Fundamental solutions for Schrodinger operators with general inverse square potentials
In this paper, we classify the fundamental solutions for a class of Schrodinger operators.
math.AP↗
arXiv subjects
Huyuan Chen
Publications and source records attributed to Huyuan Chen.
Isolated singularities of positive solutions for Choquard equations in sublinear case
Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -Δu+ u =I_α[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, % [2mm] \phantom{ } \;\; \displaystyle \lim_{|x|\to+\infty}u(x)=0, $$ where $p >0, N \geq 3, α\in (0,N)$ and $I_α[u^p](x) = \int_{\mathbb{R}^N} \frac{u^p(y)}{|x-y|^{N-α}}dy$ is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent $(p,q)$. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.
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The Dirichlet elliptic problem involving regional fractional Laplacian
In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^α_Ωu=f \qquad & {\rm in}\quad Ω,\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=g\quad & {\rm on}\quad \partial Ω, \end{array} \end{equation} where $Ω$ is a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partialΩ$, $α\in(\frac12,1)$ and the operator $(-Δ)^α_Ω$ denotes the regional fractional Laplacian. We prove that when $g\equiv0$, problem (\ref{0}) admits a unique weak solution in the cases that $f\in L^2(Ω)$, $f\in L^1(Ω, ρ^βdx)$ and $f\in \mathcal{M}(Ω,ρ^β)$, here $ρ(x)={\rm dist}(x,\partialΩ)$, $β=2α-1$ and $\mathcal{M}(Ω,ρ^β)$ is a space of all Radon measures $ν$ satisfying $\int_Ωρ^βd|ν|<+\infty.$ Finally, we provide an Integral by Parts Formula for the classical solution of (\ref{0}) with general boundary data $g$.
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Liouville theorem for the fractional Lane-Emden Equation in unbounded domain
Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-Δ)^αu = u^p+ν\quad {\rm in}\quad Ω,\qquad\ u=g\quad {\rm in}\quad \mathbb{ R}^N\setminus Ω, \end{equation} where $α\in(0,1)$, $p>0$, $Ω$ is a unbounded $C^2$ domain in $\mathbb{ R}^N$ with $N>2α$, $g\in L^1(\mathbb{ R}^N\setminus Ω,\frac{dx}{1+|x|^{N+2α}})$ nonnegative and $ν$ is a nonnegative Radon measure. We obtain that\smallskip $(i)$ if $Ω\supseteq \left(\mathbb{R}^N\setminus \overline{B_{r_0}(0)}\right)$ for some $r_0>0$ and $p<\frac{N}{N-2α}$, then fractional Lane-Emden equation has no weak solutions. $(ii)$ if $Ω\supseteq \left\{x\in \mathbb{ R}^N:\, x\cdot a>r_0\right\}$ for some $r_0\ge 0$, $a\in \mathbb{ R}^N$ and $p<\frac{N+α}{N-α}$, then fractional Lane-Emden equation has no weak solutions. Here $\frac{N+α}{N-α}$ is sharp for the nonexistence in the half space. \smallskip The above Liouville theorem could be applied to obtain nonexistence of classical solution of the fractional Lane-Emden equations $$ (-Δ)^αu = u^p\quad {\rm in}\quad Ω, \qquad u\ge 0\quad {\rm in}\quad \mathbb{ R}^N\setminus Ω, $$ where $Ω=\mathbb{R}^N\setminus B_{r_0}(0)$ with $r_0>0$ or $Ω=\mathbb{R}^{N-1}\times(0,+\infty)$.
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Classification of isolated singularities of positive solutions for Choquard equations
In this paper we classify the isolated singularities of positive solutions to Choquard equation and prove the existence of isolated singular solutions.
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On semi-linear elliptic equation arising from Micro-Electromechanical Systems with contacting elastic membrane
This paper is concerned with the nonlinear elliptic problem $-Δu=\frac{λ}{(a-u)^2}$ on a bounded domain $Ω$ of $\mathbb{R}^N$ with Dirichlet boundary conditions. This problem arises from Micro-Electromechanical Systems devices in the case that the elastic membrane contacts the ground plate on the boundary. We analyze the properties of minimal solutions to this equation when $λ>0$ and the function $a:\barΩ\to[0,1]$ satisfying $a(x)\ge κ{\rm dist}(x,\partialΩ)^γ$ for some $κ>0$ and $γ\in(0,1)$. Our results show how the boundary decay of the membrane works on the solutions and pull-in voltage $λ$.
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Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results
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 \begin{equation}\label{eq 0.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^αu=u^p+kδ_0\quad &{\rm in}\quad Ω,\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminusΩ\end{array} \end{equation} for some $k\ge0$, where $δ_0$ is the Dirac mass at the origin. In particular, when $p\ge \frac{N}{N-2α}$, we have that $k=0$; when $p< \frac{N}{N-2α}$, $u$ has removable singularity at the origin if $k=0$ and if $k>0$, $u$ satisfies $$\lim_{x\to0} u(x)|x|^{N-2α}=c_{N,α}k,$$ where $c_{N,α}>0$. Furthermore, when $p\in(1, \frac{N}{N-2α})$, we obtain that there exists $k^*>0$ such that problem (\ref{eq 0.1}) has at least two positive solutions for $k k^*$.
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Weak solutions of semilinear elliptic equation involving Dirac mass
In this paper, we study the following elliptic problem with Dirac mass \begin{equation}\label{eq 0.1} -Δu=Vu^p+k δ_0\quad {\rm in}\quad \mathbb{R}^N, \qquad \lim_{|x|\to+\infty}u(x)=0, \end{equation} where $N>2$, $p>0$, $k>0$, $δ_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\mathbb{R}^N\setminus\{0\}$ satisfying $$ V(x)\le \frac{c_1}{|x|^{a_0}(1+|x|^{a_\infty-a_0})} $$ with $a_0 a_0 $ and $c_1>0$. We obtain two positive solutions of (\ref{eq 0.1}) with additional conditions for parameters on $a_\infty, a_0$, $p$ and $k$. The first solution is a minimal positive solution and the second solution is constructed by Mountain Pass theorem.
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Fractional heat equations with subcritical absorption having a measure as initial data
We study existence and uniqueness of weak solutions to (F) $\partial\_t u+ (-Δ)^\alphau+h(t, u)=0 $ in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=ν$ in $\R^N$, where $N\ge2$, the operator $(-Δ)^α$is the fractional Laplacian with $α\in(0,1)$, $ν$ isa bounded Radon measure and $h:(0,\infty)\times\R\to\R$ is a continuous function satisfying a subcritical integrability condition.In particular, if $h(t,u)=t^βu^p$ with $β\textgreater{}-1$ and $0 \textless{} p \textless{} p^*\_β:=1+\frac{2α(1+β)}{N}$, we prove that there exists a unique weak solution $u\_k$ to (F) with $ν=kδ\_0$, where $δ\_0$ is the Dirac mass at the origin. We obtain that $u\_k\to\infty$ in $(0,\infty)\times\R^N$ as $k\to\infty$ for $p\in(0,1]$ and the limit of $u\_k$ exists as $k\to\infty$ when $1 \textless{} p \textless{} p^*\_β$, we denote it by $u\_\infty$.When $1+\frac{2α(1+β)}{N+2α}:=p^{**}\_β\textless{} p \textless{} p^*\_β$,$u\_\infty$ is the minimal self-similar solution of $(F)\_\infty$ $\partial\_t u+ (-Δ)^αu+t^βu^p=0 $ in $(0,\infty)\times\R^N$ with the initial condition $u(0,\cdot)=0$ in $\R^N\setminus\{0\}$ and it satisfies $u\_\infty(0,x)=0$ for $x\neq 0$.While if $1\textless{} p \textless{} p^{**}\_β$, then $u\_\infty\equiv U\_p$, where $U\_p$ is the maximal solution of the differential equation $y'+t^βy^p=0$ on $\R\_+$.
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On a class of semilinear fractional elliptic equations involving outside Dirac data
The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-Δ)^α u+u^p=0\ \ \ \ &\ {\rm in}\ \ B_1(e_N),\\[2mm]\phantom{(-Δ)^α +u^p} u=δ_{0}& \ {\rm in}\ \ \mathbb{R}^N\setminus B_1(e_N), \end{array} \end{equation} where $p>0$, $ (-Δ)^α$ with $α\in(0,1)$ denotes the fractional Laplacian operator in the principle value sense, $B_1(e_N)$ is the unit ball centered at $e_N=(0,\cdots,0,1)$ in $\mathbb{R}^N$ with $N\ge 2$ and $δ_0$ is the Dirac mass concentrated at the origin. We prove that problem (\ref{00}) admits a unique weak solution when $p> 1+\frac{2α}{N}$. Moreover, if in addition $p\ge \frac{N+2}{N-2}$, the weak solution vanishes as $α\to 1^-$. We also show that problem (\ref{00}) doesn't have any weak solution when $p\in[0, 1+\frac{2α}{N}]$. These results are very surprising since there are in total contradiction with the classical setting, i.e. $$ \arraycolsep=1pt \begin{array}{lll} -Δu+ u^p=0\ \ \ \ &\ {\rm in}\ \B_1(e_N),\\[2mm] \phantom{-Δ+u^{p} } u=δ_{0}& \ {\rm in}\ \ \R^N\setminus B_1(e_N), \end{array} $$ for which it has been proved that there are no solutions for $p\ge \frac{N+1}{N-1}$.
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Boundary blow-up solutions to fractional elliptic equations in a measure framework
Let $α\in(0,1)$, $Ω$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partialΩ$ and $ω$ be the Hausdorff measure on $\partialΩ$. We denote by $\frac{\partial^αω}{\partial \vec{n}^α}$ a measure $$\langle\frac{\partial^αω}{\partial \vec{n}^α},f\rangle=\int_{\partialΩ}\frac{\partial^αf(x)}{\partial \vec{n}_x^α} dω(x),\quad f\in C^1(\barΩ),$$ where $\vec{n}_x$ is the unit outward normal vector at point $x\in\partialΩ$. In this paper, we prove that problem $$ \begin{array}{lll} (-Δ)^αu+g(u)=k\frac{\partial^αω}{\partial \vec{n}^α}\quad & {\rm in}\quad \barΩ,\\[2mm] \phantom{(-Δ)^α+g(u)} u=0\quad & {\rm in}\quad Ω^c \end{array} $$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution is a classical solution of $$ \begin{array}{lll} \ \ \ (-Δ)^αu+g(u)=0\quad & {\rm in}\quad Ω,\\[2mm] \phantom{------\} \ u=0\quad & {\rm in}\quad R^N\setminus\barΩ,\\[2mm] \phantom{} \lim_{x\inΩ,x\to\partialΩ}u(x)=+\infty. \end{array} $$
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Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary
The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-Δ)^αu+εg(u)=k\frac{\partial^αν}{\partial \vec{n}^α}\quad &{\rm in}\quad\ \ \barΩ,\\[3mm] \phantom{(-Δ)^α+εg(u)} u=0\quad &{\rm in}\quad\ \ \barΩ^c, \end{array} \end{equation} where $k>0$, $ε=1$ or $-1$, $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $Ω$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $ν$ is a bounded Radon measure supported in $\partialΩ$ and $\frac{\partial^αν}{\partial \vec{n}^α}$ is defined in the distribution sense, i.e. $$ \langle\frac{\partial^αν}{\partial \vec{n}^α},ζ\rangle=\int_{\partialΩ}\frac{\partial^αζ(x)}{\partial \vec{n}_x^α}dν(x), \qquad \forallζ\in C^α(R^N), $$ here $\vec{n}_x$ denotes the unit inward normal vector at $x\in\partialΩ$.
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Elliptic equations involving general subcritical source nonlinearity and measures
In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-Δ)^αu=g(u)+σν\quad{\rm in}\quad Ω,$$ subject to the condition (E2) $$u=\varrhoμ\quad {\rm on}\quad \partialΩ \ {\rm if}\ α=1\qquad {\rm or\ \ in}\ \ Ω^c \ \ {\rm if}\ α\in(0,1),$$ where $σ,\varrho\ge0$, $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ or Laplacian operator if $α=1$, $ν,μ$ are suitable Radon measures and $g:\mathbb{R}_+\mapsto\mathbb{R}_+$ is a continuous function. We introduce an approach to obtain weak solutions for problem (E1)-(E2) when $g$ is integral subcritical and $σ,\varrho\ge0$ small enough.
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Sharp embedding of Sobolev spaces involving general kernels and its application
The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in these embeddings. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.
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Semilinear fractional elliptic equations with measures in unbounded domain
In this paper, we study the existence of nonnegative weak solutions to (E) $ (-Δ)^αu+h(u)=ν$ in a general regular domain $Ω$, which vanish in $\R^N\setminusΩ$, where $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$, $ν$ is a nonnegative Radon measure and $h:\mathbb{R}_+\to\mathbb{R}_+$ is a continuous nondecreasing function satisfying a subcritical integrability condition. Furthermore, we analyze properties of weak solution $u_k$ to $(E)$ with $Ω=\mathbb{R}^N$, $ν=kδ_0$ and $h(s)=s^p$, where $k>0$, $p\in(0,\frac{N}{N-2α})$ and $δ_0$ denotes Dirac mass at the origin. Finally, we show for $p\in(0,1+\frac{2α}{N}]$ that $u_k\to\infty$ in $\mathbb{R}^N$ as $k\to\infty$, and for $p\in(1+\frac{2α}{N},\frac{N}{N-2α})$ that $\lim_{k\to\infty}u_k(x)=c|x|^{-\frac{2α}{p-1}}$ with $c>0$, which is a classical solution of $ (-Δ)^αu+u^p=0$ in $\mathbb{R}^N\setminus\{0\}$.
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Weak and strong singular solutions of semilinear fractional elliptic equations
Let $p\in(0,\frac{N}{N-2α})$, $α\in(0,1)$ and $Ω\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $δ_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $ (-Δ)^αu+u^p=kδ_0 $ in $Ω$ which vanishes in $Ω^c$, is a classical solution of $(E_*)$ $ (-Δ)^αu+u^p=0 $ in $Ω\setminus\{0\}$ with the same outer data. When $\frac{2α}{N-2α}\leq 1+\frac{2α}{N}$, $p\in(0, 1+\frac{2α}{N}]$ we show that the $u_k$ converges to $\infty$ in whole $Ω$ when $k\to\infty$, while, for $p\in(1+\frac{2α}N,\frac{N}{N-2α})$, the limit of the $u_k$ is a strongly singular solution of $(E_*)$. The same result holds in the case $1+\frac{2α}{N}<\frac{2α}{N-2α}$ excepted if $\frac{2α}{N}
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Semilinear fractional elliptic equations with gradient nonlinearity involving measures
We study the existence of solutions to the fractional elliptic equation (E1) $(-Δ)^αu+εg(|\nabla u|)=ν$ in a bounded regular domain $Ω$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $Ω^c$, where $ε=1$ or $-1$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(1/2,1)$, $ν$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $ν$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $ε=1$.
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Self-generated interior blow-up solutions in fractional elliptic equation with absorption
In this paper we study positive solutions to problem involving the fractional Laplacian $(E)$ $(-Δ)^α u(x)+|u|^{p-1}u(x)=0 in x\inΩ\setminus\mathcal{C}$, subject to the conditions $u(x)=0$ $x\inΩ^c$ and $\lim_{x\inΩ\setminus\mathcal{C}, x\to\mathcal{C}}u(x)=+\infty$, where $p>1$ and $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $\mathcal{C}\subset Ω$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem $(E)$. Moreover, we further analyze uniqueness, asymptotic behaviour and nonexistence.
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