Semilinear equations of second order with nonnegative characteristic form (Q582442)

The authors study qualitative properties of solutions of semilinear equations of the type \(Lu=f(x,u)\), where \(L=(\partial /\partial x_ i)(a_{ij}(x)\partial /\partial x_ j),\) \(a_{ij}\in L^{\infty}({\mathbb{R}}^ n)\), \(a_{ij}\xi_ i\xi_ j\geq 0\), and f: \({\mathbb{R}}^{n+1}\to {\mathbb{R}}\) is locally bounded, \(f(x,0)=0\), \(uf(x,u)\geq | u|^{2+q},\) \(q>0.\) Let u be a weak, locally bounded solution of this equation in an open set \(G\subset {\mathbb{R}}^ n\), let \(B(0,m)=\{x\in {\mathbb{R}}^ n;\quad | x| <m\}\) and let either u vanish on \(\partial G\cap B(0,m)\) \((m>1+1/q\) or \(m=2)\) or \(G=B(0,2)\setminus K\), where \(K\subset B(0,1)\) is a compact set of Hausdorff dimension less than n-2. Then the authors derive \(L^ p\)-estimates for u restricted to \(G\cap B(0,1)\). Moreover, in the latter case, they prove also that under some additional assumptions on f and q the function u solves the equation in B(0,2). Some of these results were already announced by the authors [Usp. Mat. Nauk 42, No.5, 233-234 (1987)].