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Register a new userOptimal bilinear control of nonlinear Schr{o}dinger equations with singular potentials
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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.
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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.
In this work, we employ the $bar{partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: x=x_{0}+vt, ~x_{0}in[x_{1},x_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|mathcal{Z}(mathcal{I})|$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.
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 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 a class of Schr{o}dinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.
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