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تسجيل مستخدم جديدGround state normalized solution to Schrodinger systems with general nonlinearities and potentials
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In present paper, we prove the existence of solutions $(lambda_1,lambda_2, u_1,u_2)in R^2times H^1(R^N, R^2)$ to systems of nonlinear Schrodinger equations with potentials $$begin{cases} -Delta u_1+V_1(x)u_1+lambda_1 u_1=partial_1 G(u_1,u_2);quad&hbox{in};R^N -Delta u_2+V_2(x)u_2+lambda_2 u_2=partial_2G(u_1,u_2);quad&hbox{in};R^N 0<u_1,u_2in H^1(R^N), Ngeq 1 end{cases}$$ satisfying the normalization constraints $int_{R^N}u_1^2dx=a_1$ and $int_{R^N}u_2^2dx=a_2$, which appear in mean-field models for binary mixtures of Bose-Einstein condensates or models for binary mixtures of ultracold quantum gases of fermion atoms. The potentials $V_iota(x) (iota=1,2)$ are given functions. The nonlinearities $G(u_1,u_2)$ are considered of the form $$ begin{cases} G(u_1, u_2):=sum_{i=1}^{ell}frac{mu_i}{p_i}|u_1|^{p_i}+sum_{j=1}^{m}frac{ u_j}{q_j}|u_2|^{q_j}+sum_{k=1}^{n}beta_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}}, mu_i, u_j,beta_k>0, ~ p_i, q_j>2, ~r_{1,k}, r_{2,k}>1. end{cases} $$ Under some assumptions on $V_iota$ and the parameters, we can prove the strict binding inequality for the mass sub-critical problem and obtain the existence of ground state normalized solutions for any given $a_1>0,a_2>0$.
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