Some results on a class of nonlinear Schrödinger equations
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 \(G\in C(\mathbb R^N; \mathbb R^N)\): \[ \begin{aligned} & G_j(P):=-\frac 12 \frac{\partial V}{\partial x_j}(P) \int\limits_{\mathbb R^N} w^2_p dx + \int\limits_{\mathbb R^N} F_{x_j}(P,w_p)dx, \\ & F(x,t):= \int\limits^t_0 f(x,z) dz. \end{aligned} . \]
scientific article; zbMATH DE number 1559189