We are concerned with the following \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\)$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution \(u_{\lambda }\) of perturbation equations. First, we prove that \(u_{\lambda }\rightharpoonup u\) as \(\lambda \rightarrow 0\). Second, by using vanishing lemma, we get that \(u\) is a nontrivial solution of the original problem.

中文翻译：

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基于摄动法的变指数超线性椭圆方程

我们关注以下\（p（x）\）- \（\ mathbb {R} ^ {N} \）$$中的拉普拉斯方程$$-\ triangle _ {p（x）} u + | u | ^ {p （x）-2} u = f（x，u）\ quad \ mbox {in} \ mathbb {R} ^ {N}。$$非线性是超线性的，但不满足Ambrosetti-Rabinowitz类型的条件。我们的主要困难是（PS）序列的弱极限并不总是该问题的弱解决方案。为了克服这个困难，通过添加潜在项并使用山口定理，我们获得了扰动方程的弱解\（u _ {\ lambda} \）。首先，我们证明\（u _ {\ lambda} \ rightharpoonup u \）为\（\ lambda \ rightarrow 0 \）。其次，通过使用消失的引理，我们得到\（u \） 是原始问题的重要解决方案。