A positive answer to Ambrosetti-Malchiodi conjecture in Fractional Schr\"{o}dinger Equations

We study the following fractional Schrodinger equation \begin{equation*}\label{eq0.1} \epsilon^{2s}(-\Delta)^s u + Vu = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1),\,p\in \big(2 + 2s/(N - 2s), 2^*_s\big)$, $2^*_s=2N/(N-2s),\,N>2s$, $V\in C(\mathbb{R}^N;[0,\infty))$. We use penalized technique to show that the problem has a family of solutions concentrating at a local minimum of $V$. Our results solve the fractional version of Ambrosetti-Malchiodi conjecture(\cite{AA}) completely since the potential can decay arbitrarily. The argument used in this paper also works well to problems above with more general nonlinear terms.