Some remarks on a class of elliptic equations with degenerate coercivity (Q2735886)

In this paper the solvability of the Dirichlet problem for the equation \(-\text{div} (a(x,u) \text{grad }u)= f\) with zero boundary function and with the function \(a(x,u)\) satisfying the condition \(\alpha (1+ |u|)^{-\Theta}\leq a(x,u)\leq \beta\) where \(\alpha> 0\), \(\beta> 0\), \(\Theta\in (0,1)\) is investigated. For \(f\in L^m (\Omega)\) the existence of a solution in \(H_0^1 (\Omega)\cap L^M (\Omega)\) is proved where \(M= \infty\) if \(m> \frac{N}{2}\) and \(M = \frac{mN}{N-2m}\cdot (1-\Theta)\) if \(\frac{2N} {N+2- \Theta(N-2)}< m< \frac{N}{2}\). Here \(\Omega\) is a bounded open set in \(\mathbb{R}^N\), \(N> 2\). The solvability of the considered problem for other values of \(m\) is also discussed, in particular for \(m=1\) the definition of an entropy solution is given.