By applying Mountain Pass Lemma, Ekeland’s variational principle and Fountain Theorem, we prove the existence and multiplicity of solutions for the following Robin problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\text {div}\left( a(x)\left| \nabla u\right| ^{p(x)-2}\nabla u\right) =\lambda b(x)\left| u\right| ^{q(x)-2}u, &{} x\in \varOmega \\ a(x)\left| \nabla u\right| ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+\beta (x)\left| u\right| ^{p(x)-2}u=0, &{} x\in \partial \varOmega , \end{array}\right. \end{aligned}$$

under some appropriate conditions in the space \(W_{a,b}^{1,p(.)}\left( \varOmega \right) \).

中文翻译：

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带权p(.)-Laplacian特征值Robin问题弱解的存在性和多重性

通过应用山口引理、埃克兰变分原理和喷泉定理，我们证明了以下罗宾问题的解的存在性和多重性

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\text {div}\left( a(x)\left| \nabla u\right| ^{p (x)-2}\nabla u\right) =\lambda b(x)\left| 你\对| ^{q(x)-2}u, &{} x\in \varOmega \\ a(x)\left| \nabla你\对| ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+\beta (x)\left| 你\对| ^{p(x)-2}u=0, &{} x\in \partial \varOmega , \end{array}\right。\end{对齐}$$

在空间\(W_{a,b}^{1,p(.)}\left( \varOmega \right) \) 的一些适当条件下。