The fundamental solution of nonlinear equations with natural growth terms

Jaye, Benjamin J.; Verbitsky, Igor E.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 93-139.

We find bilateral global bounds for the fundamental solutions associated with some quasilinear and fully nonlinear operators perturbed by a nonnegative zero order term with natural growth under minimal assumptions. Important model problems involve the equations $-{\Delta }_{p}u=\sigma {\left|u\right|}^{p-2}u+{\delta }_{x_0}$, for $p>1$, and $F_k(-u)=\sigma {\left|u\right|}^{k-1}u+{\delta }_{x_0}$, for $k\ge 1$. Here ${\Delta }_p$ and $F_k$ are the $p$-Laplace and $k$-Hessian operators respectively, and $\sigma$ is an arbitrary positive measurable function (or measure). We will in addition consider the Sobolev regularity of the fundamental solution away from its pole.

Classification : 42B37, 31C45, 35J92, 42B25

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