Existence of positive solutions to the nonlinear Choquard equation with competing potentials

Abstract

This article concerns the existence of positive solutions of the nonlinear Choquard equation $$ -\Delta u+a(x)u=b(x)\Big(\frac{1}{|x|}*|u|^2\Big)u,\quad u\in H^{1}({\mathbb R}^3), $$ where the coefficients a and b are positive functions such that $a(x)\to\kappa_\infty$ and $b(x)\to \mu_\infty$ as $|x|\to\infty$. By comparing the decay rate of the coefficients a and b, we prove the existence of positive ground and bound stat solutions of Choquard equation.

Keywords

• Positive solutions
• Choquard equation
• competing coefficients
• variational methods