Existence of positive solutions for $p\&q$ equations involving vanishing potentials with exponential decay

Abstract

We show the existence of positive solutions for a class of $p\&q$ problems given by $$ \begin{cases} -div\left(a\left(|\nabla u|^p\right) |\nabla u|^{p-2}\nabla u\right)+V(z)b \left(|u|^p\right)|u|^{p-2}u = f(u)\ \ \mbox{in $\mathbb{R}^N,$} \\ u \in D^{1,p}(\mathbb{R}^{N})\cap D^{1,q}(\mathbb{R}^{N}), \end{cases} $$ $N\geq 3$ and $1 < p\leq q < N$. The potential $V$ can vanish at infinity with exponential decay and $f$ is a function with subcritical growth of class $C^1$. We use Del Pino and Felmer's arguments [18] in order to overcome the lack of compactness.