On elliptic equations with unbounded or decaying potentials involving Stein-Weiss convolution parts and critical exponential growth

Abstract

We study a class of nonlinear Schrödinger equations with Stein-Weiss convolution parts$-\Delta u+V\left(x\right)u=\left(\underset{R^2}{\int }\frac{F\left(u\right)}{|x-y{|}^{\mu }||y{|}^{\beta }}dy\right)\frac{f\left(u\right)}{|x{|}^{\beta }},x\in R^2,$ where V is an unbounded or decaying potential, $\beta >0,\mu >0$ with $0<2\beta +\mu <2$, and F denotes the primitive of f that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Via establishing a new version of the Trudinger-Moser inequality, we shall exploit the general minimax principle to demonstrate the existence of nontrivial solutions using variational method.