We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems:

$$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u+\alpha (x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\gamma -2}u=f(x,u),&{}\ \mathrm{in}\ \Omega ,\\ u=0,&{}\ \mathrm{on}\ {\partial \Omega ,} \end{array}\right. \end{aligned}$$

applying the mountain pass theorem and fountain theorem. The Ambrosetti—Rabinowitz condition as well as the monotonicity of \(f(x,t)/|t|^{q-1}\) are not assumed.

中文翻译：

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无Ambrosetti–Rabinowitz条件的双相变分问题

我们关注以下双重阶段问题的非平凡解的存在和多样性：

$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ mathrm {div}（| \ nabla u | ^ {p-2} \ nabla u + \ alpha（x）| \ nabla u | ^ {q-2} \ nabla u）+ V（x）| u | ^ {\ gamma -2} u = f（x，u），＆{} \ \ mathrm {in} \ \ Omega，\\ u = 0，＆{} \ \ mathrm {on} \ {\ partial \ Omega，} \ end {array} \ right。\ end {aligned} $$

应用山口定理和喷泉定理。不假定Ambrosetti-Rabinowitz条件以及\（f（x，t）/ | t | ^ {q-1} \）的单调性。