In this paper, we consider the quasilinear elliptic problem with potential $\left(P\right)\left\{\begin{array}{ll}-\text{div}\left(\phi \right(x,|\mathrm{\nabla }u|\left)\mathrm{\nabla }u\right)+V\left(x\right)|u{|}^{q\left(x\right)-2}u=f(x,u)& \text{in}\mathrm{\Omega },\\ u=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\right.$ where Ω is a smooth bounded domain in ${\mathbb{R}}^N$ ($N\ge 2$), V is a given function in a generalized Lebesgue space $L^{s\left(x\right)}\left(\mathrm{\Omega }\right)$, and $f(x,u)$ is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for $\left(P\right)$ in the framework of Musielak–Sobolev spaces. The main difficulty here is that the nonlinearity $f(x,u)$ considered does not satisfy the well-known Ambrosetti–Rabinowitz condition.

中文翻译：

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无 Ambrosetti-Rabinowitz 条件的拟线性椭圆问题涉及 Musielak-Sobolev 空间设置中的势

在本文中，我们考虑具有潜在的拟线性椭圆问题 $\left(磷\right)\left\{\begin{array}{ll}-\text{div}\left(\phi \right(X,|\mathrm{\nabla }你|\left)\mathrm{\nabla }你\right)+伏\left(X\right)|你{|}^{q\left(X\right)-2}你=F(X,你)& \text{在}\mathrm{\Omega },\\ 你=0& \text{在}\mathrm{\partial }\mathrm{\Omega },\end{array}\right.$ 其中 Ω 是一个光滑的有界域 ${\mathbb{电阻}}^N$ ($N\ge 2$), V是广义勒贝格空间中的给定函数${升}^{秒\left(X\right)}\left(\mathrm{\Omega }\right)$， 和 $F(X,你)$是满足适宜生长条件的 Carathéodory 函数。使用变分论证，我们研究了弱解的存在性$\left(磷\right)$在Musielak-Sobolev空间的框架中。这里的主要困难是非线性$F(X,你)$ 被认为不满足著名的 Ambrosetti-Rabinowitz 条件。