Asymptotically linear fractional Schrödinger equations

By exploiting a variational technique based upon projecting over the Poho?aev manifold, we prove existence of positive solutions for a class of nonlinear fractional Schrödinger equations having a nonhomogenous nonautonomous asymptotically linear nonlinearity.     

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.     

Soliton dynamics for fractional Schrödinger equations

We investigate the soliton dynamics for the fractional nonlinear Schrödinger equation by a suitable modulational inequality. In the semiclassical limit, the solution concentrates along a trajectory determined by a Newtonian equation depending of the fractional diffusion parameter.     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

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 ${\kappa_0 > 0}$ .     

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...     

Quasilinear asymptotically periodic Schrödinger equations with critical growth

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in ${\mathbb{R}^N}$ with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in ${H^1(\mathbb{R}^N)}$ and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.     

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.     

A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

Numerical Algorithms - In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a...