On the connection between two quasilinear elliptic problems with source terms of order 0 or 1

2008/11/20 by Haydar Abdel Hamid, Hamid, Haydar Abdel, Marie-Françoise Bidaut-Véron +1 · 1 citation
Mathematics · #35J65 #35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J65 #msc:35J70

paper · pdf · doi:10.48550/arxiv.0811.3292

arxiv created 2008/11/20 · arxiv updated 2009/12/01

Abstract

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ℝ^N. The first one, of the form -Δ_pu=β(u)| ∇ u| ^p+λf(x)+α, involves a source gradient term with natural growth, where β is nonnegative, λ>0,f(x)\geqq0, and α is a nonnegative measure. The second one, of the form -Δ_pv=λf(x)(1+g(v))^p-1+μ, presents a source term of order 0, where g is nondecreasing, and μ is a nonnegative measure. Here β and g can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.

Cited by

• A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general Hénon-type superquadratic and gradient growth

Related

• Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient
• Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data
• Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities
• Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term
• A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms