On the Existence of Ground State Solutions to a Quasilinear Schr?dinger Equation involving p-Laplacian

We consider the following quasilinear Schr?dinger equation involving p-Laplacian ■ in R^N,where N > p > 1, η ≥p/(2(p-1)), p < q < 2ηp*(μ), p*(s) =(p(N-s))/(N-p), and λ, μ, ν are parameters with λ > 0,μ, ν ∈ [0, p). Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Sign-changing Solutions for a Schrödinger Equation with Saturable Nonlinearity

We consider the nonlinear Schrödinger equation

$-\Delta{u} + V (x)u = K(x)u^3/(1 + u^2)$

in \({\mathbb {R}^N}\) , and assume that V and K are invariant under an orthogonal involution. Moreover, V and K converge to positive constants V _∞ and K _∞, as |x| → ∞. We present some results on the existence of a particular type of sign changing solution, which changes sign exactly once. The basic tool employed here is the Concentration–Compactness Principle and the interaction between translated solutions of the corresponding autonomous problem.

     

Existence and Asymptotic Behavior of Ground State Solutions for Quasilinear Schr?dinger Equations with Unbounded Potential

The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients. By applying the mountain pass theorem and the concentration compactness principle, they get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.     

Multiple Solutions for a Nonlinear Schrödinger Equation with Magnetic Fields

We study a nonlinear Schrödinger equation in presence of a magnetic field and relate the number of solutions with the topology of the set where the potential attains its minimum value. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.     

Solutions for Quasilinear Schrödinger Equations via the Nehari Method

Abstract

For a class of quasilinear Schrödinger equations we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method.     

Soliton Solutions for a Singular Schrödinger Equation with Any Growth Exponents

This paper is concerned with a kind of quasilinear Schrödinger equation with combined nonlinearities, a convex term with any growth and a singular term, in a bounded smooth domain. Multiplicity results are obtained by critical point theory together with truncation arguments and the method of upper and lower solutions.     

Ground State Solutions for Schrdinger-Poisson Systems

This paper deals with a class of Schr¨odinger-Poisson systems. Under some conditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.     

Ground State Solutions for Schr¨odinger-Poisson Systems

This paper deals with a class of Schr¨odinger-Poisson systems. Under some con-ditions, we prove that there exists a ground state solution of the system. The proof is based on the compactness lemma for the system. Our results here improve some existing results in the literature.     

Finite Morse Index Solutions of a Nonlinear Schrödinger Equation

Acta Mathematica Sinica, English Series - We prove Liouville type theorems for stable and finite Morse index H loc 1 ∩ L loc ∞ solutions of the nonlinear Schrödinger equation...     

Global Solutions of Modified One-Dimensional Schrödinger Equation

In this paper, we consider the modified one-dimensional Schrödinger equation:$(D_t-F(D))u=λ|u|^2u,$where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$when $t→+∞$.     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.     

On Multiwave Solutions of One Nonlinear Schrödinger Equation

Differential Equations - We consider a nonlinear Schrödinger equation arising in a number of physical problems. It is shown that when the real part is separated in this equation, there arises...     

Infinitely Many Sign-changing Solutions for a Schrdinger Equation in R~N

We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion equation     

Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

Single Peak Solutions for a Schr?dinger Equation with Variable Exponent

We study the following Schr?dinger equation with variable exponent■ where■ . Under certain assumptions on a vector field related to a(x), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.     

Schrödinger Equation as a Self-Consistent Field

It is well known that the Schrödinger equation can be reduced to the Hamilton–Jacobi equation in Bohmian mechanics. Corresponding new equations of the Vlasov and Lamb types are derived, and their stationary solutions are investigated.