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Register a new userSemitrivial vs. fully nontrivial ground states in cooperative cubic Schrodinger systems with $dge3$ equations
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In this work we consider the weakly coupled Schrodinger cubic system [ begin{cases} displaystyle -Delta u_i+lambda_i u_i= mu_i u_i^{3}+ u_isum_{j eq i}b_{ij} u_j^2 u_iin H^1(mathbb{R}^N;mathbb{R}), quad i=1,ldots, d, end{cases} ] where $1leq Nleq 3$, $lambda_i,mu_i >0$ and $b_{ij}=b_{ji}>0$ for $i eq j$. This system admits semitrivial solutions, that is solutions $mathbf{u}=(u_1,ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $lambda_i,mu_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $dgeq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $lambda_iequiv lambda$ and $b_{ij}equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.
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We study generic semilinear Schrodinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a general instability criterion and then apply it to some well-known models arising in several physical contexts. In particular, this mass-transfer instability allows us to exhibit $L^2$-subcritical unstable ground-states.
We give short survey on the question of asymptotic stability of ground states of nonlinear Schrodinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
We study the existence of ground states for the coupled Schrodinger system begin{equation} left{begin{array}{lll} displaystyle -Delta u_i+lambda_i u_i= mu_i |u_i|^{2q-2}u_i+sum_{j eq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i u_iin H^1(mathbb{R}^n), quad i=1,ldots, d, end{array}right. end{equation} $ngeq 1$, for $lambda_i,mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called symmetric attractive case) and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.
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 $a_1>0,a_2>0$.
We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential $V$. Under suitable assumptions on $V$, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti-Rabinowitz type condition or monotonicity assumptions.
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