In this paper, we consider a class of $p\left(x\right)$-Laplacian problems of the form: $\begin{array}{l}(-\mathrm{\Delta }{)}_{p\left(x\right)}u+a(x)|u{|}^{p\left(x\right)-2}u=f(x,u)\text{in }\mathrm{\Omega },\\ |\mathrm{\nabla }u{|}^{p\left(x\right)-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }v}+b\left(x\right)|u{|}^{q\left(x\right)-2}u=g(x,u)\text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$ is a bounded domain with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega },\left(\mathrm{\partial }/\mathrm{\partial }v\right)$ is outer unit normal derivative. The functions a, b, p, q, g and f are assumed to satisfy suitable assymptions. The existence and the multiplicity of solutions is obtained by using variational methods, and mountain pass lemma combined with Ekeland variational principle.

中文翻译：

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一些涉及 p(x)-拉普拉斯算子的 Styklov 问题解的存在性和多重性

在本文中，我们考虑一类$p\left(X\right)$- 形式的拉普拉斯问题：$\begin{array}{l}(-\mathrm{\Delta }{)}_{p\left(X\right)}你+\mathrm{一种}(X)|你{|}^{p\left(X\right)-2}你=F(X,你)\text{在 }\mathrm{\Omega },\\ |\mathrm{\nabla }你{|}^{p\left(X\right)-2}\frac{\mathrm{\partial }你}{\mathrm{\partial }v}+b\left(X\right)|你{|}^{q\left(X\right)-2}你=G(X,你)\text{在 }\mathrm{\partial }\mathrm{\Omega },\end{array}$在哪里$\mathrm{\Omega }\subset {\mathbb{R}}^{ñ}$,$ñ\ge 2$是具有 Lipschitz 边界的有界域$\mathrm{\partial }\mathrm{\Omega },\left(\mathrm{\partial }/\mathrm{\partial }v\right)$是外单位正态导数。假设函数a、b、p、q、g和f满足适当的假设。利用变分方法，结合Ekeland变分原理，得到解的存在性和多重性。