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arXiv · 1801.06659

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

Giulio Galise
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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.

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Giulio Galise. 2018-08-20. On positive solutions of fully nonlinear degenerate Lane-Emden type equations. https://doi.org/10.1016/j.jde.2018.08.014

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