Matteo Franca

On a diffusion model with absorption and production

Abstract We discuss the structure of radial solutions of some superlinear elliptic equations which model diffusion phenomena when both absorption and production are present. We focus our attention on solutions defined in R (regular) or in R ∖ { 0 } (singular) which are infinitesimal at infinity, discussing also their asymptotic behavior. The phenomena we find are present only if absorption and production coexist, i.e., if the reaction term changes sign. Our results are then generalized to include the case where Hardy potentials are considered.

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to

Radial ground states and singular ground states for a spatial-dependent p-Laplace equation

\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}

Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type

Δu+h(|x|)|x|2u+f(u,|x|)=0where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.