Positive solution of quasilinear elliptic equations in $\mathbb{R}^N$ with a bounded quasilinearity

Abstract

Consider the quasilinear elliptic equation $$ -\text{div}(\mathcal{A}(u)\nabla u) +\frac{1}{2}\mathcal{A}'(u)|\nabla u|^2+V(x)u =(I_{\alpha}\ast |u|^p) |u|^{p-2}u\quad \text{in } \R^N, $$% where $\mathcal{A}\in C^1(\R,\R)$ is a positive bounded function, $V$ is a given potential and $I_\alpha$ denotes the Riesz potential with $0< \alpha< N$. While most existing works in the literature are concerned with the case where $\mathcal{A}$ is unbounded, little is known about the case where $\mathcal{A}$ is bounded. Under some general conditions on $\mathcal{A}$ and $V$, we establish the existence of a positive solution for the above equation by variational approach.