Tommaso Leonori

Large Solutions For a Class of Nonlinear Elliptic Equations With Gradient Terms

Abstract In this paper we prove the existence, in a suitable sense, of a large solution (a solution that blows-up at the boundary) for a class of nonlinear elliptic equations whose model is -Δpu + u + u|▽u|q = f in Ω, p > 1, p - 1 < q ≤ p and f(x) ∈ L1(Ω). The main tool in order to prove it relies on approximating the problem with a more regular one, prove local (i.e. independent from the behavior on the boundary) a priori estimates and local compactness for truncations in W1,p(Ω). Such scheme of the proof is also applied in order to prove the existence of a solution for equations of the type -Δpu + u + u|▽u|q = f in ℝN, where f(x) ∈ L1loc(ℝN) without any condition at infinity. Moreover, the local summability of the solution depending on the local summability of the datum is studied.

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in

Large solutions and gradient bounds for quasilinear elliptic equations

{\mathbb{R}^N}

Existence and regularity results for some singular elliptic problems

with absorbing first order terms, whose model is

Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations

\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.

Comparison principles for p-Laplace equations with lower order terms

where