A note on regularity and existence of solutions for a class of non- uniformly degenerate elliptic equations (Q1333880)

The existence of weak solutions and its \(C^{1 + \alpha (\delta)}\) \((\Omega_ \delta)\)-regularity, \(\Omega_ \delta : = \{x \in \Omega : \text{dist} (x, \partial \Omega) > \delta\}\), is shown for the boundary value problem \[ \text{div} A(x,Du) + B(x) = 0 \quad \text{in } \Omega \] where \(\Omega \subset R^ N\) is bounded, \(B \in L^ \infty (\Omega)\), \(A(x,p) = (A^ 1, \dots, A^ N)\), \[ \biggl\{ b(x) | \xi |^{\alpha - 1} + \bigl[ 1 - b(x) \bigr] | p |^{\beta - 1} \biggr\} | \xi |^ 2 \leq {\partial A^ i \over \partial p_ j} \xi_ i \xi_ j \leq a_ 0 \biggl[ b(x) | p |^{\alpha - 1} + | p |^{\beta -1} \biggr] | \xi |^ 2, \] \[ \bigl | A(x,p) \bigr | \leq a_ 0 \biggl[ b(x) | p |^ \alpha + | p |^ \beta \biggr], \bigl | A_ x (x,p) \bigr | \leq a_ 0 \bigl( 1 + | p |^ \beta \bigr), \] \(b \in C^ \tau (\overline \Omega)\), \(0 \leq b(x) \leq 1\), \(0 < \alpha \leq \beta < \max \{{N + 1 \over N} \alpha + {1 \over N}, \alpha + 1\}\); \(u - u_ 0 \in W^{1,(1 + \alpha) {\beta \over \alpha}}_ 0 (\Omega) \cap L^ \infty (\Omega)\), where \(u_ 0 \in W^{1,(1 + \alpha) {\beta \over \alpha}} (\Omega) \cap L^ \infty (\Omega)\) is given.