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 $N\ge 2$, $p>1,p<\frac{N+2}{N-2}$ if $N\ge 3$, $a\in L_{loc}^{N/2}({\mathbb{R}}^N)$, $\inf a>0$, ${\lim }_{|x|\to \infty }a(x)=a_{\infty }$. Assuming on the potential that ${\lim }_{|x|\to \infty }[a(x)-a_{\infty }]e^{\eta |x|}=\infty \forall \eta >0$ and $ \lim_{ρ\to \infty} \sup \left\{a(ρθ_1) - a(ρθ_2) \ :\ θ_1, θ_2 \in \mathbb{R}^N,\ |θ_1|= |θ_2|=1 \right\} e^{\tildeηρ} = 0 \mbox{ for some } \ \tildeη>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].