Consider the following Schrödinger system: where either or is a smooth bounded domain. Note that the cubic nonlinearities and the coupling terms are of critical growth whenever dimension . We give a characterization of the least energy solutions when or is a smooth bounded domain of , if the coupling matrix is positively or negatively definite with and exists and satisfies for . Furthermore, when and , we obtain a nonexistence theorem about the least energy solutions provided attraction and repulsion coexist, i.e. some of are positive but some others are negative.
We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\geq 3, r\geq2, 2^*=\frac{2N}{N-2}, \lambda_j\in (0, \frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $\beta_{jk}=\beta_{kj}$; \; $\alpha_{jk}>1, \alpha_{kj}>1,$ satisfying $\alpha_{jk}+\alpha_{kj}=2^* $ for all $k\neq j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms all are critical in arbitrary dimension $N\geq3 $. The signs of the coupling constants $\bb_{ij}$'s are decisive for the existence of the ground state solutions. We show that the critical system with $r\geq 3$ has a positive least energy solution for all $\beta_{jk}>0$. However, there is no ground state solutions if all $\beta_{jk}$ are negative. We also prove that the positive solutions of the system are radially symmetric. Furthermore, we obtain the uniqueness theorem for the case $r\geq 3$ with $N=4$ and the existence theorem when $r=2$ with general coupling exponents.
Abstract We study the following elliptic system with critical exponent: Here, Ω is a smooth bounded domain of ℝ N (N ≥ 6), is the critical Sobolev exponent, 0 < λ1, λ2 < λ1(Ω) and μ1, μ2 > 0, where λ1(Ω) is the first eigenvalue of − Δ with the Dirichlet boundary condition. When β = 0, this turn to be the well-known Brézis-Nirenberg problem. We show that, for each fixed β <0, this system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive. We also study the asymptotic behavior of these solutions as β → − ∞ and phase separation appears. Precisely, two components of these solutions tend to repel each other and converge to solutions of the Brézis-Nirenberg problem in segregated regions. Keywords: Critical exponentNonlinear elliptic systemPhase separationSign-changing solution2010 Mathematics Subject Classification: 35J4735J5058J37 Acknowledgements The authors wish to thank the anonymous referees very much for careful reading and valuable comments.
This paper is concerned with the following logarithmic Schr\"{o}dinger system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \ &\ \int_{\Omega}|u_i|^2\,dx=\rho_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in H_0^1(\Omega;\mathbb R^2),\end{align}\right.$$ where $\Omega=\mathbb{R}^N$ or $\Omega\subset\mathbb R^N(N\geq3)$ is a bounded smooth domain, $\omega_i\in\mathbb R$, $\mu_i,\ \rho_i>0,\ i=1,2.$ Moreover, $p,\ q\geq1,\ 2\leq p+q\leqslant 2^*$, where $2^*:=\frac{2N}{N-2}$. By using a Gagliardo-Nirenberg inequality and careful estimation of $u\log u^2$, firstly, we will provide a unified proof of the existence of the normalized ground states solution for all $2\leq p+q\leqslant 2^*$. Secondly, we consider the stability of normalized ground states solutions. Finally, we analyze the behavior of solutions for Sobolev-subcritical case and pass the limit as the exponent $p+q$ approaches to $2^*$. Notably, the uncertainty of sign of $u\log u^2$ in $(0,+\infty)$ is one of the difficulties of this paper, and also one of the motivations we are interested in. In particular, we can establish the existence of positive normalized ground states solutions for the Br\'{e}zis-Nirenberg type problem with logarithmic perturbations (i.e., $p+q=2^*$). In addition, our study includes proving the existence of solutions to the logarithmic type Br\'{e}zis-Nirenberg problem with and without the $L^2$-mass $\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$ constraint by two different methods, respectively. Our results seems to be the first result of the normalized solution of the coupled nonlinear Schr\"{o}dinger system with logarithmic perturbation.
Abstract Note: Please see pdf for full abstract with equations. We consider the existence and asymptotic behavior on mass of positive solutions to the following system −Δ u + λ 1u = μ1u 3 + α1|u| p−2 u + βv 2 u in R 4 , −Δ v + λ 2v = μ2v 3 + α2|v| p−2 v + βu 2 v in R 4 , under the normalized mass constraints ∫ R4 u2 = a21 and ∫ R4 v2 = a22 , where a 1 , a 2 are prescribed, μ 1 , μ 2 , β > 0, α 1 , α 2 ∈ R, p ∈ (2, 4) and λ 1 , λ 2 ∈ R appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., α i < 0(i = 1, 2) . Then turning to the case of α i > 0(i = 1, 2), if 2 < p < 3 , we show that the problem admits a ground state and an excited state. Moreover, we give a precise asymptotic behavior of these two solutions as (a 1 , a 2 ) → (0, 0) and a 1 ~ a 2 . This seems to be the first contribution regarding the multiplicity as well as synchronized mass collapse behavior of normalized solutions to Schrödinger systems with Sobolev critical exponent. When 3 ≤ p < 4 , we prove an existence as well as a non-existence (p = 3) results of ground states. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively. 2010 Mathematics Subject Classification : 35J50, 35B33, 35B09, 35B40.
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