Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations

Liouville type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −aij(x)∂iju(x)− qi(x)∂iu(x) = f(x, u(x)), x ∈ R . The aim is to prove uniqueness of positive bounded solutions Liouville type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the line...

Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation ut = ∆u + |u|p−1u. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence ...

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma devel...