Existence of strong solutions to some quasilinear elliptic problems on bounded smooth domains (Q698548)

The authors study the existence of strong solutions to the following problems: \[ \begin{cases} \sum\limits_{i,j=1}^{N}a_{ij}(x,u)\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}}+\sum\limits_{i=1}^{N}b_{i}(x,u)\frac{\partial u}{ \partial x_{i}}+c(x,u)u=f(x)&\text{ \;in } \Omega , \\ u=0&\text{ in }\partial \Omega , \end{cases} \tag{1} \] and \[ \begin{cases} \sum\limits_{i,j=1}^{N}a_{ij}(x,u)\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}}+\sum\limits_{i=1}^{N}b_{i}(x,u)\frac{\partial u}{ \partial x_{i}}+c(x,u)u=f(x,u,\nabla u)&\text{ \;in }\Omega , \\ u=0&\text{ in }\partial \Omega . \end{cases}\tag{2} \] Let \(\Omega \) be a bounded smooth domain of \(\mathbb{R}^{N},N\geq 3,\) \(p>N,\) all the coefficients \(a_{ij},b_{i},c\) are Carathéodory functions and \( f\in L^{p}(\Omega ),\) then\ the authors prove that, if \(a_{ij}\in C^{0,1}(\Omega \times \mathbb{R)}\), \(a_{ij},\frac{\partial a_{ij}}{\partial x_{i}}, \frac{\partial a_{ij}}{\partial r},b_{i},c\in L^{\infty }(\Omega \times \mathbb{R}),c\leq 0\) for \(i,j=1,\cdots ,N,\) and the oscillations of \( a_{ij}(x,r)\) with respect to \(r\) are sufficiently small, the problem (1) has a strong solution \(u\in W^{2,p}(\Omega )\cap W_{0}^{1,p}(\Omega ).\) With the help of the preceding result the authors obtain the same result for the problem (2) if \(-c\geq \alpha >0\), the function \(f(x,r,\xi )\) is a Carathéodory function and satisfies \[ f(x,r,\xi)\leq C_{0}+h(r)\xi^{\theta }, \] where \(C_{0}\geq 0,h\) is a locally bounded function and \(0\leq \theta \leq 2. \)