Quasilinear elliptic problem with Hardy potential and singular term
Boumediene Abdellaoui Ahmed Attar
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
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
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
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
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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