Pavol Quittner

Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems

In this paper, we study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. Namely, we develop a general method for derivation of universal, pointwise a priori estimates of local solutions from Liouville-type theorems, which provides a simpler and unified treatment for such questions. The method is based on rescaling arguments combined with a key “doubling” property, and it is different from the classical rescaling method of Gidas and Spruck. As an important heuristic consequence of our approach, it turns out that universal boundedness theorems for local solutions and Liouville-type theorems are essentially equivalent. ∗Supported in part by NSF Grant DMS-0400702 †Supported in part by VEGA Grant 1/3021/06

Semilinear parabolic equations involving measures and low regularity data

A detailed study of abstract semilinear evolution equations of the form u + Au = μ(u) is undertaken, where -A generates an analytic semigroup and μ(u) is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝN, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions.

SYMMETRY OF COMPONENTS FOR SEMILINEAR ELLIPTIC SYSTEMS

In this paper, we give sufficient conditions ensuring that any positive classical solution

Large Time Behavior of Solutions of a Semilinear Parabolic Equation with a Nonlinear Dynamical Boundary Condition

(u,v)

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

of an elliptic system in the whole space

Universal bound for global positive solutions of a superlinear parabolic problem

\mathbb{R}^n

Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure

has the symmetry property

A nodal theorem for coupled systems of Schrödinger equations and the number of bound states

u=v

The Quenching Problem on the N-dimensional Ball

. As an application, we significantly improve the results of Li and Ma [SIAM J. Math. Anal., 40 (2008), pp. 1049--1057] on the classification of solutions of Sobolev-critical elliptic systems of Schrodinger type. Our techniques apply to some supercritical problems as well. We also obtain new Liouville-type theorems for noncooperative systems. Moreover, we provide some counterexamples which indicate that our assumptions are in a sense necessary. Our proofs are based on suitable maximum principle arguments, combined with properties of spherical means of superharmonic functions and on some appropriate auxiliary functions.