We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the solution may diverge.
The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system.
This paper presents some sufficient conditions for the validity of the comparison principle for the weak solutions of non - cooperative weakly coupled systems of elliptic second-order PDEs.
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
In this article we study the symmetry breaking phenomenon of solutions of noncooperative elliptic systems. We apply the degree for G-invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon.
In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in $\RN$, $N \geq 2 $. More precisely we prove that solutions having Morse index $j \leq N $ are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution.
In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in R^N, with N greather or equal than 2. More precisely we prove that solutions having Morse index j smaller than N are foliated Schwarz symmetric if the nonlinearity has a convex derivative and a full coupling condition is satisfied along the solution.
In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in R^N, or in the exterior of a ball. We consider the case of fully coupled systems and nonlinearities which are either convex or have a convex derivative. The solutions are shown to be foliated Schwarz symmetric if a bound on their Morse index holds. As a consequence of the symmetry results we also obtain some nonexistence theorems.
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.
We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.
We study here the behavior of the solutions to a $2\times 2$ semi-linear cooperative system involving Schr\" odinger operators (considered in its variational form): $$LU:=(-Δ+ q(x))U = AU+μU + F(x,U) \quad{\rm in}\ \mathbb{R}^N$$ $$U(x)_{|x|\rightarrow \infty} \rightarrow 0$$ where $q$ is a continuous positive potential tending to $+\infty$ at infinity; $μ$ is a real parameter varying near the principal eigenvalue of the system; $U$ is a column vector with components $u_1$ and $u_2$ and $A$ is a square cooperative matrix with constant coefficient. $F$ is a column vector with components $f_1$ a…
The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball is a hemisphere, then these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n)-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool we use the degree theory fo…
Through Morrey's spaces (plus Zorko's spaces) and their potentials/capacities as well as Hausdorff contents/dimensions, this paper estimates the singular sets of nonlinear elliptic systems of the even-ordered Meyers-Elcrat type and a class of quadratic functionals inducing harmonic maps.
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.
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -Δu+\fracμ{|x|^2} u=u^p\quad {\rm in}\quad Ω\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partialΩ. \end{equation} We classify the isolated singularities and obtain the existence, the stability of positive solutions of (\ref{0}). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
In this paper we deal with the cubic Schrödinger system $ -Δu_i = \sum_{j=1}^n β_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $β=(β_{i,j})_{ij}$ is a symmetric matrix with real coefficients and $β_{ii}\geq 0$ for every $i=1,\ldots,n$. We analyse the existence and nonexistence of nontrivial solutions in connection with the properties of the matrix $β$, and provide a complete characterization in dimensions $N=1,2$. Extensions to more general power-type nonlinearities are given.
In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=f(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ (-Δ)^αv=g(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ u=v=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $(-Δ)^α$ denotes the fractional Laplacian and $ Ω$ is a smooth bounded domain in $ \R^N $. It shown that under some assumptions on $ f $ and $ g $, the problem has at least one positive solution $(u,v)$. Our proof is based on th…
In two dimensions every weak solution to a nonlinear elliptic system $\rm{div} a(x,u,Du)=0$ has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p \geq 2$. We give an example showing that this result cannot be extended to the subquadratic case, i.e. that weak solutions are not necessarily continuous if $1< p <2$. Furthermore, we discuss related results for variational integrals.
In this paper we introduce some fully nonlinear second order operators defined as weighted partial sums of the eigenvalues of the Hessian matrix, arising in geometrical contexts, with the aim to extend maximum principles and removable singularities results to cases of highly degenerate ellipticity.nn
In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-Δ)^σu=u^{\frac{n+2σ}{n-2σ}}$ with an isolated singularity, where $σ\in (0,1)$. We prove that all the solutions are asymptotically radially symmetric. When $σ=1$, these have been proved in \cite{CGS} by Caffarelli, Gidas and Spruck.
Inspired by so many possible applications of this class of problems, we seek solution for non-cooperative elliptic systems of two Schrodinger equations. General conditions are assumed under the potentials, which produces convenient spectral properties on the elliptic operator concerned, hence the non-cooperation characterizes it as a strongly indefinite problem. An Abstract Linking Theorem developed previously by the first two authors is the main tool, since variational approach is applied. Furthermore, super and asymptotically quadratic nonlinearities are considered.
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-Δ)^σu = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).
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.
We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$, $κ_1,κ_2\in\mathbb{R}$, $μ_1,μ_2,λ>0$, $α,β>1$, and $α+ β= p\le 2^*:=\frac{2N}{N-2}$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $λ$ sufficiently large. If $p=2^*$ and $κ_1,κ_2>0$ we establish the ex…
Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is Sobolev subritical, $n\le3$ and $β\in{\mathbb R}$. We first prove a Liouville theorem for the system $$\left.\begin{aligned} -Δu &=|u|^{2q+2}u+β|v|^{q+2}|u|^q u, \\ -Δv &=|v|^{2q+2}v+β|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$ such that the number of nodal domains of $u,v,u-v,u+v$ i…
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$Δu = -χ_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss.
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.
We study the boundary value problem with measures for (E1) $-\Gd u+g(|\nabla u|)=0$ in a bounded domain $\Gw$ in $\BBR^N$, satisfying (E2) $ u=\gm$ on $\prt\Gw$ and prove that if $g\in L^1(1,\infty;t^{-(2N+1)/N}dt)$ is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When $g(r)\geq r^q$ with $q>1$ we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1
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.
The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds. Such a theory is of importance for the study of elliptic and parabolic equations on non-compact, or even incomplete manifolds, with or without boundary.
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 removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this singularity for equations in the variable exponent Sobolev space.
We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and existence or non existence and uniqueness results for solutions of some nonlinear elliptic equations on the half unit sphere.
In the present work we study elliptic operators on manifolds with singularities in the situation, when the manifold is endowed with an action of a discrete group $G$. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair, consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). Fredholm property of elliptic elements is obtained.
We consider general linear non-degenerate weakly-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in $\mathbb{R}^d$ with respect to the potential. It is shown that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. The strict monotonicity property of the principal eigenvalue is shown to be equivalent with the exponential stability of the twisted operators. An equivalence between th…
We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces.
We study operators on a singular manifold, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. The idea is to construct so-called asymptotic parametrices with flat left -over terms. Our structures are motivated by models of particle physics with singular potentials that contribute embedded singularities in $\R^N$ of higher order, according to the number of particles.
We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in $\mathbb{R}^3$ that satisfies an arbitrary elliptic Weingarten equation $W(κ_1,κ_2)=0$, and study the singularities of such examples. As global applications of this classification, we prove a sharp halfspace theorem for general elliptic Weingarten equations of finite order, and a classification of peaked elliptic Weingarten spheres with at most two singularities. In the case that $W$ is not elliptic, we give a negative answer to a question by Yau regarding the uniqueness of rota…
In this paper we give a new, less restrictive condition for removability of singular sets, $E$, of smooth solutions to the m-Hessian equation (and also for more general fully nonlinear elliptic equations) in $Ω\setminus E$, $Ω\subset \mathbb R^n$. Besides the existence and regularity results for these equations, the proof only makes use of the classical elliptic theory, i.e. the classical maximum principles and a Hopf lemma.
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…
The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(…
In this paper we study a parametrised non-cooperative symmetric semi-linear elliptic system on a sphere. Assuming that there exist critical orbits of the potential, we study the structure of the sets of solutions of the system. In particular, using the equivariant Rabinowitz Alternative we formulate sufficient conditions for a bifurcation of unbounded sets of solutions.
In this paper, we present a new distributional identity for the solutions of elliptic equations involving Hardy potentials with singularities located on the boundary of the domain. Then we use it to obtain the boundary isolated singular solutions of nonhomogeneous problems.
The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic…
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén-Lindelöf result as …
It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula …
We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $Ω$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of $\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0<μ_{s,¶}(Ω)=\inf{\int_Ω|\nabla u|^2 dx; u\in \huno \hbox{and}\int_Ω\frac{|u(x)|^{\crit(s)}}{|π(x)|^s} dx=1}$$ where $0<s<2$, $\crit(s)=\frac{2(n-s)}{n-2}$ and where $π$ is the orthogonal projection on a linear…
Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation ${\rm det} (D^2 z)=φ(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where $\mathcal{M}_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Ampère equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities o…
In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^αu=u^p\quad &{\rm in}\quad Ω\setminus\{0\},\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminusΩ, \end{array} \end{equation} where $p>1$, $Ω$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing the origin, $N\ge2$ and the fractional Laplacian $(-Δ)^α$ is defined in the principle value sense. We obtain that any classical solution $u$ of (\ref{eq 0.1}) is a weak solution of …
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.
