On polyharmonic regularizations of $k-$Hessian equations: Variational methods

Abstract

This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (-1)^\alpha \Delta^\alpha u = (-1)^k S_k[u] + \lambda f, \qquad x &\in& \Omega \subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u = \cdots = \partial_n^{\alpha-1} u = 0, \qquad x &\in& \partial \Omega, \end{eqnarray} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f$ obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We look for the optimal values of $\alpha \in \mathbb{N}$ that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results.