On the growth of positive entire solutions of elliptic PDEs and their gradients
Antonio Vitolo
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1335-1346 doi: 10.3934/dcdss.2014.7.1335
We investigate the growth of entire positive functions $u(x)$ and their gradients $Du$ in Sobolev spaces when a polynomial growth is assumed for their image $Lu$ through a linear second-order uniform elliptic operator $L$. In particular, under suitable assumptions on the coefficients, we show that if $Lu$ is bounded, then $u(x)$ may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations.
keywords: gradient estimates Entire solutions elliptic equations positive solutions viscosity solutions.
Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations
Italo Capuzzo Dolcetta Antonio Vitolo
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 539-557 doi: 10.3934/dcds.2010.28.539
In this paper we discuss some extensions to a fully nonlinear setting of results by Y.Y. Li and L. Nirenberg [25] about gradient estimates for non-negative solutions of linear elliptic equations. Our approach relies heavily on methods developed by L. Caffarelli in [3] and [4].
keywords: Maximum Principles Gradient estimates Elliptic equations Viscosity solutions.
$H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains
Antonio Vitolo
Communications on Pure & Applied Analysis 2011, 10(5): 1315-1329 doi: 10.3934/cpaa.2011.10.1315
In this paper we consider estimates of the Raleigh quotient and in general of the $H^{1,p}$-eigenvalue in quasicylindrical domains. Then we apply the results to obtain, by variational methods, existence and uniqueness of weak solutions of the Dirichlet problem for second-order elliptic equations in divergent form. For such solutions global boundedness estimates have been also established.
keywords: eigenvalues Elliptic equations Dirichlet problem pointwise estimates.
On the uniqueness of blow-up solutions of fully nonlinear elliptic equations
Antonio Vitolo Maria E. Amendola Giulio Galise
Conference Publications 2013, 2013(special): 771-780 doi: 10.3934/proc.2013.2013.771
This paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.
keywords: Elliptic equations viscosity solutions blow-up. fully nonlinear equations maximum principle

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