Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth

In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: (−Δ+m2)su+V(εx)u=f(u)+u2s*−1inRN,u∈Hs(RN),u>0inRN,\left\{\begin{array}{ll}{\left(-\Delta +{m}^{2})}^{s}u+V\left(\varepsilon x)u=f\left(u)+{u}^{{2}_{s}^{* }-1}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, s∈(0,1)s\in \left(0,1), m>0m\gt 0, N>2sN\gt 2s, 2s*=2NN−2s{2}_{s}^{* }=\frac{2N}{N-2s} is the fractional critical exponent, (−Δ+m2)s{\left(-\Delta +{m}^{2})}^{s} is the fractional relativistic Schrödinger operator, V:RN→RV:{{\mathbb{R}}}^{N}\to {\mathbb{R}} is a continuous potential, and f:R→Rf:{\mathbb{R}}\to {\mathbb{R}} is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential VV, we construct a family of positive solutions uε∈Hs(RN){u}_{\varepsilon }\in {H}^{s}\left({{\mathbb{R}}}^{N}), with exponential decay, which concentrates around a local minimum of VV as ε→0\varepsilon \to 0.