Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

Abstract

In this paper we study the existence of bounded weak solutions in unbounded domains for some nonlinear Dirichlet problems. The principal part of the operator behaves like the $p$-laplacian operator, and the lower order terms, which depend on the solution $u$ and its gradient $\D u$, have a power growth of order $p-1$ with respect to these variables, while they are bounded in the $x$ variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity

A-priori bounds for quasilinear problems in critical dimension

“…Regarding quasilinear equations, similar results have been achieved for 1 < m < N by Azizieh and Clément [3], Ruiz [26], Dall'Aglio et al [8], Lorca and Ubilla [19] and Zou [29]. In these works, the nonlinearity may depend also on x and on the gradient, nonetheless its growth at infinity with respect to u should be less than a subcritical power.…”

Nonlinear elliptic equations with natural growth in general domains

“…On the other hand, results concerning sets Ω of infinite measure and terms with growth of order p − 1 with respect to the gradients have been proved in Bottaro, Marina [4], Lions [14], [15], Chicco, Venturino [6] for the linear setting ( p = 2) and in Dall'Aglio, De Cicco, Giachetti, Puel [7] for the nonlinear one.…”

Dirichlet problems with singular and gradient quadratic lower order terms

“…Contributions to the existence of solutions of nonlinear elliptic problems with lower order terms having quadratic growth with respect to the gradient, like (1.3), can be found in some papers in collaboration with F. Murat and J.-P. Puel [14][15][16] (see also [7,19,20]), where we proved existence of bounded solutions (without the assumption g(x, s, ξ) s ≥ 0 and with the assumption f ∈ L m (Ω), m > N/2). If we look for unbounded solutions, we refer to the paper [13], in collaboration with F. Murat and J.-P. Puel and to…”