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
\[ \lim_{s\to0+}\int_s^1\sqrt{h(t)}\,dt<\infty. \]
\[ g(x,s)\leq h(s)\qquad\text{for a.e.\;}x\in\Omega,\;\forall s>0, \]
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).


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
Full Text: DOI Link


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