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Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

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  • 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 $f \in L^1(Ω)$.
    Mathematics Subject Classification: 35A01, 35D30, 35J25, 35J70, 35J60, 35J75.

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