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$-\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 $f \in L^1(Ω)$.

-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.

$ -\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.

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.

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