (in [19]). Liu and Li in [9] has got infinitely many solutions of problem (1.1) by the fountain theorem In their discussion, they supposed that f (x,t) is odd with respect to t.
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