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.

中文翻译：

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带分数\（p（\ cdot）\）- Laplacian的Schrödinger–Kirchhoff型问题的存在性和多重性\（\ mathbb {R} ^ {N} \）

我们关注以下具有可变指数的椭圆方程：$$ M \ bigl（[u] _ {s，p（\ cdot，\ cdot）} \ bigr）\ mathcal {L} u（x）+ \ mathcal { V}（x）\ vert u \ vert ^ {p（x）-2} u = \ lambda \ rho（x）\ vert u \ vert ^ {r（x）-2} u + h（x，u） \ quad \ text {in} \ mathbb {R} ^ {N}，$$其中$ [u] _ {s，p（\ cdot，\ cdot）}：= \ int _ {\ mathbb {R} ^ {N }} \ int _ {\ mathbb {R} ^ {N}} \ frac {| u（x）-u（y）| ^ {p（x，y）}} {p（x，y）| xy | ^ {N + sp（x，y）}} \，dx \，dy $，运算符$ \ mathcal {L} $是小数$ p（\ cdot）$-拉普拉斯算子$ p，r：{\ mathbb { R} ^ {N}} \ to（1，\ infty）$是连续函数，$ M \ in C（\ mathbb {R} ^ {+}）$是基尔霍夫型函数，潜在函数$ \ mathcal {V}：\ mathbb {R} ^ {N} \ to（0，\ infty）$是连续的，而$ h：\ mathbb {R} ^ {N} \ times \ mathbb {R} \ to \ mathbb { R} $满足Carathéodory条件。在关于h的适当假设下，本文的目的是说明在凹-凸非线性组合效应的情况下，对于上述问题至少存在两个非平凡的不同解。为此，我们使用山口定理和Ekeland变分原理的变体作为主要工具。