On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points (Q2735887)

The paper is devoted to the behavior of solutions of the Dirichlet problem near the conical boundary point \(\{0\}\) for the equation \(-\text{div}( |\nabla u|^{m-2}\nabla u)= -a_0(x) u|u|^{q-1}+ f(x)\) with zero boundary function and where the functions \(a_0(x)\), \(f(x)\) satisfy the inequalities \(0\leq a_0(x)\leq a_1= \text{const} |f(x)|\leq c|x|^\beta\), \(\beta> -m\). It is proved that if \(1< m\leq n\), \(q> 0\), then \(|u(x)|\leq c_0 |x|^{\lambda_0}\) for \(\lambda_0< \frac{\beta+m} {m-1}\) and \(|u(x)|\leq c_0|x|^{\frac{\beta+m}{m-1}}\) for \(\lambda_0> \frac{\beta+m}{m-1}\). Here \(\lambda_0\) is the first eigenvalue of the associated eigenvalue problem on the intersection of the conical domain with the unit sphere.