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Vicentiu Radulescu

Publications and source records attributed to Vicentiu Radulescu.

27 records · Page 2Linked to original sources
arXiv2005

Boundary blow-up in nonlinear elliptic equations of Bieberbach--Rademacher type

We establish the uniqueness of the positive solution for equations of the form $-Δu=au-b(x)f(u)$ in $Ω$, $u|\_{\partialΩ}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is \emph{not regular} (e.g., $\exp(u)-1$, $\sinh(u)$, $\cosh(u)-1$, $\exp(u)\log(u+1)$, $u^β\exp(u^γ)$, $β\in {\mathbb R}$, $γ>0$ or $\exp(\exp(u))-e$) and functions $b\geq 0$ in $Ω$ vanishing on $\partialΩ$. The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the non-regular variation of $f$ at infinity with the blow-up rate of the solution near $\partialΩ$.

math.AP↗
arXiv2005

Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach

We study the uniqueness and expansion properties of the positive solution of the logistic equation $Δu+au=b(x)f(u)$ in a smooth bounded domain $Ω$, subject to the singular boundary condition $u=+\infty$ on $\partialΩ$. The absorption term $f$ is a positive function satisfying the Keller--Osserman condition and such that the mapping $f(u)/u$ is increasing on $(0,+\infty)$. We assume that $b$ is non-negative, while the values of the real parameter $a$ are related to an appropriate semilinear eigenvalue problem. Our analysis is based on the Karamata regular variation theory.

math.AP↗
arXiv2005

On a class of sublinear singular elliptic problems with convection term

We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a singular nonlinearity, combined with a convection term. Our approach takes into account both the sign of the potential and the decay rate around the origin of the singular nonlinearity. The proofs are based on various techniques related to the maximum principle for elliptic equations.

math.AP↗
arXiv2005

Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity

We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with sublinear nonlinearity and two parameters; (ii) combined effects of asymptotically linear and singular nonlinearities in bifurcation problems; (iii) bifurcation for a class of singular elliptic problems with subquadratic convection term. In some concrete situations we also establish the asymptotic behaviour of the solution around the bifurcation point. Our analysis relies on the maximum principle for elliptic equations combined with adequate estimates.

math.AP↗
arXiv2005

Nonradial blow-up solutions of sublinear elliptic equations with gradient term

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $Δu+|\nabla u|=p(x)f(u)$ in $Ω,$ where $Ω$ is either a smooth bounded domain or $Ω=\RR^N$. If $Ω$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $Ω=\RR^N$ a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.

math.AP↗
arXiv2005

Singular elliptic problems with lack of compactness

We consider the following nonlinear singular elliptic equation $$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$ where $g$ belongs to an appropriate weighted Sobolev space, and $p$ denotes the Caffarelli-Kohn-Nirenberg critical exponent associated to $a$, $b$, and $N$. Under some natural assumptions on the positive potential $K(x)$ we establish the existence of some $\la\_0>0$ such that the above problem has at least two distinct solutions provided that $\la\in(0,\la\_0)$. The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem without the Palais-Smale condition, combined with a weighted variant of the Brezis-Lieb Lemma.

math.AP↗