Existence of radial solutions to biharmonic $k-$Hessian equations

TL;DR

This paper establishes the existence of radial solutions for a class of biharmonic $k$-Hessian equations with boundary conditions, showing solutions exist for small parameters and characterizing the range of solvable parameters.

Contribution

It provides the first existence theory for radial solutions to biharmonic $k$-Hessian equations with explicit bounds on the parameter range.

Findings

Existence of solutions for small $|mbda|$

Uniqueness of solutions near the origin

Finite solvability set with explicit bounds

Abstract

This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber \Delta^2 u = (-1)^k S_k[u] + \lambda f(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = \Delta u = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while $\lambda \in \mathbb{R}$. We prove the existence of a Carath\'eodory 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.