Existence of solutions for a large defocusing parameter in a strongly indefinite Schrödinger equation

Abstract

Let $N\ge 3$, $2<q<p<2^{\ast }$, and let $V\in L^{\infty }({\mathbb{R}}^N)$ be real-valued and ${\mathbb{Z}}^N$-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator $L=-\Delta +V$. We prove that there exists ${\lambda }_{\infty }>0$ such that, for every $\lambda \ge {\lambda }_{\infty }$, the competing-power equation $$-\Delta u+V(x)u=|u{|}^{p-2}u-\lambda |u{|}^{q-2}u\text{in }{\mathbb{R}}^N$$ has a nontrivial solution $u_{\lambda }\in H^1({\mathbb{R}}^N)$. Moreover, $$\Vert u_{\lambda }{\Vert }_{H^1({\mathbb{R}}^N)}+\Vert u_{\lambda }{\Vert }_{L^{\infty }({\mathbb{R}}^N)}\le C{\lambda }^{-1/(q-2)}.$$ We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation $Lv=-|v{|}^{q-2}v$.