Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners

Let Ω⊂Rn be a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=up in Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis–Turner exponent p*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented.