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This article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a,b>0a,b\gt 0 are given constants, ε\varepsilon is a small parameter, 2s∗=2NN−2s{2}_{s}^{\ast }=\frac{2N}{N-2s} with 04sN\gt 4s, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε\varepsilon small.

Compactness is a fundamental issue in nonlinear analysis. Assume s∈(0,1)$s\in (0,1)$ and Ω⊂RN$\Omega \subset \mathbb {R}^N$ with N>2s$N>2s$, we consider a subspace X0,Gs(Ω)$X^s_{0,G}(\Omega)$ of X0s(Ω)$X_0^s(\Omega)$, which is the space of invariant functions corresponding to a subgroup G of O(N)$O(N)$, where X0s(Ω)$X_0^s(\Omega)$ is a kind of function spaces including fractional Sobolev spaces H0s(Ω)$H_0^s(\Omega)$. We show that the embeddings X0,Gs(Ω)↪Lp(Ω)$X_{0,G}^s(\Omega)\hookrightarrow L^p(\Omega)$ are compact for p∈(2,2s∗)$p\in (2,2_s^*)$ provided Ω is compatible with G, where 2s∗=2NN−2s$2^*_s=\frac{2N}{N-2s}$ is the fractional Sobolev critical exponent. Moreover, the existence and multiplicity of radial and nonradial solutions were obtained of the following nonlocal Schrödinger equation: −Lku+V(x)u=f(x,u),x∈RN,$$\begin{equation*} -{\mathcal {L}_k}u+V(x)u=f(x,u),\ x\in \ \mathbb {R}^N, \end{equation*}$$where −Lk$-{\mathcal {L}_k}$ is an integro‐differential operator has order 2s and can be given by Lku(x):=∫RNu(x+y)+u(x−y)−2u(x)K(y)dy,x∈RN,$$\begin{equation*} {\mathcal {L}_k}u(x):=\int _{\mathbb {R}^N}{\left(u(x+y)+u(x-y)-2u(x)\right)}K(y)dy,\ x\in \mathbb {R}^N, \end{equation*}$$and the kernel K satisfies the following properties: (K1)there is θ>0$\theta > 0$ and s∈(0,1)$s\in (0,1)$ such that K(x)≥θ|x|−(N+2s)$K(x)\ge \theta |x|^{-({N+2s})}$ for any x∈RN∖{0}$x\in \mathbb {R}^N\backslash \lbrace 0\rbrace$;(K2)mK∈L1(RN)$mK\in L^1(\mathbb {R}^N)$, where m(x):=min{|x|2,1}$m(x):=\min \lbrace |x|^2,1\rbrace$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Kirchhoff equation: (a + b ∫ R 3 | (− Δ) s 2 u | d x) (− Δ) s u + V (x) u = f (u) , x ∈ R 3 , where a > 0 , b ≥ 0 , (− Δ) s denotes the fractional Laplacian operator with order s ∈ (3 4 , 1) , V is a positive continuous potential and f is supercubic but subcritical functional at infinity with some valid conditions. We prove that there exist multiple sign-changing solutions for the above problem via the method of invariant sets of descending flow. In particular, the nonlinear term includes the power-type nonlinearity f (u) = | u | p − 2 u for the less studied case p ∈ (3 , 4). Even for s = 1, our result is new and extends the existing results in the literature. [ABSTRACT FROM AUTHOR]

We study the following mean field equation on a flat torus |$T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$|⁠ : $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$ where |$ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$|⁠ , and |$|T|$| denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of |$u$| provided that |$\rho \leq 8\pi $|⁠. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if |$\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$|⁠. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a generalized quasilinear Choquard equation: − ε 2 Δ u + V (x) u − ε 2 u Δ (u 2) = ε μ − N (| x | − μ ∗ F (u)) f (u) in R N , where N ≥ 3 , ε > 0 is a parameter, μ ∈ (0 , (N + 2) / 2) , F (t) = ∫ 0 t f (s) d s. Under some assumptions of V and f, the existence, multiplicity and concentration behavior of positive solutions are obtained via dual approach. [ABSTRACT FROM AUTHOR]