Lucio Damascelli

Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle

Abstract In this paper we study the positive solutions of the equation −Δu + λu = f(u) in a bounded symmetric domain Ω in R N, with the boundary condition u = 0 on ∂Ω. Using the maximum principle we prove the symmetry of the solutions v of the linearized problem. From this we deduce several properties of v and u; in particular we show that if N = 2 there cannot exist two solutions which have the same maximum if f is also convex and that there exists only one solution if f(u) = up and λ = 0. In the final section we consider the problem −Δu = uP + μuq in Ω with u = 0 on ∂Ω, and show that if 1 N+2 N−2 , q ϵ]0,1[ there are exactly two positive solutions for μ sufficiently small and some particular domain Ω.

Some nonexistence results for positive solutions of elliptic equations in unbounded domains

We prove some Liouville type theorems for positive solutions of semilinear elliptic equations in the whole space

Symmetry Results for Cooperative Elliptic Systems via Linearization

\mathbb{R}^N

Symmetry of Ground States of p-Laplace Equations via the Moving Plane Method

,

Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method

N\geq 3

Monotonicity and symmetry results for

, and in the half space

p

\mathbb{R}^N_{+}

-Laplace equations and applications

with different boundary conditions, using the technique based on the Kelvin transform and the Alexandrov-Serrin method of moving hyperplanes. In particular we get new nonexistence results for elliptic problems in half spaces satisfying mixed (Dirichlet-Neumann) boundary conditions.

A strong maximum principle for a class of non-positone singular elliptic problems

In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in an annulus in

Symmetry results for cooperative elliptic systems in unbounded domains

\mathbb{R}^N