Existence of radial solutions to biharmonic k-Hessian equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Escudero, Carlos¹  Torres, Pedro J.² 

[1] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain

[2] Univ Granada, Dept Matemat Aplicada, E-18071 Granada, Spain

关键词: k-Hessian type equations;    Biharrnonic boundary value problems;    Existence of solutions;    Non-existence of solutions;   

DOI  :  10.1016/j.jde.2015.04.001

来源: Elsevier

【 摘 要 】

This work presents the construction of the existence theory of radial solutions to the elliptic equation Delta(2)u = (-1)S-k(k)[u] + lambda f(x), x is an element of B-1(0) subset of R-N, provided either with Dirichlet boundary conditions u = partial derivative(n)u = 0, x is an element of partial derivative B-1(0), or Navier boundary conditions u = Delta u = 0, x is an element of partial derivative B-1(0), where the k-Hessian S-k[u] is the k-th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f is an element of L-1 (B-1(0)) while lambda is an element of R. We prove the existence of a Caratheodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided \lambda\ is small enough. Moreover, we prove that the solvability set of lambda is finite, giving an explicity bound of the extreme value. (C) 2015 Elsevier Inc. All rights reserved.

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