## 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 $$\lambda$$.

### MSC:

 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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