Multiple existence of sign changing solutions for coupled nonlinear Schrödinger equations

In this paper, we consider the multiple existence of sign changing solutions of coupled nonlinear Schrödinger equations

(P)     

Asymptotically linear Schrödinger equations with sign-changing potential

In this paper, we show that the following system:

(0.1)     

Multiple positive and sign-changing solutions for a singular Schrödinger equation with critical growth

Variational methods are used to prove the existence of multiple positive and sign-changing solutions for a Schrödinger equation with singular potential having prescribed finitely many singular points. Some exact local behavior for positive solutions obtained here are also given. The interesting aspects are two. One is that one singular point of the potential V(x)$V(x)$ and one positive solution can produce one sign-changing solution of the problem. The other is that each sign-changing solution changes its sign exactly once.     

Multiple solutions for a class of nonlinear Schrödinger equations

In this paper, we find new conditions to ensure the existence of infinitely many homoclinic type solutions for the Schrödinger equation

     

Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials

We establish the existence and multiplicity of semiclassical bound states of the following nonlinear Schrödinger equation:

     

Multiple solutions of a coupled nonlinear Schrödinger system

We will consider the relation between the number of positive standing waves solutions for a class of coupled nonlinear Schrödinger system in RN and the topology of the set of minimum points of potential V(x). The main characteristics of the system are that its functional is strongly indefinite at zero and there is a lack of compactness in RN. Combining the dual variational method with the Nehari technique and using the Concentration-Compactness Lemma, we obtain the existence of multiple solutions associated to the set of global minimum points of the potential V(x) for ? sufficiently small. In addition, our result gives a partial answer to a problem raised by Sirakov about existence of solutions of the perturbed system.     

Multiple solutions for quasilinear Schrödinger equations involving critical exponent

By using Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that the (PS) condition holds locally and by minimax methods and the Krasnoselski genus theory, we establish the multiplicity of solutions for a class of quasilinear Schrödinger equations arising from physics.     

Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations −Δu+V(x)u=f(x,u), x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing.     

On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f(u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f(u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.     

Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part

We consider a class of nonlinear Schrödinger equation with indefinite linear part in RN. We prove that the problem has at least three nontrivial solutions by means of Linking Theorem and (∇)-Theorem.     

Multiple solutions and their limiting behavior of coupled nonlinear Schrödinger systems

We study the multiplicity of positive solutions and their limiting behavior as ? tends to zero for a class of coupled nonlinear Schrödinger system in R^N. We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ? sufficiently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.     

Spike-layer solutions to singularly perturbed semilinear systems of coupled Schrödinger equations

Let Ω be a bounded domain in RN(N?3), we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple Schrödinger equations in Ω

(S_ε)     

Soliton solutions for quasilinear Schrödinger equations, II

For a class of quasilinear Schrödinger equations, we establish the existence of ground states of soliton-type solutions by a variational method.     

Wave front set for solutions to Schrödinger equations

We consider solutions to Schrödinger equation on Rd with variable coefficients. Let H be the Schrödinger operator and let u(t)=e−itHu₀ be the solution to the Schrödinger equation with the initial condition u₀∈L²(Rd). We show that the wave front set of u(t) in the nontrapping region can be characterized by the wave front set of e−itH₀u₀, where H₀ is the free Schrödinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow).     

Homoclinic and heteroclinic orbits for near-integrable coupled nonlinear Schrödinger equations

In this paper, a geometrical perturbation method is employed to prove the existence of heteroclinic orbits for the kinetic system of near-integrable coupled nonlinear Schrödinger (CNLS) equations. Furthermore, we obtain the persistence of homoclinic orbits for the perturbed CNLS equations with even and periodic boundary conditions.