We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
Let $0<α<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{α/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.
Let $Δ_k$ be the Dunkl Laplacian on $\mathbb R^d$ associated with a reflection group $W$ and a multiplicity function $k$. The purpose of this paper is to establish necessary and sufficient condition under which there exists a positive solution of the equation $Δ_ku=φ(u)$ in the unit ball of $\mathbb R^d$ as well as in the whole space $\mathbb R^d$.
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
We study bounded solutions to the fractional equation $(-Δ)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).
In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates obtained in the previous studies. The second aim of this article is to apply these estimates to prove small data global well-posedness for the Cauchy problem of the equation with power nonlinearities. Especially, the estimates obtained in this paper enable us to treat more general conditions on the nonlinearities and the spatial dimension than the results in the p…
In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space $\mathbb{R}^n$: \begin{equation} (-Δ)^{α/2} u(x) = f(x_n) \,u^p(x), \quad x \in \mathbb{R}^n \label{n26} \end{equation} in the subcritical case with $1<p<\frac{n+α}{n-α}$. Instead of carrying out direct investigations on pseudo-differential equation (\ref{n26}), we first seek its equivalent form in an integral equation as below: \begin{equation} u(x)=\int_{\mathbb{R}^n}G_{\infty}(x,y)\,f(y_n)\, u^{p}(y)\,dy, \label{n27} \end{equation} where $ G_{\infty}(x,y)$ is the Green's function ass…
We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open sets.
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the non-local operator and boundary behaviour of Green and Poisson potentials.
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
We study the local propagation of conormal singularities for solutions of semilinear wave equations $\square u = P(y, u)$, where $P(y, u)$ is a polynomial of degree $N \geq 3$ in $u$ with $C^\infty(\mathbb{R}^3_y)$ coefficients. We know from the work of Melrose & Ritter and Bony that if u is conormal to three waves which intersect transversally at point $q$, then after the triple interaction $u(y)$ is a conormal distribution with respect to the three waves and the characteristic cone $Q$ with vertex at $q$. We compute the principal symbol of $u$ at the cone and away from the hypersurfaces. We …
Let $n\geq2$ and $ Ω\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$Δu + u^p = 0 \quad\hbox{in}\, Ω,$$ which vanish in a suitable trace sense on $\partialΩ,$ but which are singular at prescribed "edge" of $Ω$ if $p$ is equal or slightly above a certain exponent $p_0>1$ which depends on $Ω.$ Moreover, in the case which $Ω$ is unbounded, the solutions have fast decay at infinity.
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
We show radial symmetry of positive solutions to the Hénon equation $-Δu = |x|^{-\ell} u^q $ in $\mathbb{R}^N \setminus \{ 0\} $, where $\ell \geq 0$, $q>0$ and satisfy further technical conditions. A new ingredient is a maximum principle for open subsets of a half space. It allows to apply the Moving Plane Method once a slow decay of the solution at infinity has been established, that is $\lim _{|x|\to \infty } |x|^{γ} u(x) =L $, for some numbers $γ\in (0, N-2)$ and $L >0$. Moreover, some examples of non-radial solutions are given for $q> \frac{N+1}{N-3}$ and $N\geq 4$. We also establish radi…
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractiona…
Given a smooth domain $Ω\subset\RR^N$ such that $0 \in \partialΩ$ and given a nonnegative smooth function $ζ$ on $\partialΩ$, we study the behavior near 0 of positive solutions of $-Δu=u^q$ in $Ω$ such that $u = ζ$ on $\partialΩ\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient cond…
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the us…
We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$. Our main result establishes an $L^\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in(0,1)$ in dimensions $2 \leq n\leq 6$. It applies to all nonlinearities $f\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of …
The purpose of this paper is to study the existence of solutions for semilinear elliptic system driven by fractional Laplacian and establish some new existence results which are obtained by virtue of the local linking theorem and the saddle point theorem. To make the nonlinear scheme feasible, rigorous analysis of the function space involved and corresponding energy functional is necessary.
In this note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped domains.
We study the system $-Δ\mathbf{u}=| \mathbf{u}|^{α-1} \mathbf{u}$ with $1<α\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one in the scalar case, and also a new class of singular solutions provided that the new invariant is nontrivial. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well a…
In this article, we study the boudary blow-up solutions for semilinear fractional equations with power absorption. Our main purpose is to obtain the existence, nonuniqueness and behavior asymptotic near the boundary.
The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniquen…
Interior regularity of mixed local-nonlocal parabolic semilinear equations
Universidad Complutense de Madrid（马德里康普顿斯大学）；Facultad de Ciencias Económicas y de Administración, Universidad de la República（共和国大学经济与行政学院）；PEDECIBA Matemática（PEDECIBA数学研究所）
We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term is a Carathéodory function which is resonant with respect to any nonprincipal eigenvalue both at $\pm \infty$ and 0. Using a variant of the reduction method, we show that the problem has at least two nontrivial smooth solutions.
We study stable solutions to the equation $(-Δ)^{1/2} u = f(u)$, posed in a bounded domain of $\mathbb{R}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\leq 4$. This result, which was known only for $n=1$, follows from a new interior Hölder estimate that is completely independent of the nonlinearity $f$. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobol…
We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-Δ)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and $2^*_s=\frac{2N}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at an extreme point, we show that this problem has infinitely many non-radial positive or sign-changing solutions.
The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equation…
We obtain some existence theorems for periodic solutions to several linear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.
We show that singularities form after the interaction of three transversal semilinear conormal waves. Our results hold for space dimensions two and higher, and for arbitrary smooth nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang.
In this paper, we consider the blow-up problem of semilinear generalized Tricomi equation. Two blow-up results with lifespan upper bound are obtained under subcritical and critical Strauss type exponent. In the subcritical case, the proof is based on the test function method and the iteration argument. In the critical case, an iteration procedure with the slicing method is employed. This approach has been successfully applied to the critical case of semilinear wave equation with perturbed Laplacian or the damped wave equation of scattering damping case. The present work gives its application t…
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -Δu \le (\frac{1}{|x|^α}* v)^λ$ $0\le -Δv \le (\frac{1}{|x|^β}* u)^σ$ in $B_2(0)\setminus\{0\} \subset R^n$, $n\ge 3$, where $λ,σ\ge 0$ and $α,β\in (0,n)$. A by-product of our methods used to study these solutions will be results on the behavior near the origin of $L^1(B_1(0))$ solutions $f$ and $g$ of the system $0 \le f(x) \le C(|x|^{2-α} + \int_{|y|<1}\frac{ g(y) dy}{|x-y|^{α-2}} )^λ$ $0 \le g(x) \le C(|x|^{2-β} + \int_{|y|<1}\frac{ f(y) dy}{|x-y|^{β-2}} )^σ$ for $0<|x|<1$ where $λ,σ\ge 0…
We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-Δu -\fracμ{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<μ<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the linear problem $-Δh -\fracμ{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $μ$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinea…
We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.
We study the boundary behaviour of the of (E) $-\Gd u-\myfrac{\xk }{d^2(x)}u+g(u)=0$, where $0<\xk <\frac{1}{4}$ and $g$ is a continuous nonndecreasing function in a bounded convex domain of $\BBR^N$. We first construct the Martin kernel associated to the the linear operator $\CL_{\xk }=-\Gd-\frac{\xk }{d^2(x)}$ and give a general condition for solving equation (E) with any Radon measure $\gm$ for boundary data. When $g(u)=|u|^{q-1}u$ we show the existence of a critical exponent $q_c=q_c(N,\xk )>1$: when $0<q<q_c$ any measure is eligible for solving (E) with $\gm$ for boundary data; if $q\geq …
