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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonlinear equations
and weighted norm inequalities


Authors: N. J. Kalton and I. E. Verbitsky
Journal: Trans. Amer. Math. Soc. 351 (1999), 3441-3497
MSC (1991): Primary 35J60, 42B25, 47H15; Secondary 31B15
Published electronically: April 27, 1999
MathSciNet review: 1475688
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Abstract: We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem

\begin{equation*}\begin{split} -& \Delta u = v \, u^{q} + w, \quad u \ge 0 \quad \text{on} \quad \Omega , \\ &u = 0 \quad \text{on} \quad \partial \Omega , \end{split}\end{equation*}

on a regular domain $\Omega $ in $\mathbf{R}^{n}$ in the ``superlinear case'' $q > 1$. The coefficients $v, w$ are arbitrary positive measurable functions (or measures) on $\Omega $. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities.

Our characterizations of the existence of positive solutions take into account the interplay between $v$, $w$, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on $v$ and $w$; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v \equiv 1$ and $\Omega $ is a ball or half-space.

The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called $3 G$-inequality by an elementary ``integration by parts'' argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.


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Additional Information

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: nigel@math.missouri.edu

I. E. Verbitsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: igor@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02215-1
PII: S 0002-9947(99)02215-1
Received by editor(s): March 1, 1997
Received by editor(s) in revised form: August 15, 1997
Published electronically: April 27, 1999
Additional Notes: The first author was partially supported by NSF grant DMS-9500125, and the second by NSF grant DMS-9401493 and the University of Missouri Research Board grant RB-96029.
Article copyright: © Copyright 1999 American Mathematical Society