Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].     

Fractional Schrödinger Equations with Logarithmic and Critical Nonlinearities

Acta Mathematica Sinica, English Series - In this paper, we study a class of the fractional Schrödinger equations involving logarithmic and critical nonlinearities. By using the Nehari...     

A System of Schrödinger Equations in the Critical Case

A system of nonlinear Schrödinger equations u _k } / t=ia _k u _k+f _k (u,u ^*), t>0, k=1,... ,m; u _k (0,x)=u _k0 (x), where f _k are homogeneous functions of order 1+4/n, is considered. Sufficient conditions for the globality of the solution are obtained. The existence of the explicit blow-up solution is proved.     

Schrödinger Equations with Fractional Laplacians

It is shown that the unique solution of } can be represented as { } where X=(X _t , t≥ 0) is a stable process whose generator is (-Δ) ^α/2 with X ₀ =0 . Accepted 24 July 2000. Online publication 13 November 2000.     

Spherical Perturbations of Schrödinger Equations

We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights and on , for which the a priori inequality holds with constant independent of . Here is the Laplace-Beltrami operator on the sphere . Due to the relation between and the tangential component of the gradient, , we obtain some "Morawetz-type" estimates for on . As a consequence we establish some new estimates for the free Schr?dinger propagator , which may be viewed as certain refinements of the -(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the dimensional Schr?dinger equation.     

Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity

In this paper, we investigate the following critical fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^su+V(x)u=|u|^{2_s^*-2}u+\lambda K(x)f(u), \ x \in \mathbb {R}^N, \end{aligned}$$

where \(\lambda >0\), \(0<s<1\), \((-\Delta )^s\) denotes the fractional Laplacian of order s, \(V, \ K\) are nonnegative continuous functions satisfying some conditions and f is a continuous function, \(N>2s\) and \(2_s^*=\frac{2N}{N-2s}\). We prove that the equation has a positive solution for large \(\lambda \) by the variational method.

     

Bound states of nonlinear Schrödinger equations with magnetic fields

We study bound states of the following nonlinear Schr?dinger equation in the presence of a magnetic field: $$ \left\{\begin{array}{l} \left(-i\hbar\nabla+A(x)\right)^2u+V(x)u=g(x,|u|)u \\ |u|\in H^1(\mathbb{R}^N) \end{array} \right. $$ where ${A: \mathbb{R}^N\to\mathbb{R}^N, V: \mathbb{R}^N\to\mathbb{R}}$ and ${g: \mathbb{R}^N\times\mathbb{R}\to [0,\infty)}$ . We prove that if V is bounded below with the set ${\{x\in\mathbb{R}^N: V(x) < b\}\not=\emptyset}$ having finite measure for some b?>?0, inf V???0, and g satisfies some growth conditions, then for any integer m when ${\hbar >0 }$ is sufficiently small the problem has m geometrically different solutions.     

On a Family of Solvable Multiparticle Schrödinger Equations with Pair Potential

Mathematical Notes -     

Bound states for a coupled Schrödinger system

We consider the existence of bound states for the coupled elliptic system

On the Absence of a First Order COrrection for the Number of Bound States of a Schrödinger operator with Coulomb Singularity

Contents

1 Introduction 616

2 Regularizaton of the Singular Operator 620

3 Functional Calculus with Projections 621

A The Kutaanheimo–Stiefel Transform and All That 625

B Some Remarks on a Result of Petkov and Robert on the Counting Function in the Context of Compact Group Actions 627

B.1 Introduction and Principal Statements 627

B.2 Sketch of a Proof of Theorem 10 632

B.3 On the Initial Conditions Yielding Periodic Orbits of Regularized and Unregularized Hamiltonians 634

B.4 On the Vanishing of the Second Coefficient 635     

Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well

Science China Mathematics - This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation: $$left{ {begin{array}{*{20}{l}} { -...     

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure on a Brelot space (X,) with Green function G, it is well known that -harmonic functions (i.e., in the classical case, finely continuous versions of solutions to u–u=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, ) admitting a Green function T (G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric :=(G ^–1+^* G ^–1)/2 yields -Hölder continuity for quotients of -harmonic functions as well.     

Coherent States for Systems of L 2-Supercritical Nonlinear Schrödinger Equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, at leading order, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.