Positive ground state solutions of the critical nonlinear Schrödinger system with the harmonic potentials for the cooperative case
In this paper, we study the following two coupled nonlinear Schrödinger system with critical exponent: equation* cases -Δ u+|x-x_0|^2u-λ_1u=μ_1u^2^*-1+β u^2^* /2-1v^2^* /2 , & xR^N, -Δ v+|x-y_0|^2v-λ_2v=μ_2v^2^*-1+β v^2^* /2-1u^2^* /2 , & xR^N, u> 0,v> 0,& xR^N, u(x)→0,v(x)→0, &as |x|→+∞, cases equation* where N≥3, 2^*=2N/(N-2), μ_1,μ_2,β> 0, x_0,y_0R^N and λ_1,λ_2(λ_*,N) with λ_*=1 for N=3 and λ_*=0 for N≥4. By the variational method, we show that the existence of a positive ground state solution for the above system under the cooperative case. Furthermore, we also partially reveal the influence mechanisms of x_0, y_0, λ_i and β on the existence of positive ground state solutions., number=2, journal=Topological Methods in Nonlinear Analysis, author=Wu, Zhiguo and Wu, Yuanze, year=2025, month=Dec., pages=755–782