Positive bound states of systems of nonlinear Schrödinger equations

We study the existence of positive bound states of non-autonomous systems of nonlinear Schrödinger equations. Both the singular case and the regular case are discussed. The proof is based on a nonlinear alternative principle of Leray–Schauder.     

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.     

Normalized solutions of nonlinear Schrödinger equations

We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L ²-unit sphere, and we show the existence of infinitely many solutions.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Classical solutions of nonlinear Schrödinger equations

We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iu_t–u+q(|u|²)u=0 in iu_t - u + (|u|²)u = in (t, x)xⁿ for 6n11.     

Time dependent nonlinear Schrödinger equations

We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u+iAu+f(|u|²)u or u+Au+if(|u|²)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n.     

On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation on . We present existence and non-existence results and investigate qualitative properties of the solutions when they exist. Mathematics Subject Classification (2000) 35Q55, 35G25 Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday.     

Three nonlinear integrable couplings of the nonlinear Schrödinger equations

Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville.     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).     

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Given \(\rho >0\), we study the elliptic problem

$$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain and \(p>1\) is Sobolev-subcritical, searching for conditions (about \(\rho \), N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is \(L^2\)-subcritical, i.e. \(1<p<1+4/N\), the problem admits solutions for every \(\rho >0\). In the \(L^2\)-critical and supercritical case, i.e. when \(1+4/N \le p < 2^*-1\), we show that, for any \(k\in {\mathbb {N}}\), the problem admits solutions having Morse index bounded above by k only if \(\rho \) is sufficiently small. Next we provide existence results for certain ranges of \(\rho \), which can be estimated in terms of the Dirichlet eigenvalues of \(-\Delta \) in \(H^1_0(\Omega )\), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.