The impact of a lower order term in a Dirichlet problem with a singular nonlinearity

Abstract

In this paper we study the existence and regularity of solutions to the following Dirichlet problem        −div(a(x)|∇u| p−2 ∇u) + u|u| r−1 = f (x) u θ in Ω, u > 0 in Ω, u = 0 on ∂Ω proving that the lower order term u|u| r−1 has some regularizing effects on the solutions.

Nonlinear Elliptic Equations With Variable Exponents Involving Singular Nonlinearity

“…We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in [2].…”

“…For costant-exponent cases (i.e., p(x) = p, r(x) = r and γ(x) = γ), the existence and regularity of solutions to problem (1.1) are studied in [2,14,28,29]. They proved that the solution was in W 1,q 0 (Ω) and u r+γ belongs to L 1 (Ω), where q = pr p+1−γ .…”

A singular elliptic equation and a related functional

“…In order to avoid the use of strong maximum principle and monotonicity assumption on g(x, •), a new definition of nonnegative solutions have been provided and existence, stability and uniqueness results for these notions are proved in [23,24,25]. Further contributions to semilinear elliptic equations having this type of singularity are contained for example in [1,2,3,5,14,15,16,28,29].…”

Regularizing effect of absorption terms in singular and degenerate elliptic problems

“…Let us briefly recall the mathematical framework concerning problem (1.1) we start with the case a(x, u) := a(x), θ = 0 and f lies just in L 1 (Ω) has been studied in [2].…”