## Existence of strong solutions to some quasilinear elliptic problems on bounded smooth domains.(English)Zbl 1064.35045

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 $$f\in L^{p}(\Omega ),$$ thenthe 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 $$0\leq \theta \leq 2.$$
Reviewer: C. Bouzar (Oran)

### MSC:

 35J60 Nonlinear elliptic equations 35B50 Maximum principles in context of PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35D05 Existence of generalized solutions of PDE (MSC2000)

### Keywords:

quasilinear elliptic problem; strong solution.
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