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

By Haydar Abdel Hamid, Marie-Franccoise Bidaut-V'eron. 2008

52 citations recorded by the source corpus.

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \] involves a source gradient term with natural growth, where $\beta$ is nonnegative, $\lambda>0,f(x)\geqq0$, and $\alpha$ is a nonnegative measure. The second one, of the form \[ -\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}+\mu, \] presents a source term of order $0, $where $g$ is nondecreasing, and $\mu$ is a nonnegative measure. Here $\beta$ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dcfaa10a4965825c839c2ef6105c2d710c52eaaa

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient (2010, methodology): “1), with H being for instance in the form H(x, s, ξ) = β(s)|ξ|(2) for some real function β, we refer to Boccardo, Gallouët, and Murat [9], as well as to [23], [3], [1, 2].”
Morrey global bounds and quasilinear Riccati type equations below the natural exponent (2014, background): “2) in the natural growth case q = p, see for example [1, 4, 5, 12, 13, 16, 28, 29, 35, 36, 43, 44, 45, 49, 57].”