Existence and nonexistence of solutions for singular quadratic quasilinear equations.(English)Zbl 1173.35051

The paper is concerned with quasilinear elliptic problems of the form
$\begin{cases}-\text{div}(M(x,u)\nabla u)+g(x,u)|\nabla u|^2=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{1}$
Here $$\Omega\subseteq\mathbb{R}^N$$ is open and bounded, and $$N\geq3$$. The coefficients are Caratheodory functions, and such that the principal part is uniformly elliptic with bounded coefficients. For the nonhomogeneous part $$f$$ it is assumed that it lies in a suitable Lebesgue space, and that it is uniformly bounded from below by positive constants on compact subsets of $$\Omega$$.
The main interest lies in nonnegative functions $$g$$ with a singularity in $$u=0$$ that is uniform in $$x$$. Suppose that $$h:(0,\infty)\to[0,\infty)$$ is continuous, nonincreasing in a neighborhood of zero, and satisfies
$\lim_{s\to0+}\int_s^1\sqrt{h(t)}\,dt<\infty.$
If
$g(x,s)\leq h(s)\qquad\text{for a.e.\;}x\in\Omega,\;\forall s>0,$
then it is proved that (1) has a weak positive solution in $$H^1_0(\Omega)$$.
Conversely, a nonexistence result is given in the case that $$g$$ grows faster in $$s$$ than a function $$h$$ whose square root is not integrable near $$0$$. For the model problem
$\begin{cases}-\Delta u+\frac{|\nabla u|^2}{u^\gamma}=f &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega. \end{cases}\tag{2}$
this amounts to the following assertion: Suppose that $$\gamma>0$$. Then Eq. (2) has a positive solution if and only if $$\gamma<2$$.
Existence of a solution to (1) is proved by applying classical results for quasilinear equations to a family of problems with truncated coefficients and then passing to the limit.
The regularity of solutions to (1) is also considered. Moreover, the authors treat a general semilinear variant of (1).

MSC:

 35J60 Nonlinear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 35B45 A priori estimates in context of PDEs
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References:

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