Existence of positive solutions for singular Dirichlet problems. (English) Zbl 1021.35031

Hirano, Norimichi; Shioji, Naoki

Differ. Integral Equ. 14, No. 12, 1531-1540 (2001).

Summary: We study the existence of positive solutions for singular Dirichlet problems. By the variational method, we show that if \(\lambda_i\) is contained in some interval, then a singular Dirichlet problem has a radially symmetric, positive solution on an annulus \(\Omega\), where \(\lambda_1\) is the first eigenvalue of \(-\Delta\) on \(\Omega\) with homogeneous Dirichlet boundary condition. Since we consider a problem with singularity, we have difficulties that either the functional \(I\) associated with the problem is not differentiable even in the sense of Gâteaux or the strong maximum principle is not applicable. But we show that a function which attains some minimax value for \(I\) is a solution. We also treat the case that \(\Omega\) is a ball.

MSC:

35J25

Boundary value problems for second-order elliptic equations

Keywords:

variational method; singularity