On a Schrödinger equation involving fractional (N/s1,q)-Laplacian with critical growth and Trudinger–Moser nonlinearity

Abstract

A nonlinear Schrödinger equation of fractional $(N/s_1,q)$-Laplacian is considered with the Rabinowitz potential, critical Sobolev growth and Trudinger–Moser nonlinearity in $R^N$ ${\left(-\Delta \right)}_{N/s_1}^{s_1}u+{\left(-\Delta \right)}_q^{s_2}u+V\left(\varepsilon x\right)\left({\left(u\right)}^{\frac{N}{s_1}-2}u+{\left(u\right)}^{q-2}u\right)=\lambda f\left(u\right)+{\left(u\right)}^{q_{s_2}^{\ast }-2}u.$We establish the global existence of nonnegative ground-state solution for suitable parameter values primarily through variational analysis, fractional Trudinger–Moser inequality and mountain pass approach. It is a crucial ingredient to handle three aspects concerning the limiting setting $s_{1}p=N$, the critical Sobolev growth and Trudinger–Moser nonlinearity.