Concentrating ground state for linearly coupled Schrodinger systems involving critical exponent cases

We study the following linearly coupled Schrodinger system: {-Delta u + (mu V-1(x) + a) u = f (x)vertical bar u vertical bar(p-2) u + beta(x)v, x is an element of R-N, {-Delta v + (mu V-2(x) + b) v = g (x)vertical bar v vertical bar(2*-2) v + beta(x)v, x is an element of R-N, where N >= 3, 2 < p <= 2*, V-1, V-2 is an element of C(R-N, R+) are potential wells with bottoms Omega(i) = intV(i)(1) (0), the parameters a > -lambda(1)(Omega(1)), b > -lambda(1)((Omega)) and lambda(1)(Omega(i)) is the first eigenvalue of - Delta in H-0(1) (Omega(i)). Under some suitable assumptions on beta(x) which relate to the potentials V-1, V-2 and the parameters a, b, the existence of positive ground states is obtained by variational method. Some interesting phenomena are that we relax the upper control condition of the coupling function beta(x) and we do not need the weight function f (x) to be integrable or bounded in the subcritical case 2 < p < 2*. Moreover, from the concentration phenomenon of solutions, we obtain some results of the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain, which extend the recent results of Peng, et al. (2017), [25]. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this study, we investigate a linearly coupled Schrodinger system and establish the existence of positive ground states under suitable assumptions and by using variational methods. We also relax some of the conditions and provide some results on the existence of positive ground states to a linearly coupled Schrodinger system in a bounded domain.