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.
Breaking of resonance for elliptic problems with strong degeneration at infinity
Francesco Della Pietra Ireneo Peral
In this paper we study the problem

-div$(\frac{Du}{(1+u)^\theta})+|Du|^q =\lambda g(x)u +f$ in $\Omega,$

$u=0$ on $\partial \Omega, $

$u\geq 0$ in $\Omega,$

where $\Omega$ is a bounded open set of $R^n$, $1 < q \leq 2$, $\theta\geq 0$, $f\in L^1(\Omega)$, and $f>0$. The main feature is to show that even for large values of $\theta$ there is solution for all $\lambda>0$.
The problem could be seen as a reaction-diffusion model which produces a saturation effect, that is, the diffusion goes to zero when $u$ go to infinity.

keywords: Non-coercive non-linear elliptic equations degeneration at infinity existence and nonexistence.
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 remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential
Lucio Boccardo Luigi Orsina Ireneo Peral
We study the effect of a zero order term on existence and optimal summability of solutions to the elliptic problem

$ -\text{div}( M(x)\nabla u)- a\frac{u}{|x|^2}=f \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega$,

with respect to the summability of $f$ and the value of the parameter $a$. Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ containing the origin.

keywords: Laplace equation Hardy potential summability of solution.
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations
Tommaso Leonori Ireneo Peral Ana Primo Fernando Soria
In this work we consider the problems $$ \left\{\begin{array}{rcll} \mathcal{L \,} u&=&f &\hbox{ in } \Omega,\\ u&=&0 &\hbox{ in } \mathbb{R}^N\setminus\Omega, \end{array} \right. $$ and $$ \left\{\begin{array}{rcll} u_t +\mathcal{L \,} u&=&f &\hbox{ in } Q_{T}\equiv\Omega\times (0, T),\\ u (x,t) &=&0 &\hbox{ in } \big(\mathbb{R}^N\setminus\Omega\big) \times (0, T),\\ u(x,0)&=&0 &\hbox{ in } \Omega, \end{array} \right. $$ where $\mathcal{L \,}$ is a nonlocal differential operator and $\Omega$ is a bounded domain in $\mathbb{R}^N$, with Lipschitz boundary.
    The main goal of this work is to study existence, uniqueness and summability of the solution $u$ with respect to the summability of the datum $f$. In the process we establish an $L^p$-theory, for $p \geq 1$, associated to these problems and we prove some useful inequalities for the applications.
keywords: summability of the solutions. elliptic equations Nonlocal operators parabolic equations

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