An application of the perturbation theorem for m-accretive operators. II
This paper is concerned with the Dirichlet problem for a nonlinear elliptic equation \[ (*)\quad -\sum^{n}_{j=1}\frac{\partial}{\partial x_ j}(| \frac{\partial u}{\partial x_ j}|^{p- 2}\frac{\partial u}{\partial x_ j})+\beta (x,u)=f\text{ on } \Omega,\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with smooth boundary. Let \(\beta \in C^ 1(\Omega \times J)\), where J is an open interval on \({\mathbb{R}}\) containing the origin. We assume that (i) \(\beta (x,0)=0\) for every \(x\in \Omega\), \(\partial \beta /\partial s\geq 0\) on \(\Omega\) \(\times J;\) (ii) \(\beta\) (x,\(\cdot):J\to {\mathbb{R}}\) is surjective for every \(x\in \Omega;\) (iii) there are nonnegative constants c, a and b \([b<\frac{p^ p}{(p- 1)^{p-1}}]\) such that on \(\Omega\) \(\times J\) \[ (**)\quad \sum^{n}_{j=1}| \frac{\partial \beta}{\partial x_ j}(x,s)|^ p\leq \{c+as^ 2+b[\beta (x,y)]^ 2\}[\frac{\partial \beta}{\partial s}(x,s)]^{p-1}. \] Then for every \(f\in L^ 2(\Omega)\) there exists a unique solution \(u\in D(\partial \phi)\cap D(\partial \psi)\) of (*), where \(\partial \phi\) and \(\partial \psi\) are respectively the subdifferential of proper lower semicontinuous convex functions \(\phi (u)=1/p\sum^{n}_{j=1}\int_{\Omega}| \partial u/\partial x_ j|^ pdx\) and \(\psi (u)=\int_{\Omega}\int^{u(x)}_{0}\beta (x,y)dsdx\). Note that \(\partial \phi\) is strictly accretive (monotone). For the case in which \(\Omega ={\mathbb{R}}^ n\), \(p=2\) and \(c=0\) in (**) see part I, ibid. 59, 88-90 (1983; Zbl 0523.47036).
proper lower semicontinuous convex functions
scientific article; zbMATH DE number 3967221