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

Abstract

Let $N\geq 3$, $2 0$ such that, for every $λ\geqλ_\infty$, the competing-power equation $$ -Δu+V(x)u=|u|^{p-2}u-λ|u|^{q-2}u \quad\text{in }\mathbb{R}^N $$ has a nontrivial solution $u_λ\in H^1(\mathbb{R}^N)$. Moreover, $$ \|u_λ\|_{H^1(\mathbb{R}^N)}+\|u_λ\|_{L^\infty(\mathbb{R}^N)} \leq Cλ^{-1/(q-2)}. $$ 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 $Lv=-|v|^{q-2}v$.