Ester Giarrusso

On Blow Up Solutions of a Quasilinear Elliptic Equation

Given a bounded regular domain Ω in ℝN, we study existence and asymptotic behaviour of the solutions of the equation Δu + |Du|q = f(u) in Ω, which diverge on ∂Ω. We extend and complete some results contained in [4].

Asymptotic behaviour of large solutions of an elliptic quasilinear equation in a borderline case

Given a bounded smooth domain Ω in RN, we study the asymptotic behaviour close to the boundary ∂Ω of the large solutions of the equation Δu−|Du|q=f(u), where 1<q<2 and f(u)uq/(q−2) converges to a positive number, as u tends to ∞. Existence and asymptotic behaviour of large solutions of this equation are studied also in [2] for a general f(u). However, the assumptions considered in [2] do not apply to the case studied in this Note. As a consequence of the asymptotic behaviour we also show an uniqueness result.

Estimates for solutions of elliptic equations in a limit case

Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝ n . We prove, that if the right hand side of the equation is an element of H −1, n (Ω), then u belongs to the Orlicz space L Φ where Φ( t ) = exp(| t | n /( n −1) ) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.

Estimates for Blow-Up Solutions to Nonlinear Elliptic Equations with p-Growth in the Gradient

In this paper we deal with blow-up solutions to p-Laplacian equations with a nonlinear gradient term. We prove comparison results for the solutions in terms of the solutions to suitable symmetrized problems defined in a ball. We analyze two cases where the form of the term which depends on the gradient plays different roles.

On a method of K. Uhlenbeck for proving partial regularity for solutions of certain nonlinear elliptic systems

We extend a method of K. Uhlenbeck for proving the partial regularity of solutions of nonlinear elliptic systems. We do not employ “reverse Hölder” type inequalities.