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Register a new userNormalized ground states for the fractional nonlinear Schr{o}dinger equations
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In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr{o}dinger equation begin{equation} ipartial_{t}psi=(-Delta)^{s}psi-f(psi), qquad (0.1)end{equation} where $0<s<1$, $f(psi)=|psi|^{p}psi$ with $frac{4s}{N}<p<frac{4s}{N-2s}$ or $f(psi)=(|x|^{-gamma}ast|psi|^2)psi$ with $2s<gamma<min{N,4s}$. To this end, we look for normalized solutions of the associated stationary equation begin{equation} (-Delta)^s u+omega u-f(u)=0. qquad (0.2) end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.
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We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.
In this paper, we study important Schr{o}dinger systems with linear and nonlinear couplings begin{equation}label{eq:diricichlet} begin{cases} -Delta u_1-lambda_1 u_1=mu_1 |u_1|^{p_1-2}u_1+r_1beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+kappa (x)u_2~hbox{in}~mathbb{R}^N, -Delta u_2-lambda_2 u_2=mu_2 |u_2|^{p_2-2}u_2+r_2beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+kappa (x)u_1~ hbox{in}~mathbb{R}^N, u_1in H^1(mathbb{R}^N), u_2in H^1(mathbb{R}^N), onumber end{cases} end{equation} with the condition $$int_{mathbb{R}^N} u_1^2=a_1^2, int_{mathbb{R}^N} u_2^2=a_2^2,$$ where $Ngeq 2$, $mu_1,mu_2,a_1,a_2>0$, $betainmathbb{R}$, $2<p_1,p_2<2^*$, $2<r_1+r_2<2^*$, $kappa(x)in L^{infty}(mathbb{R}^N)$ with fixed sign and $lambda_1,lambda_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $Ngeq 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4, r_1=r_2=2$.
In this paper, we study the nonlinear Schr{o}dinger equation $ -Delta u+V(x)u=f(x,u) $on the lattice graph $ mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from $mathbb{Z}^{N}$ to quasi-transitive graphs.
In this paper, we consider an optimal bilinear control problem for the nonlinear Schr{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in cite{Sp} in several aspects.
In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schr{o}dinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on $b_1$ and $b_2$, we prove the existence of normalized solutions. When $N=3$, the energy functional is always unbounded on the product of $L^2$-spheres. We show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as $betarightarrow 0$. Finally, we deal with the high dimensional cases $Ngeq 4$. Several non-existence results are obtained if $beta<0$. When $N=4$, $beta>0$, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case $beta=0$, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schr{o}dinger system but also leads to a stabilization of the related evolution system.
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