Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 100, pp. 1-24.

Title: Existence results for nonlinear Schrodinger equations involving the
 fractional (p,q)-Laplacian and critical nonlinearities
       
Authors: Huilin Lv (Beijing Jiaotong Univ., Beijing, China)
         Shenzhou Zheng (Beijing Jiaotong Univ., Beijing, China)
         Zhaosheng Feng (Univ. of Texas Rio Grande Valley, Edinburg, TX, USA)

Abstract:
In this article, we consider the existence of  ground state positive solutions for
nonlinear Schrodinger equations of the fractional (p,q)-Laplacian with
Rabinowitz potentials defined in R<sup>n</sup>,
$$
( -\Delta ) _p^{s_1}u+( -\Delta )
_q^{s_2}u+V( \epsilon x) ( | u|^{p-2}u+| u| ^{q-2}u) =\lambda f( u)
+\sigma | u| ^{q_{s_2}^{\ast }-2}u.
$$
We prove existence by confining different ranges of the parameter &lambda; under the
subcritical or critical nonlinearities caused by &sigma;=0 or 1, respectively.
In particular, a delicate calculation for the critical growth is provided so as to
avoid the failure of a global Palais-Smale condition for the energy functional.

Submitted May 18, 2021. Published December 20, 2021.
Math Subject Classifications: 35R11, 35A15, 58E05.
Key Words: Nonlinear Schrodinger equations; nonlocal (p,q)-Laplacian;
           critical growth; Rabinowitz potentials; Nehari manifold.