On the Double Phase Variational Problems Without Ambrosetti–Rabinowitz Condition

Abstract

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.