Salomón Alarcón

Existence and non-existence of solutions to elliptic equations with a general convection term

In this paper we consider the nonlinear elliptic problem −Δ u + αu = g (∣∇ u ∣) + λh ( x ) in Ω , u = 0 on ∂Ω , where Ω is a smooth bounded domain of ℝ N , α ≥ 0, g is an arbitrary C 1 increasing function and h ∈ C 1 ( ) is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ . We show that there exist solutions for every λ when α = 0 and the integral 1/ g ( s )d s = ∞, or when α > 0 and the integral s / g ( s )d s = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω , existence depends on the size of λ . Moreover, non-existence holds for large λ . Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ .

Double-spike solutions for a critical inhomogeneous elliptic problem in domains with small holes

In this paper we construct solutions which develop two negative spikes as

Keller–Osserman type conditions for some elliptic problems with gradient terms

\varepsilon\to0^+

Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation

for the problem

Liouville Type Theorems for Elliptic Equations with Gradient Terms

-\Delta u=|u|^{4/(N-2)}u+\varepsilon f(x)

Nonexistence of positive supersolutions to some nonlinear elliptic problems

in

Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian

\varOmega

Large number of fast decay ground states to Matukuma-type equations

,

Nonexistence results for elliptic equations with gradient terms

u=0

The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations: the power like case

on