Infinitely many positive solutions of nonlinear Schrodinger equations

Abstract

The paper deals with the equation $-\Delta u+a(x) u =|u|^{p-1}u $, $u \in H^1(\mathbb{R}^N)$, with $N\ge 2$, $p>1,\ p<{N+2\over N-2}$ if $N\ge 3$, $a\in L^{N/2}_{loc}(\mathbb{R}^N)$, $\inf a>0$, $\lim_{|x| \to \infty} a(x)= a_\infty$. Assuming on the potential that $\lim_{|x| \to \infty}[a(x)-a_\infty] e^{\eta |x|}= \infty \ \ \forall \eta>0$ and $ \lim_{\rho \to \infty} \sup \left\{a(\rho \theta_1) - a(\rho \theta_2) \ :\ \theta_1, \theta_2 \in \mathbb{R}^N,\ |\theta_1|= |\theta_2|=1 \right\} e^{\tilde{\eta}\rho} = 0 \mbox{ for some } \ \tilde{\eta}>0$, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers [8,12,15,17,28].