Boumediene Abdellaoui

Abou Bakr Belkaid University of Tlemcen · Département de Maths

In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem $$\begin{aligned} {\left\{ \begin{array}{ll} (w_t-\Delta w)^s=\frac{\lambda }{|x|^{2s}} w+w^p +f, &{}\quad \text {in }\mathbb {R}^N\times (0,+\infty ),\\ w(x...

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem (wt − ∆w) s = λ |x| 2s w + w p + f, in R N × (0, +∞), w(x, t) = 0, in R N × (−∞, 0], where N > 2s, 0 < s < 1 and 0 < λ < Λ N,s , the optimal constant in the f...

In the present work we study the existence and non-existence of nonnegative solutions to a class of deterministic KPZ system with nonlocal gradient term. More precisely we will consider the system \begin{equation*} \left\{ \begin{array}{rcll} (-\Delta)^{s} u & =&|\mathbb{D}_{s} v|^q + \rho f\,, & \quad \textup{in } \Omega,\\ (-\Delta)^{s} v & =& |\...

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω⊂RN, N≥2, be a bounded domain with boundary ∂Ω of class C2. For s∈(0,1), we consider problems of the form (-Δ)su=μ(x)|D(u)|q+λf(x),inΩ,u=0,inRN\Ω,(KPZ)where q>1 a...

Our aim in this paper is to analyze the existence and regularity of solutions for the following nonlocal parabolic problem involving the fractional Laplacian with singular nonlinearity $$\begin{aligned} \left\{ \begin{array}{rlll} u_t+(-\Delta )^{s} u &{}=\frac{f(x,t)}{u^{\gamma }} &{} \text{ in } \Omega _T:=\Omega \times (0,T),\\ u &{}=0 &{} \text...

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar-Parisi-Zhang type equations involving non-local "gradient terms". More precisely, let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial \Omega$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form...

In this work we analyze the existence of solutions to the fractional quasilinear problem,(P){ut+(−Δ)su=|∇u|α+f in ΩT≡Ω×(0,T),u(x,t)=0 in (RN∖Ω)×[0,T),u(x,0)=u0(x) in Ω, where Ω is a C1,1 bounded domain in RN, N>2s and 12<s<1. We will assume that f and u0 are non negative functions satisfying some additional hypotheses that will be specified later o...

In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem \begin{document}$ \left\{ \begin{array}{rcll} (-\Delta )^s u & = &\lambda \dfrac{u}{|x|^{2s}}+ (\mathfrak{F}(u)(x))^p+ \rho f & \text{ in } \Omega,\\ u&>&0...

The aim of this paper is to study a nonlocal elliptic problem in (Formula presented.) (denoted as (Formula presented.) below) involving the fractional Laplacian, a linear Hardy potential term and a critical nonlinear term. According to suitable assumptions on the set of extremal points of the functional coefficients, we prove that (Formula presente...

We obtain a global fractional Calder\'on-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $\Omega \subset \mathbb{R}^N$, $N \geq 2$, a bounded domain with boundary $\partial \Omega$ of class $C^2$, $s \in (0,1)$ and $f \in L^m(\Omega)$ for some $m \geq 1$, we consider the problem $$ \left. \begin{aligned} (-\Delta)^...

We consider the problem (P)ut+(-Δ)su=λupδ2s(x)inΩT≡Ω×(0,T),u(x,0)=u0(x)inΩ,u=0in(IRN\Ω)×(0,T),where Ω⊂IRN is a bounded regular domain (in the sense that ∂Ω is of class C0,1), δ(x)=dist(x,∂Ω), 00, λ>0. The purpose of this work is twofold. First We analyze the interplay between the parameters s, p and λ in order to prove the existence or the nonexist...

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely,\begin{equation*}u_t+(-\Delta)^s u=\lambda\dfrac{u}{|x|^{2s}}+u^{p}\;{\rm in}\;{{\boldsymbol R}^N}, u(x,0)=u_{0}(x)\;{\rm in}\;{{\boldsymbol R}^N},\end{equation*}where $N> 2s$, $0<s<1...

In this work we analyze the existence of solutions to the fractional quasilinear problem, (P)    ut + (−∆) s u = |∇u| α + f in Ω T ≡ Ω × (0, T), u(x, t) = 0 in (R N \ Ω) × [0, T), u(x, 0) = u 0 (x) in Ω, where Ω is a C 1,1 bounded domain in R N , N > 2s and 1 2 < s < 1. We will assume that f and u 0 are non negative functions satisfying some add...

In this paper, we investigate the existence of solutions to a nonlinear parabolic system, which couples a non-homogeneous reaction-diffusion-type equation and a non-homogeneous viscous Hamilton–Jacobi one. The initial data and right-hand sides satisfy suitable integrability conditions and non-negative. To simplify the presentation of our results, w...

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*} \left\lbrace \begin{array}{lll} (-\Delta)^{s}u + |\nabla u|^{p} &= & f \quad\text{ in } \Omega\\ \qquad \qquad \,\,\,\,\,\:\: u & = & 0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus \Omega, \quad s \in (1/2, 1). \\ \en...

In this work we study the existence of positive solution to the fractional quasilinear problem, (−Δ)su=λu|x|2s+|∇u|p+μf in Ω,u>0 in Ω,u=0 in (RN∖Ω),where Ω is a C1,1 bounded domain in RN, N>2s,μ>0, 12<s<1, and 0<λ<ΛN,s is defined in (3). We assume that f is a non-negative function with additional hypotheses. As we will see, there are deep differenc...

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0 & \inn(\mathbb{R}^N\setminus\Omega), \end{array}\right. $$ where $\Omega$ is a $C^{1,1}$ bounded domain in $\m...

Let $\Omega \subset \mathbb{R}^{N} $ , N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ > 0 is a real parameter, f belo...

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-\Delta)^s u=\lambda\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\ u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0<s<1$, $(-\Delta)^s$ is the fractional laplac...

\begin{abstract} In this work we analyze the existence of solution to the fractional quasilinear problem, $$ (P) \left\{ \begin{array}{rcll} u_t+(-\Delta )^s u &=&|\nabla u|^{\alpha}+ f &\inn \Omega_T\equiv\Omega\times (0,T),\\ u(x,t)&=&0 & \inn(\mathbb{R}^N\setminus\Omega)\times [0,T),\\ u(x,0)&=&u_{0}(x) & \inn\Omega,\\ \end{array}\right. $$ wher...

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-\Delta)^s u & = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x)\,, & \quad \mbox{in} \Omega,\\ u & = 0\,, & \quad \mbox{in} \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \] where $\lambda >...

The goal of this paper is to study the following non-local superlinear elliptic problem {(-Δ)su=|u|p-σϕ1inΩ,u=0inRN\Ω,u>0inΩ,where (- Δ) s is the fractional Laplace operator, Ω ⊂ RN is an open domain with Lipschitz boundary, σ> 0 , p∈(1,2s∗-1) with 2s∗=2NN-2s and ϕ1 is the first positive eigenfunction of the fractional Laplacian with Dirichlet boun...

The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.

In this paper we consider a fractional Kirchhoff problem with Hardy potential, The main goal of this work is to get the existence of solution for the largest class of f without any assumption on λ.

The aim goal of this paper is to treat the following problem \begin{equation*} \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u &=&\dyle \l \dfrac{u^{p-1}}{|x|^{ps}} & \text{ in } \O_{T}=\Omega \times (0,T), \\ u&\ge & 0 & \text{ in }\ren \times (0,T), \\ u &=& 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u(x,0)&=& u_0(x)& \mbox{ in }\O, \end{arr...

Ces notes ont été préparées pour le cours de master M2 : Analyse non linéaire et application à la résolution des EDP elliptiques non linéaires. Le but de ce cours est d'introduire des techniques variationnelles pour la résolution des problèmes elliptiques non linéaires.

In this work, we investigate by analysis the possibility of a solution to the fractional quasilinear problem: where is a bounded regular domain ( is sufficient), , 1 < q and f is a measurable non-negative function with suitable hypotheses. The analysis is done separately in three cases: subcritical, 1 < q < 2s; critical, q = 2s; and supercritical,...

The aim of this paper is to study the following problem (P) ≡{ (-Δ)su = uq + up in ; u > 0 inΩ; Bsu = 0 in RN/Ω with 0 < q < 1 < p, N > 2s, > 0,Ω⊂ RN is a smooth bounded domain, (-Δ)su(x) = aN;s P:V: Z RN u(x) -u(y)/x-y/N+2s dy; aN;s is a normalizing constant, and Bsu = uX∑1 + NsuX∑2 : Here, ∑1 and ∑2 are open sets in RN/Ω such that∑2 ∩ ∑2 = ; and...

In this work we will consider a class of non local parabolic problems with nonlocal initial condition, more precisely we deal with the problem where is a bounded domain, the function a can be a singular potential, g is a suitable function which will be specified later and p, θ Abstract. In this work we will consider a class of non local parabolic...

Let $\Omega\subset \mathbb{R}^N$ be a bounded regular domain, $0<s<1$ and $N>2s$. We consider $$ (P)\left\{ \begin{array}{rcll} (-\Delta)^s u &= & \frac{u^{q}}{d^{2s}} & \text{ in }\Omega , \\ u &> & 0 & \text{in }\Omega , \\ u & = & 0 & \text{ in }\mathbb{R}^N\setminus\Omega ,% \end{array}% \right. $$ where $0<q\le 2^*-1$, $0<s<1$ and $d(x) = dist...

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \textit{fractional Hardy inequality } $$\Lambda_{N}\equiv\Lambda_{N}(\Omega):=\inf_{\{\phi\in \mathbb{E}^s(\Omega, D), \phi\neq 0\}} \dfrac{\frac{a_{d,s}}{2} \displaystyle\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \dfrac{|\phi(x)-\phi(y)|^2}...

In this work, we analyse the existence of solutions to the nonlinear elliptic system: (Formula presented.) where (Formula presented.) is a bounded domain of (Formula presented.) and (Formula presented.), (Formula presented.) with (Formula presented.). f, g are nonnegative measurable functions with additional hypotheses and (Formula presented.). As...

We discuss the existence and uniqueness of solutions of nonlinear space-fractional diffusionequations subject to an initial condition of integral type. Our approach shall rely on fixed point theorems.

In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solut...

In this article we present a survey of the Ph.D. theses that have been completed under the advice of Ireneo Peral. Following a chronological order, we summarize the main results contained in the works of the former students of Ireneo Peral.

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }\O, \end{array}% \right. $$ where $\O...

The aim of this paper is to study the following problem: \left\{\begin{aligned} \displaystyle(-\Delta)^{s}_{p,\beta}u&\displaystyle=f(x% ,u)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is a smooth bounded domain of {\mathbb...

The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge 2$ For all $q<p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int_{{\mathbb R...

In this paper we consider the following nonlinear parabolic problem where Ω ⊂ ℝ N , N > 2, is a bounded domain with 0 ∈ Ω and 1 < p < 3. The main goal of this work is to analyze the influence of the gradient term in order to obtain the existence of a positive solution for the largest class of data ( f, u 0 ) and for all λ > 0.

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary. More precisely, we consider the problem Put-Δu=λupd2inΩT≡Ω×(0,T),u>0inΩT,u(x,0)=u0(x)>0inΩ,u=0on∂Ω×(0,T),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{am...

The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems $$ \left\{\begin{array}{rcll} (-\Delta)^s u-\lambda \dfrac{u}{|x|^{2s}}&=&f(x,u) &\hbox{ in } \Omega,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,\\ u&>&0 &\hbox{ in }\Omega, \end{array}\ri...

Let $0<s<1$ and $1<p<2$ be such that $ps<N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality: given $1\le q<p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \dint_{\mathbb{R}^N}\dint_{\mathbb{R}^N} \, \dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,d...

Let $0<s<1$ and $1<p<2$ be such that $ps<N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality: given $1\le q<p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \dint_{\re^N}\dint_{\re^N} \, \dfrac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N...

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c f\mbox{ in } \Omega\times (0,T),\\ u(x,t)&>&0\inn \Omega\times (0,T),\\ u(x,t)&=&0\inn (\ren\setminus\Omega)\time...

In this article we analyze the dynamics of the problem $$\displaylines{ x'(t)=-(\delta+\beta(x(t)))x(t)+\theta\int_{0}^{\tau}f(a)x(t-a)\beta(x(t-a))da, \quad t> \tau, \cr x(t)=\phi(t),\quad 0 \leq t\leq \tau, }$$ where $\delta,\theta$ are positive constants, and $\beta, \phi, f$ are positives continuous functions. The main results obtained in this...

In this article we consider the following family of nonlinear elliptic problems, [equation presented] We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains Ώ = RN, N ≥ 3, containing the pole of the potential. Recall that [equation presented] is the optim...

We prove in this note the following sharpened fractional Hardy inequality: Let N >= 1, 0 < s < 1, N > 2s, and Omega subset of R-N a bounded domain. Then for all 1 < q < 2, there a positive constant C = C (Omega, q, N, s) such that for all u is an element of C-0(infinity) (Omega) a(N,S) integral(RN) integral(RN) (u(x) - u(y))(2)/vertical bar x - y v...

In this paper we deal with the following quasilinear parabolic problem $$\left\{\begin{array}{l@{\quad}l} (u^\theta)_t - \Delta_p {u} = \lambda \frac{u^{p - 1}}{|x|^{p}} + u^q + f,\,\, u \geq 0 \quad {\rm in} \;\;\Omega \times (0, T),\\ u(x, t) = 0 \quad\qquad\qquad\qquad\qquad\qquad\qquad {\rm on}\; \partial \Omega \times(0, T),\\ u(x, 0) = u_0(x)...

Let Ω ⊂ ℝN be a bounded regular domain of ℝN and 1 < p < ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all , we have where d(x) = dist(x, ∂Ω), and C is a positive constant depending only on p, N and Ω. The optimality of the exponent of the logarithmic...

We consider the following quasilinear elliptic problem -Delta(p)u = lambda u(p-1)/vertical bar x vertical bar p + h/u(gamma) in Omega, where 1 < p < N, Omega subset of R-N is a bounded regular domain such that 0 is an element of Omega, gamma > 0 and h is a nonnegative measurable function with suitable hypotheses. The main goal of this work is to an...

In this paper we consider the problem $$(P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n u|^q +\,f(x,t),&\quad u\ge 0 \hbox{in} \Omega_T\equiv \Omega\times (0,T), u(x,t)&=&0 &\quad \hbox{on} \partial\Omega\times (0,T) u(x,0)&=&u_0(x),&\quad x\in \Omega {array}. $$ where $\O\subset \ren$, $N\ge 2$, is a bounded regular domain, $1<q\le 2$, and $f\ge 0$, $...

We will consider the following obstacle problem \int_{\Omega}\nabla u\nabla T_{k}(v-u)dx +\int_{\Omega }h(u)\left\vert \nabla u\right\vert ^{q}T_{k}(v-u)dx\geq \int_{\Omega }\left(g(x,u)+f\right) T_{k}(v-u)dx, with the condition that u\geq\psi a.e in \Omega . Under suitable condition relating g , h and q , we show the existence of a solution for al...

In this paper we consider the problem (P){−Δu=uqα|∇u|q+λf(x)in Ωu=0on ∂Ω, where Ω⊂RN is a bounded domain, 1q≤2, α∈R and f≥0. We prove that: (1)If qα−1, then problem (P) has a distributional solution for all f∈L1(Ω), and all λ>0.(2)If −1≤qα0, then problem (P) has a solution for all f∈Ls(Ω), where s>Nq if N≥2, and without any restriction on λ.(3)If q...

In this paper we consider age structured equation with diffusion under nonlocal boundary condition and nonnegative initial data. We prove existence, uniqueness and the positivity of the solution to the above problem. Our main result is to get an exponential decay of the solution for large times toward such a study state. To this end we prove a weig...

We deal with the following parabolic problem{ut−Δu=|∇u|p+λu|x|2+f,u>0in Ω×(0,T),u(x,t)=0on ∂Ω×(0,T),u(x,0)=u0(x),x∈Ω, where Ω⊂RN, N⩾3, is a bounded regular domain such that 0∈Ω or Ω=RN, p>1, λ⩾0 and f⩾0, u0⩾0 are in a suitable class of functions.There are deep differences with respect to the heat equation (λ=0). The main features in the paper are t...

In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum: where 1 < p < N, Ω is a bounded open subset of ℝ N (N ≥ 2), Δ p u = div(|∇u| p−2 ∇u) is the so called p-Laplacian operator, sign s ., ϕ(ν 0 ) ∈ L ¹ (Ω), μ is a finite Radon measure and f ∈ L ∞ (Ω×(0, T)) for every T...

We deal with the following parabolic problem,{ut−Δu+|∇u|p=λu|x|2+f,u>0inΩ×(0,T),u(x,t)=0on∂Ω×(0,T),u(x,0)=u0(x),x∈Ω, where Ω⊂RN, N⩾3, is a bounded regular domain such that 0∈Ω or Ω=RN, 1p⩽2, λ>0 and f⩾0, u0⩾0 are in a suitable class of functions. For p>p∗≡NN−1, we will show that the above problem has a solution for all λ>0, f∈L1(ΩT) and u0∈L1(Ω). W...

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem where Ω⊂ℝ N , N ≥3, is a bounded regular domain such that 0∈Ω, p >1, and u 0 ≥0, f ≥0 are in a suitable class of functions. There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there e...

We study the existence of different types of positive solutions to problem $\left\{{lll} -\Delta u - \lambda_1\dfrac{u}{|x|^2}-|u|^{2^*-2}u = \nu\,h(x)\alpha\,|u|^{\alpha-2}|v|^{\beta}u, &{\rm in}\,{\mathbb{R}}^{N},\\ &\qquad\qquad\qquad\qquad x \in {\mathbb{R}}^N,\quad N \geq 3,\\ -\Delta v - \lambda_2\dfrac{v}{|x|^2}-|v|^{2^*-2}v = \nu\,h(x)\be...

In the case where g(u) appears as an absorption term, then under some additional hypotheses on g we prove that the main problem has a solution for all λ > 0 and for all positive μ ∈ L 1 (Ω) . In the case where g appears as a reaction term, then we prove that the main problem has at least two positive solutions under suitable hypotheses on μ . The a...

In this paper we study the problem: where Ω is a bounded regular domain in RN, β is a positive nondecreasing function and f, u0 are positive functions satisfying some hypotheses of summability. Besides some regularity properties of all weak solutions, the main result is wild nonuniqueness theorem, which connects, via a change of unknown function, a...

In this article we study the problem \text{(P)}\: \begin{cases} −\mathrm{\Delta }u + |\mathrm{∇}u|^{q} = \lambda g(x)u + f(x) & \text{in }\Omega , \\ u > 0 & \text{in }\Omega , \\ u = 0 & \text{on }\partial \Omega , \end{cases} with 1⩽q⩽2 and f,g are positive measurable functions. We give assumptions on g with respect to q for which for all \lambda...

The main result of this work is to get the existence of infinitely many radial positive solutions to the problem -Δ p u = |▽u| q + λf(x) in Ω, u|aΩ = 0, where Ω = B 1 (0) and f is a radial positive function. Since, in general when q ≠ p, a Hopf-Cole type change can not be used, we will consider just the existence and multiplicity of positive radial...

In this work we study the problem $$ \left\{ {\begin{array}{lll} {u_t-{\rm div}(|x|^{-p \gamma} |\nabla u|^{p-2}\nabla u) = \lambda\frac{u^{\alpha}}{|x|^{p(\gamma + 1)}}+f\,{\rm in}\, \Omega \times (0,T),} \\ {u \geq 0\,{\rm in}\,\Omega \times (0,T)\,{\rm and}\,u = 0\,{\rm on}\, \partial\Omega \,\times (0,T),} \\ {u(x,0) = u_{0}(x)\,{\rm in}\,\Omeg...

In this article we analyze existence and nonexistence of positive solutions to problem(P±) - Δ u ± | ∇ u |2 = λ frac(u, | x |2) + f in Ω, u = 0 on ∂ Ω . The main results are the following: (i)If the quadratic term in the gradient appears in the equation as a reaction term (- | ∇ u |2) and λ > 0, then there is no solution to problem (P-) (even in a...

In this paper we deal with the following mixed Dirichlet–Neumann elliptic problems(1){−div(|x|−pγ|∇u|p−2∇u)=λup−1|x|p(γ+1)+ur|x|(r+1)γ,u>0inΩ,u=0onΣ1,|x|−pγ|∇u|p−2∂u∂ν=0onΣ2 where Ω⊂RN (N⩾3) is a bounded domain such that 0∈Ω and with different choices of the parameters 1pN, p−1r⩽p∗−1, −∞γN−pp and 0⩽λ⩽Λ which is a critical value to the existence of...

Cited By (since 1996):6, Export Date: 1 December 2013, Source: Scopus

We will consider the following problem -Delta u-lambda u/vertical bar x vertical bar(2)=vertical bar del u vertical bar(p)+c f, u>0 in Omega, where Omega subset of R-N is a domain such that 0 is an element of Omega, N >= 3, c>0 and lambda>0. The main objective of this note is to study the precise threshold p(+)=p(+)(lambda) for which there is no ve...

This article is concerned with the regularity of the entropy solution of where Ω is a smooth bounded domain Ω of ℝ N such that 0 ∈ Ω, 1 < p < N. and γ < (N-p)/p. Assuming f ∈ L q (Ω, |x| α(q-1) dx) for some q ≥ 1 and N γp /(N-p) ≤ α ≤ (γ+1)p, we obtain estimates for the entropy solution u and its weak gradient in Lebesgue spaces with weights. More...

In this paper we deal with the problem { -div(a(x, u)del u) + g(x, u, del u) = lambda h(x)u + f in Omega, u = 0 on partial derivative Omega. The main goal of the work is to get hypotheses on a, g and h such that the previous problem has a solution for all lambda > 0 and f is an element of L-1(Omega). In particular, we focus our attention in the mod...

The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case relate to Hardy inequality, providing a description of the be...

In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem(1)where β is a continuous nondecreasing positive function and f belongs to some suitable Lebesgue spaces.

We study the following parabolic problem u t -div(|x| -pγ |∇u| p-2 ∇u)=λf(x,u),u≥0inΩ×(0,T),B(u)=0on∂Ω×(0,T),u(x,0)=φ(x)ifx∈Ω, where Ω⊂ℝ N is a smooth bounded domain with 0∈Ω, B(u)≡uχ Σ 1 ×(0,T) +|x| -pγ |∇u| p-2 ∂u ∂νχ Σ 2 ×(0,T) and -∞<γ<N-p p. The boundary conditions over ∂Ω×(0,T) verify hypotheses that will be precised in each case. Mainly, we...

The paper analyzes the influence on the meaning of natural growth in the gradient, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem. The main results are: i) Optimal summability of the data to have weak...

This work deals with the study of the optimal constants of Sobolev and Hardy-Sobolev inequalities with weights and their relations with the behavior of some mixed Dirichlet-Neumann boundary conditions. More precisely, we analyze the attainability of the Sobolev constant where ω ⊂ IRN, N ≥ 3, is a smooth bounded domain such that 0 ∈ ω, -∞ < γ < N-2/...

We consider the following elliptic problem: (1){-div( x -2γ∇u) + u/ x -2(γ+1) = f(x, u), u ≥ 0 Ω, u ∂Ω = 0 and the corresponding parabolic version (2){ut - div( x -2γ∇u) + u/ x -2(γ+1) = f(x, u), u ≥ 0 Ω × (0, T), u (x, t) = 0 for (x, t) ∈ ∂ Ω × (0, T), u (x, 0) = φ (x) if x ∈ Ω, Ω ⊂ ℝN is a smooth bounded domain with 0 ∈ Ω and - ∞ < γ < N-2/2. We...

Using a perturbation argument based on a finite dimensional reduction, we find positive solutions to the following class of perturbed degenerate elliptic equations with critical growt

For 1 < p < N and \(-\infty < \gamma < \frac{N-p}{p}\) we obtain the following improved Hardy-Sobolev Inequalities \( \int\limits_\Omega \vert\nabla \phi\vert^p\vert x\vert^{-p\gamma}dx -\left(\frac{N-p(\gamma +1)}{p}\right)^p \int\limits_\Omega \frac{\vert\phi\vert^p}{\vert x\vert^{p(\gamma + 1)}}dx \) \( \ge C(p,q,r,\gamma,\vert\Omega \vert)\left...

We study the problems where is a smooth bounded domain with 0 ∈ Ω, and , mixed boundary condition. Mainly we will be interested in the behavior of the solutions to (0.1) close to the critical constant in the Hardy-Sobolev inequality, Λ N,γ (Ω,Σ 1 ).

This paper deals with the following parabolic equations: (1)where p>2, 0<(γ+1)<N/p, is a regular bounded domain containing the origin, and f is a function with suitable hypothesis that we will precise later in order to obtain some weak Harnack inequality.

We study the problem u t -div|x| -2γ ∇ u=λu α |x| 2(γ+1) +finΩ×(0,T),u≥0inΩ×(0,T),u=0on∂Ω×(0,T),u(x,0)=u 0 (x)inΩ, Ω⊂ℝ N (N≥2) is a bounded regular domain such that 0∈Ω, λ>0, α>0, -∞<γ<(N-2)/2, f and u 0 are positive functions such that f∈L 1 (Ω×(0,T)) and u 0 ∈L 1 (Ω). The main points under analysis are: (i) spectral instantaneous and complete blo...

The present work is devoted to analyze the Dirichlet problem for quasilinear elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Precisely the problem under study is, -div(\x\-pgamma\delu\p-2delu)=f(x,u)is an element ofL(1)(Omega), xis an element ofOmega u(x) = 0 on partial derivativeOmega, where -infinity < gamma < N - p/p, O...

This paper deals with the existence and nonexistence results for quasilinear elliptic equations of the form -pu=f(x, u), where p:=div(|u|p-2u), p>1, and the solutions are understood in the sense of renormalized or, equivalently, entropy solutions. In particular we prove nonexistence results in the case f(x,u)=up|x|-p, that is related to a classical...

In order to obtain solutions to problem $$ {{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplici...

This paper is devoted to the study of the elliptic problems with a critical potential,

Using a perturbation argument based on a finite dimensional reduction, we find positive solutions to the following class of perturbed degenerate elliptic equations with critical growth