For any $u \in H^2(\mathbb R^n)$, consider operator $$ Lu = -\frac{1}{2}\Delta u + u - 3u_0^2u $$ where $u_0$ is a solution of $$ -\frac{1}{2}\Delta u + u - u ^3 =0 $$ how to show $L$ has only one negative eigenvalue ?
In fact, I know $u_0$ satisfy $$ |D^\alpha u _0|\le Ce^{-|x|}, |\alpha|\le 2 $$ But I don't know how to show only one negative eigenvalue $L$ has.