Quasilinear double phase problems in the whole space via perturbation methods

We are concerned with nonlinear double phase equations in the whole space. The nonlinearity is superlinear 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 equation. To overcome this difficulty, we add a potential term and, using the mountain pass theorem, we get weak solutions uλ of the perturbed equations. First, we prove that uλ weaklyconverges to u as λ tends to 0. Then, via a vanishing lemma à la Lions, we get that u is a nontrivial solution of the original equation.