In this paper, we study the following nonlinear fractional Schr\documentclass[12pt]{minimal}\begin{document}$\ddot{\mbox{o}}$\end{document}ödinger equation with critical exponent \documentclass[12pt]{minimal}\begin{document}$h^{2\alpha }(-\Delta )^{\alpha }u + V(x)u= |u|^{2_{\alpha }^{*}-2}u + \lambda |u|^{q-2}u, x\in \mathbb {R}^{N}$\end{document}h2α(−Δ)αu+V(x)u=|u|2α*−2u+λ|u|q−2u,x∈RN, where h is a small positive parameter, 0 < α < 1, \documentclass[12pt]{minimal}\begin{document}$2< q < 2_{\alpha }^{*}$\end{document}2<q<2α*, \documentclass[12pt]{minimal}\begin{document}$2_{\alpha }^{*} = \frac{2N}{N- 2\alpha }$\end{document}2α*=2NN−2α is the critical Sobolev exponent, and N > 2α, λ > 0 is a parameter. The potential \documentclass[12pt]{minimal}\begin{document}$V: \mathbb {R}^{N} \rightarrow \mathbb {R}$\end{document}V:RN→R is a positive continuous function satisfying some natural assumptions. By using variational methods, we obtain the existence of solutions in the following case: if \documentclass[12pt]{minimal}\begin{document}$2< q< 2_{\alpha }^{*}$\end{document}2<q<2α*, there exists λ0 > 0 such that for all λ ⩾ λ0, we show that it has one nontrivial solution and there exist at least \documentclass[12pt]{minimal}\begin{document}$cat_{\Lambda _{\delta }}(\Lambda )$\end{document}catΛδ(Λ) nontrivial solutions; if \documentclass[12pt]{minimal}\begin{document}$\max \lbrace 2, \frac{4\alpha }{N-2\alpha }\rbrace < q < 2_{\alpha }^{*}$\end{document}max{2,4αN−2α}<q<2α*, then there is one nontrivial solution and there exist at least \documentclass[12pt]{minimal}\begin{document}$cat_{\Lambda _{\delta }}(\Lambda )$\end{document}catΛδ(Λ) nontrivial solutions for all λ > 0.

In this paper we study the concentration phenomenon of solutions for the nonlinear fractional Schrödinger equationwhere ε is a positive parameter, s ∈ (0, 1), N ≥ 2 and 1 < p < N+2s N−2s , V (x) and K(x) are positive smoothUnder certain assumptions on V (x) and K(x), we show existence and multiplicity of solutions which concentrate near some critical points of Γ (x) by a perturbative variational method.

This paper is devoted to study the following Choquard equationwhere 0 < α, β < 2, 1 ≤ p < ∞, and n ≥ 2. Using a direct method of moving planes, we prove the symmetry and nonexistence of positive solutions in the critical and subcritical case respectively.

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In this work, we study the following critical problem involving the fractional Laplacian:s , the sharp constant of the Hardy-Sobolev inequality.For suitable assumptions on g(x) and K(x), we consider the existence and multiplicity of positive solutions depending on the value of p. Moreover, we obtain an existence result for the problem when = 0.