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Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. (English) Zbl 0903.35018

The authors consider the nonlinear Schrödinger equation \[ i\hbar{\partial\psi\over\partial t}=-\hbar^2\triangle\psi+U(x)\psi-| \psi| ^ {p-2}\psi,\;x\in{\mathbb{R}}^N, \] where \(p>2\) if \(N=1,\;2\) and \(2<p<2N/(N-2)\) if \(N\geq 3\). Standing waves are solutions of the type \(\psi(t,x)=\exp(-i\lambda\hbar^{-1}t)u(x)\), where \(\lambda\in{\mathbb{R}}\) and \(u\) is a real valued function. Set \(\epsilon=\hbar\), then \(u\) satisfies \[ -\epsilon^2\triangle u+V(x)u=| u| ^{p-2}u,\;x\in{\mathbb{R}}^N,\;u>0,\tag{\(P_\epsilon\)} \] where \(V(x)=u(x)+\lambda\). Solutions of \((P_\epsilon)\), \(\epsilon>0\) small, are said to be semiclassical solutions of the Schrödinger equation. The first proof of the existence of a solution was given by Floer and Weinstein in 1989 in the case \(N=1\) and \(p=4\). Then Rabinowitz proved in 1992 the existence of a solution under the assumption \(\liminf_{| x| \to\infty} V(x)>\inf_{x\in{\mathbb{R}}^N} V(x)\) by using his mountain-pass theorem. Set \(V_0=\inf_{x\in{\mathbb{R}}^N},\;M=\{x\in{\mathbb{R}}^N:\;V(x)=V_0\}\) and \(M_\delta=\{x\in{\mathbb{R}}^N:\;d(x,M)\leq \delta\}\). The authors prove the following multiplicity result: Assume that \(V\) is in \(C({\mathbb{R}}^N)\) and that \(\liminf_{| x| \to\infty}V(x)>V_0>0\). Then, for any \(\delta>0\), there exists \(\epsilon_\delta>0\) such that \((P_\epsilon)\) has at least cat\(_{M_\delta}(M)\) solutions, for any \(\epsilon <\epsilon_\delta\). The proof is based on generalizations of the Lyusternik-Schnirelman category method due to Benci and Cerami.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
49J35 Existence of solutions for minimax problems
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