A non-periodic indefinite variational problem in ℝN with critical exponent

Abstract

Abstract We consider the non-linear Schrödinger equation (Pμ)\begin{equation*} \begin{array}{lc} -\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, & \end{array} \end{equation*}in $\mathbb{R}^N$, $N\geq3$, where V changes sign and $f(s)/s$, s ≠ 0, is bounded, with V non-periodic in x. The existence of a solution is established employing spectral theory, a general linking theorem due to [12] and interaction between translated solutions of the problem at infinity with some qualitative properties of them.