We are concerned with the following fractional p-Laplace equation(−Δp)su=ups*−1inRn,where 0 < s < 1, max{2nn+2s,1}<p<ns and ps*=npn−ps. We establish decay estimates at infinity for non-trivial nonnegative weak solutions. Based on this, we derive that any nonnegative weak solution is radially symmetric and monotone decreasing about some point in Rn. As a result, the optimal decay at infinity for nonnegative weak solutions follows.

In this paper, we are concerned with the following mixed order conformally invariant system with exponential Hartree nonlinearity and cubic nonlinearity: {[ (-Δ )^1/2 u(x)=( 1/|x|^2*e^2pv(x)) e^pv(x), x∈ℝ^3,; (-Δ )^3/2 v(x)= u^3(x), x∈ℝ^3, ]. where p>0 , u≥ 0 , v may change sign and u satisfies the finite total curvature condition ∫ _ℝ^3 u^3(x)dx<+∞ . Under extremely mild assumptions, we prove that, the classical solution (u, v) must take the unique form: u(x)=2^4/3p^-1/3μ/1+μ ^2|x-x_0|^2, v(x)=1/pln [( 2^7/3p^-1/3π ^2/I^2(1)) ^1/3μ/1+μ ^2|x-x_0|^2 ] for some μ >0 and x_0∈ℝ^3 , where I(1):=π ^3/2Γ (1/2)/Γ (2) .

We study the following equation involving higher-order fractional Laplacian: (-Delta)(p+ alpha)2u(x) = u(+)(gamma)(x)in & Ropf;(n), where u(+) =max{u, 0}, n > 2p + alpha, 0 < alpha < 2 and 1 <= p is an element of & Zopf;. Under either one of the two integral constraints u(+)(gamma) is an element of L-1(& Ropf;(n)),gamma is an element of( 1, n/n - 2p - alpha] and u(+)(q) is an element of L-1(& Ropf;(n)),q = n(gamma - 1)/ 2p + alpha ,gamma is an element of (1, n + 2p + alpha/ n - 2p - alpha), we establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this, we prove that these solutions are radially symmetric about some point in & Ropf;n and monotone decreasing in the radial direction via method of moving planes in integral forms. We also establish a Moser-Trudinger-type inequality associated to the equation - Delta u = u(+)(gamma) in bounded domain for gamma is an element of (1, n/n-2).

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian (-partial derivative xx)su(x)+del F(u(x)) = 0 in R, where F : R2 -> R is a smooth double-well potential. For the case that F is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark's theorem. For the case that F is only even in its the second variable and the origin is a saddle critical point of F, we give two periodic solutions by using Morse theory.

We consider the following semi-linear equations (-Δ )^pu=u^γ _+ in ℝ^n, where γ∈ (1,n+2p/n-2p) , n>2p>0 , u_+=max{u,0} , and 2≤ p∈ℕ or p∈ (0,1) . Subject to the integral constraint u_+^γ∈ L^1(ℝ^n), we obtain the classification of solutions to the above polyharmonic equation for any γ

We are concerned with periodic solutions of the fractional Laplace equation \begin{equation*} {(-\partial_{xx})^s}u(x)+F'(u(x))=0 \quad \mbox{in }\mathbb{R}, \end{equation*} where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at $+1$ and $-1$. We show that the value of least positive period is $2{\pi}\times({1}/{-F''(0)})^{{1}/({2s})}$. The axial symmetry of odd periodic solutions is obtained by moving plane method. We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution of the same equation as periods $T\rightarrow+\infty$.

We consider periodic solutions of the following problem associated with the fractional Laplacian: (-∂x⁢x)s⁢u⁢(x)+∂u⁡F⁢(x,u⁢(x))=0{(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in ℝ{\mathbb{R}}. The smooth function F⁢(x,u){F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at +1{+1} and -1 for any x∈ℝ{x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.