Least energy solutions for nonlinear Schrödinger equations involving the half Laplacian and critical growth

In this paper, we study a class of nonlinear Schrödinger equations involving the half Laplacian and critical growth. We assume that the potential of the nonlinear Schrödinger equation includes a parameter \({\lambda}\). Moreover, the potential behaves like a potential well when the parameter \({\lambda}\) is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, for \({\lambda}\) large, localizes near the bottom of the potential well. Moreover, if the zero set int \({V^{-1}(0)}\) of \({V(x)}\) includes more than one isolated component, then \({u_{\lambda}(x)}\) will be trapped around all the isolated components. However, in Laplacian case when \({s = 1}\), for \({\lambda}\) large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \({V^{-1}(0)}\). This is the essential difference with the Laplacian problems since the operator \({(- \Delta)^{1/2}}\) is nonlocal.     

Existence of solutions for a nonperiodic semilinear Schrödinger equation

In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem.     

Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities

We study the following nonlinear Schrödinger equations $$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$ for ${w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}$ , where g(|w|)w is super linear and subcritical, 2* = 2N/(N ? 2) if N > 2 and = ∞ if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V <  τ ₀ for some τ₀ > 0, or ${\max W > \kappa_{0}}$ for some ${\kappa_0 > 0}$ .     

Positive solutions of a Schrödinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

Positive solutions of a Schr?dinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation -Du+la(x)u=mu+u^2*-1,  u ? \mathbbR^N,-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N}, with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth■ where s ∈(0,1),N 4 s,and λ 0 is a parameter,2_s~*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0 λ~*+∞such that the problem has exactly two positive solutions if λ∈(0,λ~*),no positive solutions for λλ~*,a unique solution(λ~*,u_(λ~*))if λ=λ~*,which shows that(λ~*,u_(λ~*)) is a turning point in H~s(R~N) for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.     

Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation

In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

Stability of solutions for nonlinear Schrdinger equations in critical spaces

We consider the Cauchy problem for nonlinear Schrdinger equation iut + Δu = ±|u|pu,4/d< p <4 /d-2 in high dimensions d 6. We prove the stability of solutions in the critical space H˙xsp , where sp = d/2-p/2 .     

Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth

We study the energy-critical focusing nonlinear Schrödinger equation with an energy-subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig and Merle (Invent Math 166:645–675, 2006).     

Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the multiplicity of standing wave solutions of nonlinear Schr?dinger equations with electromagnetic fields