On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation $-\Delta u+V(x)u = K(x)\vert u\vert^{2^{*}-2}u+g(x,u)$, $u\in W^{1,2}(\mathbf{R}^{N})$, where $N\ge 4$, $V,K,g$ are periodic in $x_{j}$ for $1\le j\le N$, $K>0$, $g$ is of subcritical growth and 0 is in a gap of the spectrum of $-\Delta+V$. We show that under suitable hypotheses this equation has a solution $u\ne 0$. In particular, such a solution exists if $K\equiv 1$ and $g\equiv 0$.