In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems:
\begin{equation} \label{eq_abst_1}-Δu = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation}
\begin{equation} \label{eq_abst_2} -Δu = | \nabla u|^p \qquad \mbox{ in } Ω, \qquad u = 0 \mbox{ on } \;\; \partial Ω. \end{equation}
In the first problem $B_1$ is the unit ball in $ \mathbb{R}^N$ and in the second $Ω$ is a bounded smooth domain in $ \mathbb{R}^N$. In both cases we assume $ N \ge 3$, $ \frac{N}{N-1}<p<2$ and in the first problem we assume $ g \ge 0$ is a Hölder continuous function with $g(0)=0$. We obtain positive singular solutions in both cases. \\ For the second equation we also consider the case of $Ω$ an exterior domain $ \mathbb{R}^N$ where $N \ge 3$ and $ p >\frac{N}{N-1}$. We prove the existence of a bounded positive classical solution with the additional property that $ \nabla u(x) \cdot x>0$ for large $|x|$.
In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(λ(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior at infinity.
We consider the supercritical elliptic problem -Δu = λe^u, λ> 0, in an exterior domain $Ω= \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal than 3. We prove that, for λsmall, this problem admits infinitely many regular solutions.
Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.
In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combin…
This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.
In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found…
A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in $\mathbb{R}^n\backslash K$, where $K$ is a compact set, provided the structure of this equation and the dimension $n$ are related. This result is a correction of a previous one established by Serrin, since some additional hypotheses are necessary. Theoretical and numerical examples are given. Furthermore, a comparison result and the uniqueness of solution are obtained for such equations in exterior domains.
We study the semilinear elliptic inequality $-Δu\geqφ(δ_K(x))f(u)$ in $R^N\setminus K,$ where $φ, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $δ_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $φ$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.
We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(λ(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniquen…
This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-Δ)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $Ω$},\\ u \geq 0 \quad \mbox{in} \quad Ω\quad \mbox{and} \quad u \not\equiv 0, \\ u=0 \quad \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation*} involving the fractional Laplacian operator $(-Δ)^{s}$, where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is an exterior domain with (non-empty) smooth boundary $\partial Ω$ and $p\in (2, 2_{s}^{*})$. The main technical approach is based on variational a…
We study the validity of an extension of Frobenius theorem on integral manifolds for some classes of Pfaff-type systems of partial differential equations involving multidimensional "rough" signals, i.e. "differentials" of given Hölder continuous functions interpreted in a suitable way, similarly to Young Differential Equations in Rough Paths theory. This can be seen as a tool to study solvability of exterior differential systems involving rough differential forms, i.e. the forms involving weak (distributional) derivatives of highly irregular (e.g. Hölder continuous) functions; the solutions (i…
We study existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a non-standard cone to establish existence of solutions by means of fixed point index theory.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $Ω$, $u=g$ in $\mathbb R^N\setminusΩ$, in non-smooth domains $Ω$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right hand side, for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,α}$, and establish the optimal Hölder regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $Ω$.
In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: \begin{equation}\label{GPDE0} (-Δ)^{\fracα{2}}u(x)=f(x,u) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\,\, Ω_{r}:=\{x\in\mathbb{R}^{n}\,|\,|x|>r\} \end{equation} with arbitrary $r>0$, where $n\geq2$, $0<α\leq 2$ and $f(x,u)$ satisfies some assumptions. A typical case is the Hardy-Hénon type equations in exterior domains. We first derive the equivalence between \eqref{GPDE0} and the corresponding integral equations \begin{equation}\label{GIE0} u(x)=\int_{Ω_{r}}G_α(x,y)f(y,u(y)…
We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^θ\nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+θ>2$. In particular, for $N' > 2$ and $τ:=\ell-θ>-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2τ}{N'-2}$, while for any $ 0<p \leq p_s$, the only no…
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is a bounded set with smooth boundary, $(-Δ)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus Ω…
We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
In this work, first we prove that for any compact set $K\subset\mathbb{R}^{n}$ and any continuous function $ϕ$ defined on $\partial K$, there exists a bounded weak solution in $C(\bar{\mathbb{R}^{n}\backslash K}) \cap C^1(\mathbb{R}^{n}\backslash K)$ to the exterior Dirichlet problem $$ \begin{cases} -{\rm div}\big(\,|\nabla u|^{p-2}A(\,|\nabla u|\,)\nabla u\,\big)=f & \text{ in }\, \mathbb{R}^n \backslash K \;\;\;\;\;\; u = ϕ& \text{ on } \partial K \end{cases} $$ provided $p > n$, $A$ satisfies some growth conditions and $f\in L^{\infty}(\mathbb{R}^n)$ meets a suitable pointwise decay rate. …
We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - Δ+ (-Δ)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been prese…
We prove new results on the existence, non-existence, localization and multiplicity of nontrivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to nonlocal, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do {Ó}, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.
We establish the Hölder estimate and the asymptotic behavior at infinity for $K$-quasiconformal mappings over exterior domains in $\mathbb{R}^2$. As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in $\mathbb{R}^2$.
The paper concerns with positive solutions of problems of the type $-Δu+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $Ω\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $Ω$ can be an exterior domain, i.e. $\mathbb{R}^N\setminusΩ$ bounded, or the whole of $\mathbb{R}^N$. The potential $a\in L^{N/2}_{\rm loc}(\mathbb{R}^N)$ is assumed to be strictly positive and such that there exists $\lim_{|x|\to\infty}a(x):=a_\infty$, with $a_\infty>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge …
The Unified Transform provides a novel method for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The numerical implementation of this method to linear elliptic PDEs formulated in the {\it interior} of a polygon has been investigated by several authors (see the article by Iserles, Smitheman, and one of the authors in this book). Here, we show that the Unified Transform also yields a novel numerical technique for computing the solution of linear elliptic PDEs in the {\it exterior} of a polygon. One of the advantages of this new technique is that it actually yield…
We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.
We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous Dirichlet boundary conditions.
We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type $\cA=\sum_{0\le |α|,|β|\le m}(-1)^αD^αa_{α, β}(x)D^β$, $a_{α, β}(\cdot)\in C^{\infty}({\overlineΩ})$, on smooth (bounded or unbounded) domains in $\bbR^n$ with compact boundary. Using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of these closed realizations of $\cA$ and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization…
In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric $α$-stable process and probabilistic representation of the super-solutions.
The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the Cauchy problem for parabolic equations are established. In application, the well-posedness of Wentzell-Robin type mixed probem for parabolic equation, Cauchy problem for anisotropic parabolic equations and system of parabolic equations are derived
For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it i…
We obtain a quantitative high order expansion at infinity of solutions for a family of fully nonlinear elliptic equations on exterior domain, refine the study of the asymptotic behavior of the Monge-Ampère equation, the special Lagrangian equation and other elliptic equations, and give the precise gap between exterior maximal (or minimal) gradient graph and the entire case.
Conforming and Non-Conforming Functional A Posteriori Error Estimates for Elliptic Boundary Value Problems in Exterior Domains: Theory and Numerical Tests
We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-Δ)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient ca…
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
We prove the existence of ground state solution to the following problem. \begin{align*} (-Δ)^{s}u+u&=λ|u|^{-γ-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminusΩ\\ N_su(x)&=0,~\text{in}~Ω\end{align*} where $N\geq2$, $λ>0$, $0<s,γ<1$, $p\in(1,2_s^*-1)$ with $2_s^*=\frac{2N}{N-2s}$. % $0<s^-=\underset{(x,y)\inΩ\timesΩ}{\inf}\{s(x,y)\}\leq s(x,y)\leq s^+=\underset{(x,y)\inΩ\timesΩ}{\sup}\{s(x,y)\}<1$, $0<γ^-=\underset{x\inΩ}{\inf}\{γ(x)\}\leq γ(x)\leq γ^+=\underset{x\inΩ}{\sup}\{γ(x)\}<1$, $1-γ^-<1<p^-=\underset{x\inΩ}{\inf}\{p(x)\}\leq p(x)\leq p^+=\underset{x\inΩ}{\sup}\{p(x)\}<2_{s^-}^*=\und…
We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
We examine the Hénon equation $ -Δu =|x|^αu^p$ in $ Ω\subset \mathbb{R}^N$ with $u=0$ on $ \partial Ω$ where $ 0 < α$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_α(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_α(N)$ such that for any $ \frac{N+2}{N-2} \le p < p_α(N)$, which avoids $ \{p_k\}_k $, there exists a positive classical solution of the Hénon equation, provided $ Ω$ is a sufficiently small perturbation of the unit ball. We also examine the Lane-Emden-Fowler equation in the case of an exterior domain; ie. $ -Δu = u^p$ in $ Ω$, an exterior domain, with …
We establish the existence of multiple solutions for a nonvariational elliptic systems involving $p(x)$-Laplacian operator. The approach combines the methods of sub-supersolution and Leray--Schauder topological degree.
In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
In this paper we discuss the existence and regularity of solutions of strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain $Ω$ in $\R^n$.
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
