Abstract In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented Neumann boundary conditions, when source term equation belongs a Lebesgue space, under various integrability regimes. Our method is based an integral refinement Bochner identity, and leads “semilinear Calderón...

3 Semilinear equations with absorption 19 3.1 The Marcinkiewicz spaces approach . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Admissible measures and the ∆2-condition . . . . . . . . . . . . . . . . . . . 26 3.3 The duality method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Bessel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Sharp...

We discuss regularity issues for minimizers of three nonlinear elliptic problems. They concern minimal cones, minimizing harmonic maps into a hemisphere, and radial local minimizers of semilinear elliptic equations. We describe the strong analogies among the three regularity theories. They all use a method originated in a paper of J. Simons on the area minimizing properties of cones.

We study boundary blow-up solutions of semilinear elliptic equations Lu = up + with p > 1, or Lu = e with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation −∇ · (|x|∇u)−B|x|u = C|x|u in cone–like domains of R (N ≥ 2), for the full range of parameters A,B, σ, p ∈ R and C > 0. We provide a complete characterization of the set of (p, σ) ∈ R such that the equation has no positive (super-) solutions, depending on the values of A,B and the p...

Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation Lu = uα with α > 1 in a bounded smooth domain E ⊂ Rd. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation ∆u = ...

The main goal of the present paper is to define the solution operator (ξ, g) 7→ u associated to the evolution equation du = (Au)dt + dg, u(0) = ξ, where A generates a C0-semigroup in a Banach space X, ξ ∈ X, g ∈ BV ([ a, b ];X), and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution opera...