N.T. Chung, H.Q. Toan On the solvability in $H^2(\Omega)$, p. 273-284

Abstract:

In this paper we study the existence of solutions in $H^2(\Omega)$ for a class of semilinear elliptic equations with dependence on the gradient of the form

$$\begin{cases} -\Delta u = f(x,u,\nabla u) + g(x), \quad x\in \Omega, \\ u = 0, \quad x\in \partial \Omega, \end{cases}$$

where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3,$ $g(x)\in H^s_{loc}(\Omega)$, $f(x,s,t)$ satisfies the Hölder condition on $s$ and $t$. The technique is based on the theory on a family of domains smoothly depending on a parameter in S.G. Krein's sense and an interative method.

Key Words: Semilinear elliptic equations, Dependence on the gradient

2010 Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60, 58E05