Abstract
In this paper we study a Dirichlet problem for an elliptic equation withdegenerate coercivity and a singular lower order term with natural growth with respect to the gradient.The model problem is$$\begin{equation}\left\{\begin{array}{11}-div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in $\Omega$,} \\u = 0 & \mbox{on $\partial\Omega$,}\end{array}\right.\end{equation}$$where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$.The source $f$ is a positive function belonging to some Lebesgue space.We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.
Highlights
In this paper we study the following problem: (1)−div (1 b(x) + |u|)p + B |∇u|2 |u|θ =f u=0 in Ω, on ∂Ω, where Ω is an open bounded set of RN, N ≥ 3, B, p > 0 and θ > 0
These results show that if q is sufficiently large, there exists a distributional solution for any source; this is not the case for problem (3)
In the literature we find several papers about elliptic coercive problems with lower order terms having a quadratic growth with respect to the gradient
Summary
These results show that if q is sufficiently large, there exists a distributional solution for any source; this is not the case for problem (3). By Theorem 2.1 the solutions un of the above approximating problems are bounded H01(Ω) non-negative functions, since f is assumed to be positive and the lower order term has the same sign as un.
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