Nonlinear elliptic equations with singular nonlinearities

Article type: Research Article

Authors: De Cave, Linda Maria

Affiliations: Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy. E-mail: linda.decave@sbai.uniroma1.it

Abstract: In this paper we study nonlinear elliptic boundary value problems with singular nonlinearities whose simplest example is −div (|∇u|p−2∇u)=f/uγ in Ω, u=0 on ∂Ω, where Ω is a bounded open set in RN (N≥2), γ>0, 1<p<N, 0≤f∈Lm(Ω), m≥1. The main difficulty is due to the right hand side f(x)/uγ, since u=0 on the boundary. In order to overcome this “obstacle”, we approach our above model problem thanks to the smooth Dirichlet problems un∈W1,p0(Ω): −div (|∇un|p−2∇un)=min(f(x),n)/(|un|+1/n)γ and we prove that there exists a solution u as limit (in a suitable topology) of the sequence {un}. To be more precise, we prove that the above model problem has a suitable solution u for every f in L1(Ω) and for every γ>0 and how the regularity of u depends on the summability of f, on p and on γ.

Keywords: nonlinear elliptic equations, singular elliptic equations, quasilinear elliptic equations with p-Laplacian

DOI: 10.3233/ASY-131173

Journal: Asymptotic Analysis, vol. 84, no. 3-4, pp. 181-195, 2013