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标题：Existence and multiplicity of solutions for Schrödinger–Kirchhoff type problems involving the fractional p ( ⋅ ) $p(\cdot )$ -Laplacian in R N $\mathbb{R}^{N}$
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作者：In Hyoun Kim ; Yun-Ho Kim ; Kisoeb Park 等
期刊名称：Boundary Value Problems
印刷版ISSN：1687-2762
电子版ISSN：1687-2770
出版年度：2020
卷号：2020
期号：1
页码：1
DOI：10.1186/s13661-020-01419-z
出版社：Hindawi Publishing Corporation
摘要：We are concerned with the following elliptic equations with variable exponents: $$ M \bigl([u]_{s,p(\cdot,\cdot)} \bigr)\mathcal{L}u(x) +\mathcal {V}(x) \vert u \vert ^{p(x)-2}u =\lambda\rho(x) \vert u \vert ^{r(x)-2}u + h(x,u) \quad \text{in } \mathbb {R}^{N}, $$ where $[u]_{s,p(\cdot,\cdot)}:=\int_{\mathbb {R}^{N}}\int_{\mathbb {R}^{N}} \frac{ u(x)-u(y) ^{p(x,y)}}{p(x,y) x-y ^{N+sp(x,y)}} \,dx \,dy$ , the operator $\mathcal{L}$ is the fractional $p(\cdot)$ -Laplacian, $p, r: {\mathbb {R}^{N}} \to(1,\infty)$ are continuous functions, $M \in C(\mathbb {R}^{+})$ is a Kirchhoff-type function, the potential function $\mathcal {V}:\mathbb {R}^{N} \to(0,\infty)$ is continuous, and $h:\mathbb {R}^{N}\times\mathbb {R} \to\mathbb {R}$ satisfies a Carathéodory condition. Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities. To do this, we use the mountain pass theorem and variant of the Ekeland variational principle as the main tools.
关键词：35B33 ; 35D30 ; 35J20 ; 35J60 ; 35J66