×

Some results on a class of nonlinear Schrödinger equations. (English) Zbl 0970.35039

There is considered the following problem: \[ \left\{ \begin{aligned} & -h^2\Delta u+ V(x)u=f(x,u)\quad\text{in}\quad \mathbb R^N, N\geq 1, h\in \mathbb R^1, \\ & u(x) >0\quad\text{for all} \quad x\in\mathbb R^N, \end{aligned} \right. \] where the potential \(V(x)\in C^1(\mathbb R^N;\mathbb R^+)\cap L^\infty(\mathbb R^N;\mathbb R)\) and the nonlinearity \(f(x,t)\) satisfy some structural and smoothness assumptions, particularly, the following one: For any \(P\in \mathbb R^N\) the problem: \[ \left\{\begin{aligned} & -\Delta w+ V(P)w=f(P,w)\quad\text{in}\quad\mathbb R^N, \\ & w>0 \quad \text{in} \quad \mathbb R^N, \\ & w(x)\to 0 \quad \text{as} \quad |x|\to\infty, \end{aligned}\right. \] has a unique solution \(w_p(x)\) which is nondegenerate in the space of the radial functions. The main result states the existence of \(h_0>0\) such that for \(0<h<h_0\) the problem under consideration admits a family of solutions \(u_h\in C^2(\mathbb R^N)\) whose unique maximum point \(Q_h\) satisfies \(\text{dist}(Q_h, Z)\to 0\) as \(h\to 0\). Here \(Z\) is some stable bounded set of zeros of vector field \(G\in C(\mathbb R^N; \mathbb R^N)\): \[ \begin{aligned} & G_j(P):=-\frac 12 \frac{\partial V}{\partial x_j}(P) \int\limits_{\mathbb R^N} w^2_p dx + \int\limits_{\mathbb R^N} F_{x_j}(P,w_p)dx, \\ & F(x,t):= \int\limits^t_0 f(x,z) dz. \end{aligned} . \]

MSC:

35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35J10 Schrödinger operator, Schrödinger equation
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI