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Register a new userWavelets, Multiplier spaces and application to Schr{o}dinger type operators with non-smooth potentials
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In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,tau}_{r,p}(mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $tau$ of $M^{t,tau}_{r,p}(mathbb{R}^{n})$ is sharp. As an application, we consider a Schrodinger type operator with potentials in $M^{t,tau}_{r,p}(mathbb{R}^{n})$.
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In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr{o}dinger operators $(-Delta+m^{2})^{s}$ with $sin(0,1)$ and mass $m>0$. As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators $(-Delta+m^{2})^{s}$ in bounded domains, epigraph or $mathbb{R}^{N}$, including pseudo-relativistic Schrodinger equations, 3D boson star equations and the equations with De Giorgi type nonlinearities.
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.
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.
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 $N=3$, $p_1=p_2=4, r_1=r_2=2$.
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.
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