Quasilinear elliptic problem with Hardy potential and singular term
Boumediene Abdellaoui Ahmed Attar
Communications on Pure & Applied Analysis 2013, 12(3): 1363-1380 doi: 10.3934/cpaa.2013.12.1363
We consider the following quasilinear elliptic problem \begin{eqnarray*} -\Delta_pu =\lambda\frac{u^{p-1}}{|x|^p}+\frac{h}{u^\gamma} \quad in \quad\Omega, \end{eqnarray*} where $1 < p < N, \Omega\subset R^N$ is a bounded regular domain such that $0\in \Omega, \gamma>0$ and $h$ is a nonnegative measurable function with suitable hypotheses.
The main goal of this work is to analyze the interaction between the Hardy potential and the singular term $u^{-\gamma}$ in order to get a solution for the largest possible class of the datum $h$. The regularity of the solution is also analyzed.
keywords: comparison principle singular Hardy-Sobolev potential existence and nonexistence results. Quasilinear elliptic problems
Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential
Boumediene Abdellaoui Daniela Giachetti Ireneo Peral Magdalena Walias
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 1747-1774 doi: 10.3934/dcds.2014.34.1747
In this article we consider the following family of nonlinear elliptic problems,
                         $-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x). $
We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential.
    Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality.
    1.For $0 < m \le 2$ we prove the existence of a critical exponent $q_+ \le 2$ such that for $q > q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments.
    Moreover if $q = q_+ > 1$ we get the following alternative results.
    (a) If $m < 2$ and $q=q_+$ there is no solution.
    (b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions.
    2. If $m > 2$ we obtain some partial results on existence and nonexistence.
We emphasize that if $q(\frac{1}{m}-1)<-1$ and $1 < q \le 2$, there exists positive solutions for any $f \in L^1(Ω)$.
keywords: Elliptic equations dependence on the gradient renormalized solutions critical exponents singular non linearities Hardy-Leray potential.
Attainability of the fractional hardy constant with nonlocal mixed boundary conditions: Applications
Boumediene Abdellaoui Ahmed Attar Abdelrazek Dieb Ireneo Peral
Discrete & Continuous Dynamical Systems - A 2018, 0(0): 1-29 doi: 10.3934/dcds.2018131
The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the fractional Hardy inequality
$\Lambda_{N}\equiv \Lambda_{N}(\Omega): = \inf\limits_{\{\varphi\in \mathbb{E}^{s}(\Omega, D), \varphi \ne0\}}\dfrac{\frac{a_{d, s}}{2}\displaystyle\int_{\mathbb R^d}\int_{\mathbb R^d}\dfrac{|\varphi(x)-\phi(y)|^{2}}{|x-y|^{d+2s}}dx dy}{\displaystyle\int_{\Omega}\frac{\varphi^2}{|x|^{2s}}\, dx}, $
is a bounded domain of
$\mathbb R^d$
$D\subset \mathbb R^d\setminus \Omega$
a nonempty open set,
$N = (\mathbb R^d\setminus \Omega)\setminus\overline{D}$
$\mathbb{E}^{s}(\Omega, D) = \{ u \in H^s(\mathbb R^d):\, u = 0 \text{ in } D\}.$
The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian, that is,
${P_\lambda } \equiv \left\{ {\begin{array}{*{20}{l}}{{{\left( { - \Delta } \right)}^s}u\;\;\; = \;\;\;\lambda \frac{u}{{|x{|^{2s}}}} + {u^p}}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;\;\;\;u\;\;\; > \;\;\;0}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;{{\cal B}_s}u\;\;\;: = \;\;u{\chi _D} + {{\cal N}_s}u{\chi _N} = 0}&{{\rm{in}}\;{{\mathbb {R}}^d}\backslash \Omega ,}\end{array}} \right.$
open sets in
$\mathbb R^{d}\backslash\Omega$
such that
$N \cap D = \emptyset$
$\overline{N}\cup \overline{D} = \mathbb R^{d}\backslash\Omega$
$\lambda> 0$
$<p\le 2_s^*-1_s^* = \frac{2d}{d-2s}$
. We emphasize that the nonlinear term can be critical.
The operators
$(-\Delta)^s $
, fractional Laplacian, and
, nonlocal Neumann condition, are defined below in (7) and (8) respectively.
keywords: Fractional Laplacian mixed boundary condition Hardy inequality doubly-critical problem
A nonlocal concave-convex problem with nonlocal mixed boundary data
Boumediene Abdellaoui Abdelrazek Dieb Enrico Valdinoci
Communications on Pure & Applied Analysis 2018, 17(3): 1103-1120 doi: 10.3934/cpaa.2018053
The aim of this paper is to study the following problem
$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$
$λ> 0$
$Ω \subset \mathbb{R}^{N}$
is a smooth bounded domain,
$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$
is a normalizing constant, and
$\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$
are open sets in
$\mathbb{R}^{N}\backslash Ω$
such that
$Σ_{1} \cap Σ_{2} = \emptyset$
$\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$
In this setting,
can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and
is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (
) for suitable ranges of
and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
keywords: Integro differential operators fractional Laplacian weak solutions mixed boundary condition multiplicity of positive solution
An improved Hardy inequality for a nonlocal operator
Boumediene Abdellaoui Fethi Mahmoudi
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1143-1157 doi: 10.3934/dcds.2016.36.1143
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 $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality (1.1).
keywords: Fractional Sobolev spaces nonlocal problems. weighted Hardy inequality

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