DCDS
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.
CPAA
Breaking of resonance for elliptic problems with strong degeneration at infinity
Francesco Della Pietra Ireneo Peral
Communications on Pure & Applied Analysis 2011, 10(2): 593-612 doi: 10.3934/cpaa.2011.10.593
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.
DCDS
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, 38(12): 5963-5991 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}, $
where
$\Omega$
is a bounded domain of
$\mathbb R^d$
,
$0<s<1$
,
$D\subset \mathbb R^d\setminus \Omega$
a nonempty open set,
$N = (\mathbb R^d\setminus \Omega)\setminus\overline{D}$
and
$\mathbb{E}^{s}(\Omega, D) = \{ u \in H^s(\mathbb R^d):\, u = 0 \text{ in } D\}.$
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,
${P_\lambda } \equiv \left\{ {\begin{array}{*{20}{l}}{{{\left( { - \Delta } \right)}^s}u\;\;\; = \;\;\;\lambda \frac{u}{{|x{|^{2s}}}} + {u^p}}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;\;\;\;u\;\;\; > \;\;\;0}&{{\rm{in}}\;\Omega ,}\\{\;\;\;\;\;\;{{\cal B}_s}u\;\;\;: = \;\;u{\chi _D} + {{\cal N}_s}u{\chi _N} = 0}&{{\rm{in}}\;{{\mathbb {R}}^d}\backslash \Omega ,}\end{array}} \right.$
with
$N$
and
$D$
open sets in
$\mathbb R^{d}\backslash\Omega$
such that
$N \cap D = \emptyset$
and
$\overline{N}\cup \overline{D} = \mathbb R^{d}\backslash\Omega$
,
$d>2s$
,
$\lambda> 0$
and
$<p\le 2_s^*-1_s^* = \frac{2d}{d-2s}$
. We emphasize that the nonlinear term can be critical.
The operators
$(-\Delta)^s $
, fractional Laplacian, and
$\mathcal{N}_{s}$
, nonlocal Neumann condition, are defined below in (7) and (8) respectively.
keywords: Fractional Laplacian mixed boundary condition Hardy inequality doubly-critical problem
DCDS
A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential
Lucio Boccardo Luigi Orsina Ireneo Peral
Discrete & Continuous Dynamical Systems - A 2006, 16(3): 513-523 doi: 10.3934/dcds.2006.16.513
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.

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