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Marius Ghergu

Publications and source records attributed to Marius Ghergu.

At least 37 records · Page 2Linked to original sources
arXiv2019

Singular solutions of elliptic equations with iterated exponentials

We construct positive singular solutions for the problem $-Δu=λ\exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Differential Equations \textbf{264} (2018), 2684--2707] for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.

math.AP↗
arXiv2019

Isolated Singularities for Semilinear Elliptic Systems with Power-Law Nonlinearity

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 as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On the technical level, we adopt the method of the moving spheres and the balanced-energy-type monotonicity functionals.

math.AP↗
arXiv2019

Isoperimetric inequalities for Bergman analytic content

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $Ω$ measures the $L^{p}(Ω)$-distance between $\overline{z}$ and the Bergman space $A^{p}(Ω)$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

math.CA↗
arXiv2018

Analytic content and the isoperimetric inequality in higher dimensions

This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in $\mathbb{R}^{N}$ to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory.

math.CA↗
arXiv2018

Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} Δ_{p} u&=v^{m}|\nabla u|^α&&\quad\mbox{ in }Ω,\\ Δ_{p} v&=v^β|\nabla u|^{q} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $Ω\subset\R^N$ $(N\geq 2)$ is either a ball or the whole space, $1 0$, $α\geq 0$, $0\leq β\leq m$ and $(p-1-α)(p-1-β)-qm\neq 0$. We first classify all the positive radial solutions in case $Ω$ is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if $0\leq α<p-1$ and $mq< (p-1-α)(p-1-β)$. Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.

math.AP↗
arXiv2018

Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

We study the semilinear elliptic equation \begin{equation*} -Δu=u^α|\log u|^β\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < α< \frac{n+2}{n-2}$ and $-\infty<β<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{α-1}} \left(\log \frac{1}{|x|}\right)^{-\fracβ{α-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2}{α-1}\right)^{1-β}\left(n-2-\frac{2}{α-1}\right)\right]^{\frac{1}{α-1}}. \end{equation*}

math.AP↗
arXiv2017

On a class of mixed Choquard-Schrödinger-Poisson system

We study the system $$ \left\{ -Δu+u+K(x) ϕ|u|^{q-2}u&=(I_α*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -Δϕ&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $α\in (0,N)$, $p,q>1$ and $K\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,α$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.

math.AP↗
arXiv2015

Pointwise Bounds and Blow-up for Choquard-Pekar Inequalities at an Isolated Singularity

We study the behavior near the origin in $\mathbb{R}^n ,n\geq3$, of nonnegative functions \begin{equation}\label{0.1} u\in C^2 (\mathbb{R}^n \backslash \{0\})\cap L^λ(\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-Δu\leq(|x|^{-α}*u^λ)u^σ\quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $α\in(0,n),λ>0,$ and $σ\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $α,λ$, and $σ$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin.

math.AP↗
arXiv2015

Pointwise Bounds and Blow-up for Systems of Semilinear Parabolic Inequalities and Nonlinear Heat Potential Estimates

We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] in $Ω\times (0,1)$, where $λ$ and $σ$ are nonnegative constants and $Ω$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $λ$ and $σ$ such that solutions of this system satisfy pointwise bounds in compact subsets of $Ω$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.

math.AP↗
arXiv2015

Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities

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 $λ,σ\ge 0$ and $α, β\in (2,n+2)$.

math.AP↗
arXiv2014

Nonexistence of Positive Supersolutions of Nonlinear Biharmonic Equations without the Maximum Principle

We study classical positive solutions of the biharmonic inequality $-Δ^2 v \geq f(v)$ in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$ and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in any exterior domain of $\mathbb R^n$. Similar results were obtained by Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011-2047] for $-Δv\ge f(v)$ using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of $-Δ^2u \ge 0$ in a punctured neighborhood of the origin in $\mathbb{R}^n$.

math.AP↗
arXiv2014

Pointwise Bounds and Blow-up for Systems of Semilinear Elliptic Inequalities at an Isolated Singularity via Nonlinear Potential Estimates

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -Δu\leq f(v)$ $0\leq -Δv\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if $\log^+ f$ or $\log^+g$ grow at most linearly at infinity. In dimension $n\geq 3$ and under the assumption $f(t)=O(t^λ)$, $g(t)=O(t^σ)$ as $t\to \infty$, ($λ, σ\geq 0$), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context.

math.AP↗
arXiv2013

Semilinear elliptic inequalities in the exterior of a compact set

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 $K$.

math.AP↗
arXiv2013

A new critical curve for the Lane-Emden system

We study stable positive radially symmetric solutions for the Lane-Emden system $-Δu=v^p$ in $\R^N$, $-Δv=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.

math.AP↗
arXiv2012

Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls

We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence region from non-existence. In the critical case $p=p^*$ we show that the existence of a singular solution depends on the rate at which the coefficients stabilize at zero and we provide some optimal conditions in this setting.

math.AP↗
arXiv2012

Entire large solutions for semilinear elliptic equations

We analyze the semilinear elliptic equation $Δu=ρ(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $ρ$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.

math.AP↗
arXiv2010

Isolated Singularities of Polyharmonic Inequalities

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-Δ^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.

math.AP↗