arXiv subjects

Marius Ghergu

Publications and source records attributed to Marius Ghergu.

At least 19 recordsLinked to original sources

arXiv2026

Double phase inequalities with convolution nonlinearity in exterior domains

We discuss the existence of $C^1$-solutions for two related double phase inequalities: \begin{equation*} {\mathcal L}_g u\pm \Delta_s u\geq (|x|^{-\alpha}*u^p)u^q \quad\mbox{ in }\mathbb R^N\setminus \overline B_1, N\geq 1,\tag{$P^\pm$} \end{equation*} in which $\Delta_s u:={\rm div}\big(|\nabla u|^{s-2}\nabla u\big)$ is the $s$-Laplace operator, $s>1$, and $$ {\mathcal L}_g u:= -{\rm div}\Big(|\nabla u|^{m-2}g(|\nabla u|)\nabla u\Big),\quad m>s>1, $$ where $g:[0, \infty)\to (0, \infty)$ is a $C^1(0, \infty)\cap C[0, \infty)$ non-increasing function with some specific behaviour near the origin. In the above context, the general form of ${\mathcal L}_g u$ includes the case of $m$-Laplace and $m$-mean curvature operator. Our study reveals a sharp distinction between $(P^+)$ and $(P^-)$. Precisely, we show that the inequality $(P^+)$ has solutions for all $m>s>1$ and $q>s-1$. In contrast, $(P^-)$ has solutions if and only if $p$ and $q$ are sufficiently large. We also link the solvability of $(P^-)$ with that of the corresponding equation ${\mathcal L}_g u- \Delta_s u= (|x|^{-\alpha}*u^p)u^q$ in $\mathbb R^N\setminus \overline B_1$, for which we derive optimal conditions in terms of $p, q, \alpha, s$ and $N$. The approach combines integral estimates with a new sub and supersolution method that accounts for the presence of the convolution term.

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arXiv2025

Isolated singularities for elliptic equations with convolution terms in a punctured ball

The purpose of this article is two-fold. First, we investigate the inequality $$ -Δu+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we provide optimal conditions for which any solution $0\leq u\in \mathcal{C}^2(B_1\setminus\{0\})$ of the above inequality satisfies $u, Δu, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -Δu + λV(x) u = (K_{α, β} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, ν}( \overline B_1\setminus\{0\})$, $0<ν<1$, $λ, p, q>0$ and $$K_{α, β}(x) = |x|^{-α}\log^β\frac{2e}{|x|}, \quad\text{where } 0 \leq α< N, β\in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $α, β, p, q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x) \simeq |x|^{2-N}$ near the origin. For $N = 2$, we provide a classification of the existence and boundedness of solutions based on the local behavior of the potential $V(x)$ near the origin.

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arXiv2025

The Gierer-Meinhardt system in the entire space with non-local proliferation rates

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: $$ \begin{cases} \displaystyle -Δu+λu=\frac{J*u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 1,\\[0.1in] \displaystyle -Δv+μv=\frac{J*u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N.\\[0.1in] \end{cases} $$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modelling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J*u^p, J*u^m$ where the $*$-symbol denotes the convolution operation in $\mathbb{R}^N$ with a kernel $J\in C^1(\mathbb{R}^N\setminus\{0\})$. In the system, we assume that $0<ρ\in C^{0, γ}(\mathbb{R}^N)$ with $γ\in (0,1)$, while the parameters satisfy $λ, μ, q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel $J$, we establish the existence and non-existence of classical positive solutions in the function space $C^{2, δ}_{loc}(\mathbb{R}^N).$ These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.

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arXiv2025

Superharmonic functions in the upper half space with a nonlocal boundary condition

We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-Δu=μ$ for some Radon measure $μ$, so that $u$ satisfies the nonlocal boundary condition $$ \frac{\partial u}{\partial n}(x',0)=λ\int\limits_{\mathbb{R}^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\mbox{ on }\partial \mathbb{R}^N_+, $$ where $p,λ>0$ and $k\in (0, N-1)$. First, we show that no solutions exist if $0 p^*$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{**}=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $α$-lifting operator and fixed point theorems.

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arXiv2024

Steady-states of the Gierer-Meinhardt system in exterior domains

We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu=\frac{u^p}{v^q}+λρ(x) \,, u>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -Δv=\frac{u^m}{v^s} \,, v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\; \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 &\quad\mbox{ on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as }|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a compact set, $ρ\in C^{0,γ}_{loc}(\overline{\mathbb{R}^N\setminus K})$, $γ\in (0,1)$, is a nonnegative function and $p,q,m,s, λ>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents.

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arXiv2024

Asymptotically homogeneous solutions of the supercritical Lane-Emden system

We consider the Lane-Emden system-$Δ$u = |v| p-1 v,-$Δ$v = |u| q-1 u in R d. When p $\ge$ q $\ge$ 1, it is known that there exists a positive radial stable solution (u, v) $\in$ C 2 (R d) if and only if d $\ge$ 11 and (p, q) lies on or above the so-called Joseph-Lundgren curve introduced in [5]. In this paper, we prove that for d $\le$ 10, there is no positive stable solution (or merely stable outside a compact set and (p, q) does not lie on the critical Sobolev hyperbola), while for d $\ge$ 11, the Joseph-Lundgren curve is indeed the dividing line for the existence of such solutions, if one assumes in addition that they are asymptotically homogeneous (see Definition 1 below). Most of our results are optimal improvements of previous works in the litterature.

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arXiv2023

Steady state solutions for the Gierer-Meinhardt system in the whole space

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu+λu=\frac{u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -Δv+μv=\frac{u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N,\\[0.1in] \end{cases} $$ which satisfy $u(x), v(x)\to 0$ as $|x|\to \infty$. In the above system $p,q,m,s>0$, $λ, μ\geq 0$ and $ρ\in C(\mathbb{R}^N)$, $ρ\geq 0$. It is a known fact that posed in a smooth and bounded domain of $\mathbb{R}^N$, the above system subject to homogeneous Neumann boundary conditions has positive solutions if $p>1$ and $σ=\frac{mq}{(p-1)(s+1)}>1$. In the present work we emphasize a different phenomenon: we see that for $λ, μ>0$ large, positive solutions with exponential decay exist if $0< σ\leq 1$. Further, for $λ=μ=0$ we derive various existence and nonexistence results and underline the role of the critical exponents $p=\frac{N}{N-2}$ and $p=\frac{N+2}{N-2}$.

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arXiv2023

Elliptic inequalities with nonlinear convolution and Hardy terms in cone-like domains

We study the inequality $ -Δu - \fracμ{|x|^2} u \geq (|x|^{-α} * u^p)u^q$ in an unbounded cone $\mathcal{C}_Ω^ρ\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $Ω$ of the unit sphere $S^{N-1}\subset \mathbb{R}^N,$ $p, q, ρ>0$, $μ\in \mathbb{R}$ and $0\leq α< N$. In the above, $|x|^{-α} * u^p$ denotes the standard convolution operator in the cone $\mathcal{C}_Ω^ρ$. We discuss the existence and nonexistence of positive solutions in terms of $N, p, q, α, μ$ and $Ω$. Extensions to systems of inequalities are also investigated.

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arXiv2022

Higher order evolution inequalities with nonlinear convolution terms

We are concerned with the study of existence and nonexistence of weak solutions to $$ \begin{cases} &\displaystyle \frac{\partial^k u}{\partial t^k}+(-Δ)^m u\geq (K\ast |u|^p)|u|^q \quad\mbox{ in } \mathbb R^N \times \mathbb R_+,\\[0.1in] &\displaystyle \frac{\partial^i u}{\partial t^i}(x,0) = u_i(x) \,\, \text{ in } \mathbb R^N,\, 0\leq i\leq k-1,\\ \end{cases} $$ where $N,k,m\geq 1$ are positive integers, $p,q>0$ and $u_i\in L^1_{\rm loc}(\mathbb{R}^N)$ for $0\leq i\leq k-1$. We assume that $K$ is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, $K\ast |u|^p$ denotes the standard convolution operation between $K(|x|)$ and $|u|^p$. We obtain necessary conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our analysis emphasizes the role played by the sign of $\displaystyle \frac{\partial^{k-1} u}{\partial t^{k-1}}$.

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arXiv2021

Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_α\ast\big((I_β\ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $α, β\in (0,N)$, $p, q>0$ and $I_α$, $I_β$ denote the Riesz potentials of order $α$ and $β$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $α$, $β$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

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arXiv2021

Polyharmonic inequalities with nonlocal terms

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm Δ^m u \geq \big(Ψ(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $Δ^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $Ψ>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

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arXiv2021

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $$ where $u_0\in L^1_{loc}({\mathbb R}^N)$, $L_{\mathcal{A}}$ denotes a weakly $m$-coercive operator, which includes as prototype the $m$-Laplacian or the generalized mean curvature operator, $p,\,q>0$, while $K\ast u^p$ stands for the standard convolution operator between a weight $K>0$ satisfying suitable conditions at infinity and $u^p$. For problem $(P^-)$ we obtain a Fujita type exponent while for $(P^+)$ we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.

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arXiv2021

Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.

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arXiv2020

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

We study the quasilinear elliptic inequality $$ -Δ_m u - \fracμ{|x|^m}u^{m-1} \geq (I_α*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, μ\in \mathbb{R}$, $m>1$ and $I_α$ is the Riesz potential of order $α\in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.

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arXiv2020

Radial regular and rupture solutions for a MEMS model with fringing field

We investigate radial solutions for the problem \[ \begin{cases} \displaystyle -ΔU=\frac{λ+δ|\nabla U|^2}{1-U},\; U>0 & \textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset \mathbb{R}^N$ $(N\geq 2)$ denotes the open unit ball and $λ, δ>0$ are real numbers. Two classes of solutions are considered in this work: (i) {\it regular solutions}, which satisfy $0 0$ is also discussed.

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arXiv2020

Radial single point rupture solutions for a general MEMS model

We study the initial value problem $$ \begin{cases} r^{-(γ-1)}\left(r^α|u'|^{β-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0 0 & \textrm{for}\ 0 α>β\geq 1$ and $f\in C[0,\bar u)\cap C^2(0,\bar u)$, $f(0)=0$, $f(u)>0$ on $(0, \bar u)$ and $f$ satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution $u^*$ to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions $u(\,\cdot\,,a)$ (with $u(0,a)=a$) as $a\to 0$. In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.

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arXiv2020

Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator

We are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }Ω,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\in C^2(Ω)$ ($1\leq k\leq N$) and $Ω\subset\mathbb{R}^N$ $(N\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\geq 0$, $p\geq s\geq 0$ and $(k-m)(k-s)\neq pq$. In the case where $Ω$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $Ω=\mathbb{R}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\leq m<k$ and $pq < (k-m)(k-s)$. Using arguments from three component cooperative and irreducible dynamical systems we deduce the behavior at infinity of such solutions.

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arXiv2020

Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms

We study the inequality $$ {\rm div}\big(|x|^{-α}|\nabla u|^{m-2}\nabla u\big)\geq (I_β\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $α>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_β$ denotes the Riesz potential of order $β\in(0, N)$. We obtain sharp conditions in terms of these parameters for which positive singular solutions exist. We further establish the asymptotic profile of singular solutions to the double inequality $$ a(I_β\ast u^p)u^q\geq {\rm div}\big(|x|^{-α}|\nabla u|^{m-2}\nabla u\big)\geq b(I_β\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $a\geq b>0$ are constants.

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