Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Abstract

This is a continuation of Ikoma and Ishii (Ann Inst H Poincaré Anal Non Linéaire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, Communicated by Neil Trudinger.

C1, regularity for fully nonlinear elliptic equations with superlinear growth in the gradient

“…If, in addition, we have W 2,p regularity of solutions, we can extend theorem 5.2 even further, allowing an unbounded first order coefficient. Eigenvalues for fully nonlinear operators with such coefficients have been previously studied, to our knowledge, only for radial operators and eigenfunctions, in [22] and [23]. As a particular case of theorem 5.2, we obtain the existence of positive eigenvalues with a nonnegative unbounded weight for the extremal Pucci's operators with unbounded coefficients.…”

Methods of the Regularity Theory in the Study of Partial Differential Equations With Natural Growth in the Gradient

“…If, in addition, we have W 2,p regularity of solutions, we can extend theorem 4.2 even further, allowing an unbounded first order coefficient. Eigenvalues for fully nonlinear operators with such coefficients have been previously studied, to our knowledge, only for radial operators and eigenfunctions, in [76] and [77]. As a particular case of theorem 4.2, we obtain the existence of positive eigenvalues with a nonnegative unbounded weight for the extremal Pucci's operators with unbounded coefficients.…”