Multiplicity of solutions of nonlinear scalar field equations

Abstract

In this Note we present new multiplicity results for the solutions of nonlinear elliptic problems of the form $-\Delta u+a(x)u=|u{|}^{p-1}u$ in ${\mathbb{R}}^N$, $u\in H^1({\mathbb{R}}^N)$, where $N\ge 2$, $p>1$, $p<\frac{N+2}{N-2}$ if $N\ge 3$, $a\in L_{\mathrm{loc}}^{N/2}({\mathbb{R}}^N)$, ${\inf }_{{\mathbb{R}}^N}a>0$. In particular, we have infinitely many positive solutions when there exists $a_{\infty }>0$ such that ${\lim }_{|x|\to \infty }a(x)=a_{\infty }$ and ${\lim }_{|x|\to \infty }[a(x)-a_{\infty }]e^{\eta |x|}=+\infty$ $\forall \eta >0$.