EXISTENCE OF POSITIVE GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A SUBCRITICAL AND CRITICAL COUPLED SYSTEM OF KIRCHHOFF-SCHRODINGER EQUATIONS IN R3

In this paper, we consider the following Kirchhoff-Schrodinger system in R (3) : { - ( a(1)+b(1)integral(R3)|del u|(2)dx) triangle u+V(1)u=mu|u|(p-2)u+lambda v, inR(3), - ( a(2)+b(2)integral(R3)|del v|(2)dx) triangle v+V(2)v=|v|(q-2)v+lambda u, inR(3), u,v is an element of H-1(R-3), where a(1), a(2), b(1), b(2), V-1, V-2, lambda are positive constants and mu is a nonnegative constant. For the case 4 < p <= q < 6 and 4 < p < q = 6, Albuquerque-O - Figueiredo(Topol. Methods Nonlinear Anal. 53(2019)) obtained a ground state solution by using Nehari manifold. However, the Nehari manifold seems not to work for the case where p, q <= 4, and the existence of a nontrivial solution has been unclear in this case. In this paper, for the case where 3 < p <= q < 6 or 3 < p < q = 6 under certain assumptions on V-1, V (2) , and lambda , we obtain a positive ground state solution by using the Nehari-Pohozaev manifold inspired by the work of Li-Ye(J. Differential Equations 257(2014)), Guo(J. Differential Equations 259(2015)), and Tang-Chen(Calc. Var. Partial Differential Equations 56(2017)). In particular for the critical case 3 < p < q = 6, to obtain the ground state solution, we have to take mu large enough in order to control the energy level of the ground state solution.