Standing waves for supercritical nonlinear Schrödinger equations. (English) Zbl 1124.35082

The authors treat standing waves of the nonlinear Schrödinger equation in \(\mathbb{R}^N\) with a supercritical nonlinearity, namely solutions of the problem \[ \Delta u-V(x)u+u^p=0,\qquad u>0,\;\lim_{| x| \to\infty}u(x)=0.\tag{1} \] Here it is assumed that \(V\) is bounded and nonnegative, \(N\geq3\), and \(p>(N+2)/(N-2)\). To prove existence, in the case \(N\geq4\) and \(p>(N+1)/(N-3)\), the only additional assumption on \(V\) is that of superquadratic decay at \(\infty\). In the general supercritical case the authors have to assume a somewhat faster decay of \(V\) at \(\infty\). Under these hypotheses it is proved that there exists a continuum of small solutions of (1).
This result stands in sharp contrast to the subcritical case, where one only expects solutions if \(V\) decays slower that quadratic at \(\infty\). Moreover, it is remarkable that a continuum of solutions is presented without employing a singular perturbation parameter. The method rests on the existence of a scaled family \(w_\lambda(x)=\lambda^{\frac{2}{p-1}}w(\lambda x)\) of radially symmetric positive solutions of the problem \[ \Delta w +w^p=0\qquad\text{in }\mathbb{R}^N. \] By a fixed point argument it is shown that a solution of (1) exists near some \(w_\lambda(\cdot-\xi)\) if \(\lambda\) is sufficiently small. For \(p> (N+1)/(N-3)\) the center of symmetry \(\xi\) can be chosen arbitrarily within expanding domains of \(\mathbb{R}^N\), essentially giving an \(N+1\)-dimensional set of solutions. In the general case \(\xi\) needs to be chosen from a fixed point set depending on \(\lambda\).


35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q51 Soliton equations
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