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

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