American Institute of Mathematical Sciences

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

In this paper, we study a class of integro-differential elliptic operators \begin{document} $L_{σ}$ \end{document} with kernel \begin{document} $k(y) = a(y)/|y|^{d+σ}$ \end{document}, where \begin{document} $d≥2, σ∈(0,2)$ \end{document}, and the positive function \begin{document} $a(y)$ \end{document} is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution \begin{document} $Φ$ \end{document} of \begin{document} $L_{σ}$ \end{document} if \begin{document} $a(y)$ \end{document} satisfies a natural cancellation assumption and \begin{document} $|a(y)-1|$ \end{document} is small. Furthermore, we show that the fundamental solution \begin{document} $Φ$ \end{document} is \begin{document} $-α^{*}$ \end{document} homogeneous and Lipschitz continuous, where the constant \begin{document} $α^{*}∈(0,d)$ \end{document}. A Liouville-type theorem demonstrates that the fundamental solution \begin{document} $Φ$ \end{document} is the unique nontrivial solution of \begin{document} $L_{σ}u = 0$ \end{document} in \begin{document} $\mathbb{R}^{d}\setminus\{0\}$ \end{document} that is bounded from below.

In this paper, we consider a fractional equation with indefinite nonlinearities

\begin{document}$(-\vartriangle )^{α/2} u = a(x_1) f(u) $ \end{document}

for \begin{document} $0<α<2$ \end{document}, where \begin{document} $a$ \end{document} and \begin{document} $f$ \end{document} are nondecreasing functions. We prove that there is no positive bounded solution. In particular, in the case \begin{document} $a(x_1) = x_1$ \end{document} and \begin{document} $f(u) = u^p$ \end{document}, this remarkably improves the result in [15] by extending the range of \begin{document} $α$ \end{document} from \begin{document} $[1,2)$ \end{document} to \begin{document} $(0,2)$ \end{document}, due to the introduction of new ideas, which may be applied to solve many other similar problems.

In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with subcritical or critical growth:

\begin{document} $ \begin{equation*}-Δ u+W(x)u^{2α-1}-ul'(u^2)Δ l(u^2) = f(u) \ \mbox{or}\ h(u)+u^{2^*-1},\ x∈\mathbb{R}^N,\end{equation*} $ \end{document}

where \begin{document}$W(x):\mathbb{R}^N \to \mathbb{R} $\end{document} is a given potential and \begin{document}$ l,h,f $\end{document} are real functions, \begin{document}$ u>0,$\end{document} \begin{document}$ 2^* = 2N/(N-2), $\end{document} \begin{document}$ N≥3 $\end{document}. Our results cover physical models \begin{document}$ l(s) = s^{\frac{α}{2}}, $\end{document} \begin{document}$ \frac{1}{2}<α<1. $\end{document}

It is known that the supercritical Hardy-Littlewood-Sobolev (HLS) systems with an integer power of Laplacian admit classic solutions. In this paper, we prove that the supercritical HLS systems with fractional Laplacians \begin{document}$ (-Δ)^s $\end{document}, \begin{document}$ s∈(0,1) $\end{document}, also admit classic solutions.

The Lane-Emden conjecture says that the subcritical Lane-Emden system admits no positive solution. In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy-type a priori estimate. The necessity of the condition we found can be easily checked. However, a major difficulty lies in the sufficiency. The proof is quite involving, but the benefit is that it reduces the longstanding problem to obtaining the a priori estimate of energy type.

In this paper, we consider a nonlinear equation involving fractional Laplacian of higher order on the whole space. We establish the equivalence between the pseudo-differential equation and an integral equation by applying the maximum principle and the Liouville theorem. For positive solutions to the equation, we obtained non-existence by applying the method of moving planes.

In this paper, we are concerned with the fractional order equations (1) with Hartree type \begin{document}$ \dot{H}^{\frac{α}{2}} $\end{document}-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions \begin{document}$ u $\end{document} to (1) and (3) are radially symmetric about some point \begin{document}$ x_{0}∈\mathbb{R}^{d} $\end{document} and derive the explicit forms for \begin{document}$ u $\end{document} (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).

We are concerned with Hölder regularity estimates for weak solutions \begin{document}$u$\end{document} to nonlocal Schrödinger equations subject to exterior Dirichlet conditions in an open set \begin{document}$\Omega\subset \mathbb{R}^N$\end{document}. The class of nonlocal operators considered here are defined, via Dirichlet forms, by symmetric kernels \begin{document}$K(x, y)$\end{document} bounded from above and below by \begin{document}$|x-y|^{-N-2s}$\end{document}, with \begin{document}$s\in (0, 1)$\end{document}. The entries in the equations are in some Morrey spaces and the domain \begin{document}$\Omega$\end{document} satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When \begin{document}$K$\end{document} defines a nonlocal operator with sufficiently regular coefficients, we obtain Hölder estimates, up to the boundary of \begin{document}$ \Omega$\end{document}, for \begin{document}$u$\end{document} and the ratio \begin{document}$u/d^s$\end{document}, with \begin{document}$d(x) = \text{dist}(x, \mathbb{R}^N\setminus\Omega)$\end{document}. If the kernel \begin{document}$K$\end{document} defines a nonlocal operator with Hölder continuous coefficients and the entries are Hölder continuous, we obtain interior \begin{document}$C^{2s+\beta}$\end{document} regularity estimates of the weak solutions \begin{document}$u$\end{document}. Our argument is based on blow-up analysis and compact Sobolev embedding.

In this paper, by using the Alexandrov-Serrin method of moving plane combined with integral inequalities, we obtained the complete classification of positive solution for a class of degenerate elliptic system.

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions for a fractional p-Laplacian equation possess certain Hölder continuity up to the boundary.

Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak finite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the finite energy solution. Based on these results, we prove that the weak finite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.

In this paper, we consider the fractional Laplacian system

\begin{document}$\left\{\begin{array}{ll}(-\triangle)^{\frac{\alpha}{2}}u+\sum^{N}_{i = 1}b_{i}(x)\frac{\partial u}{\partial x_{i}}+C(x)u = f(x,v), \;\;x\in \Omega,\\(-\triangle)^{\frac{\beta}{2}}v+\sum^{N}_{i = 1}c_{i}(x)\frac{\partial v}{\partial x_{i}}+D(x)v = g(x,u),\;\; x\in \Omega,\\u>0, v>0, \;\; x\in \Omega,\\u = 0, v = 0, \;\; x\in \mathbb R^{N}\setminus \Omega,\end{array}\right.$ \end{document}

where \begin{document}$Ω$\end{document} is a smooth bounded domain in \begin{document}$\mathbb R^{N}$\end{document}, \begin{document}$α ∈ (1,2)$\end{document}, \begin{document}$β ∈ (1,2)$\end{document}, \begin{document}$N>\max\{α, β\}$\end{document}. Under some suitable conditions on potential functions and nonlinear terms, we use scaling method to obtain a priori bounds of positive solutions for the fractional Laplacian system with distinct fractional Laplacians.

In this paper, we study the radial symmetry of the solution to the following system of integral form:

\begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_i}(x) = \int_{{{\bf{R}}^n}} {\frac{1}{{|x - y{|^{n - \alpha }}}}} {f_i}(u(y))dy,\;\;x \in {{\bf{R}}^n},\;\;i = 1, \cdots ,m,}\\{0 < \alpha < n,\;{\rm{ and }}\;u(x) = ({u_1}(x),{u_2}(x), \cdots ,{u_m}(x)).}\end{array}} \right.\;\;\;\;\;\;\left( 1 \right)$ \end{document}

Here \begin{document}$f_i(s)∈ C^1(\mathbf{R^m_+})\bigcap$\end{document}\begin{document}$ C^0(\mathbf{\overline{R^m_+}})$\end{document}\begin{document}$(i = 1,2,···,m)$\end{document} are real-valued functions, nonnegative and monotone nondecreasing with respect to the variables \begin{document}$s_1$\end{document}, \begin{document}$s_2$\end{document}, \begin{document}$···$\end{document}, \begin{document}$s_m$\end{document}. We show that the nonnegative solution \begin{document}$u = (u_1,u_2,···,u_m)$\end{document} is radially symmetric in the general condition that \begin{document}$f_i$\end{document} satisfies monotonicity condition which contains the critical and subcritical homogeneous degree as special cases. The main technique we use is the method of moving planes in an integral form. Due to our condition here is more general, the more subtle method is needed to deal with this difficulty.

In this paper, we investigate the following fractional elliptic system

\begin{document}$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$ \end{document}

where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $ α = β$, a Liouville theorem is established.

We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in \begin{document}$\mathbb{R}^n$\end{document}. By using a direct method of moving planes, we show that \begin{document}$u_i(x)$\end{document} (\begin{document}$i = 1,2,···,m$\end{document}) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of \begin{document}$\frac{\partial f_i}{\partial u_i}$\end{document}. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.

In this paper, we consider the fractional p-Laplacian equation

\begin{document}$( - \Delta )_p^su(x) = f(u(x)), $ \end{document}

where the fractional p-Laplacian is of the form

\begin{document}$( - \Delta )_p^su(x) = {C_{n, s, p}}PV\int_{{\mathbb{R}^n}} {\frac{{{{\left| {u(x) - u(y)} \right|}^{p - 2}}(u(x) - u(y))}}{{{{\left| {x - y} \right|}^{n + sp}}}}} dy.$ \end{document}

By proving a narrow region principle to the equation above and extending the direct method of moving planes used in fractional Laplacian equations, we establish the radial symmetry in the unit ball and nonexistence on the half space for the solutions, respectively.

In this paper, we show the following equation

\begin{document}$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$ \end{document}

has at most one positive radial solution for a certain range of \begin{document}$λ>0$\end{document}. Here \begin{document}$p>1$\end{document} and \begin{document}$Ω$\end{document} is the annulus \begin{document}$\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$\end{document}, \begin{document}$0<a<b$\end{document}. We also show this solution is radially non-degenerate via the bifurcation methods.

In this paper, we consider the following Schrödinger systems involving pseudo-differential operator in \begin{document}$ R^n$\end{document}

\begin{document}$\left\{ {\begin{array}{*{20}{l}}{{{( - \Delta )}^{\frac{\alpha }{2}}}u(x) = {u^{{\beta _1}}}(x){v^{{\tau _1}}}(x),}&{{\rm{in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\\{{{( - \Delta )}^{\frac{\gamma }{2}}}v(x) = {u^{{\beta _2}}}(x){v^{{\tau _2}}}(x),}& \ \ \ {{\rm{ in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ \end{document}

where \begin{document}$ α$\end{document} and \begin{document}$ γ$\end{document} are any number between 0 and 2, \begin{document}$ α$\end{document} does not identically equal to \begin{document}$ γ$\end{document}.

We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.

In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.