Quasilinear double phase problems in the whole space via perturbation methods

Abstract

We are concerned with the following double phase problems in the whole space $$ \begin{aligned} -{\rm div}(|\nabla u|^{p-2}\nabla u + & \mu(x)|\nabla u|^{q-2}\nabla u) \\ & +|u|^{p-2}u+\mu(x)|u|^{q-2}u= f(x,u)\;{\rm in}\; \mathbb{R}^N. \end{aligned}$$ The nonlinearity is super-linear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main difficulty is that weak limits of $(PS)$ sequences are not always weak solutions of the problem. To overcome this difficulty, we add {a} potential term and, using the mountain pass theorem, we get weak solutions $u_\lambda$ of the perturbed equations. First, we prove that $u_\lambda\rightharpoonup u$ as $\lambda\rightarrow 0$. Then, via a vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.