On positive solutions of fully nonlinear degenerate Lane-Emden type equations

TL;DR

This paper establishes existence and uniqueness of positive viscosity solutions for fully nonlinear degenerate elliptic equations involving truncated Laplacians, identifying critical exponents that determine solution existence.

Contribution

It introduces new results on positive solutions for degenerate elliptic equations with truncated Laplacians, including explicit critical exponents for solution existence.

Findings

Existence and uniqueness of positive viscosity solutions proven.

Critical exponent for solution existence explicitly identified.

New phenomena observed compared to semilinear cases.

Abstract

We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either $\mathcal{P}^-_{k}(D^2u)$ or $\mathcal{P}^+_{k}(D^2u)$, some sort of \lq\lq truncated Laplacians\rq\rq, given respectively by the smallest and the largest partial sum of $k$ eigenvalues of the Hessian matrix. New phenomena with respect to the semilinear case occur. Moreover, for $\mathcal{P}^-_{k}$, we explicitely find the critical exponent $p$ of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero Dirichlet boundary condition.