The paper investigates the following fractional Schrödinger equation: (-) s u + V(x)u = K(x)f (u), x ∈ R N , where 0 < s < 1, 2s < N, (-) s is the fractional Laplacian operator of order s. V(x), K(x) are nonnegative continuous functions and f (x) is a continuous function satisfying some conditions. The existence of infinitely many solutions for the above equation is presented by using a variant fountain theorem, which improves the related conclusions on this topic. The interesting result of this paper is the potential V(x) vanishing at infinity, i.e., lim |x|→+∞ V(x) = 0.
In this paper, we investigate the fractional Schödinger equation involving a critical growth. By using the principle of concentration compactness and the variational method, we obtain some new existence results for the above equation, which improve the related results on this topic.
In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1<p<+∞, a,b>0 are constants, s(·):RN×RN→(0,1) is a continuous and symmetric function with N>s(x,y)p for all (x,y)∈Ω×Ω, λ>0 is a parameter, (−Δ)ps(·) is a fractional p-Laplace operator with variable-order, V(x):Ω→R+ is a potential function, and f(x,ξ):Ω×RN→R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.
In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this paper is to study impulsive fractional differential equations with p-Laplacian operators.
In this paper, we investigate a class of asymptotically periodic fractional Schrödinger equation with critical growth (−Δ) u + V(x)u = (x, u) + |u| 2 * −2 u, x ∈ R N , where ∈ (0, 1), N > 2 , 2 * = 2N N−2 , (−Δ) denotes the fractional Laplacian of order. V and f are asymptotically periodic functions. Based on the principle of concentration compactness and variational methods, we obtain some new existence results for the above equation, which improve the related conclusions on this topic.
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