Transactions of the American Mathematical Society

Author(s): N. J. Kalton; I. E. Verbitsky
Journal: Trans. Amer. Math. Soc. 351 (1999), 3441-3497.
MSC (1991): Primary 35J60, 42B25, 47H15; Secondary 31B15
Posted: April 27, 1999
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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''

. The coefficients

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

, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on

and

; 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

-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.

1.: D. R. Adams, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), 73-94. MR 88m:31011
2.: D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1996. MR 97j:46024
3.: D. R. Adams and M. Pierre, Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier, Grenoble 41 (1991), 117-135. MR 92m:35074
4.: H. Aikawa and M. Essén, Topics in Potential Theory, Lecture Notes, Uppsala Univ., no. 1, 1994.
5.: A. Ancona, Comparison of Green's functions for elliptic operators on manifolds or domains, Preprint 96-11, Université de Paris-Sud, 1996, pp. 1-42.
6.: P. Baras and M. Pierre, Critère d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré, Analyse Non Linéaire 2 (1985), 185-212. MR 87j:45032
7.: R. F. Bass, Probabilistic Techniques in Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1995. MR 96e:60001
8.: H. Brezis and L. Nirenberg, Positive solutions of nonlinear equations involving critical Sobolev exponents, Comm. Pure. Appl. Math. 36 (1983), 437-477. MR 84h:35059
9.: H. Brezis and X. Cabre, Some simple nonlinear PDE's without solutions, preprint (1997), 1-37; Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 223-262.
10.: L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Math. Studies, no. 13, Van Nostrand, Princeton, 1967. MR 37:1576
11.: R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 56:6264
12.: M. Cranston, E. Fabes and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), 171-194. MR 90a:60135
13.: K. L. Chung and Z. Zhao, From Brownian motion to Schrödinger's equation, Springer-Verlag, Berlin-Heidelberg-New York, 1995. MR 96f:60140
14.: E. B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), 1185-1262. MR 94j:60156
15.: E. B. Dynkin and S. E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math. 49 (1996), 125-176. MR 97m:60114
16.: C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206.
17.: M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS-AMS Regional Conf. Series, vol. 79, 1991.MR 92m:42021
18.: J. Garcia-Cuerva and J.-L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Studies, vol. 116, North-Holland, Amsterdam, 1985. MR 87d:42023
19.: J. Garnett, Bounded Analytic Functions, Academic Press, New York-London-Toronto, 1981.
20.: A. E. Gatto, C. Segovia and S. Vági, On fractional differentiation on spaces of homogeneous type, Revista Mat. Iberoamer. 12 (1996), 1-35. MR 97k:42028
21.: K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77-102. MR 81j:31007
22.: K. Hansson, V. G. Mazya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati's equations, to appear in Arkiv för Matem.
23.: L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), 161-187. MR 85f:31015
24.: J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford-New York-Tokyo, 1993. MR 94e:31003
25.: H. Hueber and M. Sieveking, Uniform bounds for quotients of Green functions on $C^{1, 1}$ -domains, Ann. Inst. Fourier, Grenoble 32 (1982), 105-117. MR 84a:35063
26.: N. J. Kalton and L. Tzafriri, Legendre and Jacobi polynomials in $L_{p}$ spaces and the $\Lambda _{p}$ problem, in preparation.
27.: R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-228. MR 88b:35150
28.: J. Komlos, A generalization of a problem of Steinhaus, Acta Math. Sci. Hungar. 18 (1967), 217-229. MR 35:1071
29.: M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 85b:47057
30.: J.-F. Le Gall, A class of path-valued Markov processes and its applications to superprocesses, Probab. Theory Related Fields 95 (1993), 25-46. MR 94f:60093
31.: J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II, Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR 81c:46001
32.: P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24 (1982), 441-467. MR 84a:35093
33.: B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans un espaces $L^{p}$ , Astérisque, vol. 11, Soc. Math. de France, Paris, 1974. MR 49:9670
34.: V. G. Mazya, On the theory of the $n$ -dimensional Schrödinger operator, Izv. Akad. Nauk SSSR, Ser. Matem. 28 (1964), 1145-1172.
35.: V. G. Mazya, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985. MR 87g:46056
36.: V. G. Mazya, Beurling's theorem on a minimum principle for positive harmonic functions, Zapiski Nauchn. Sem. LOMI 30 (1972), 76-90; English transl. in. J. Soviet Math. 4 (1972), 367-379.
37.: V. G. Mazya and I. E. Verbitsky, Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Arkiv för Matem. 33 (1995), 81-115. MR 96i:26021
38.: L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble 7 (1957), 183-281.MR 20:6608
39.: E. M. Nikishin, Resonance theorems and superlinear operators, Russian Math. Surveys 25 (1970), 124-187.
40.: N. K. Nikolskii, Treatise on the Shift Operator, Springer-Verlag, Berlin-Heidelberg-New York, 1986. MR 87i:47042
41.: C. Pérez, Two weighted norm inequalities for Riesz potentials and uniform $L_{p}$ -weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31-44. MR 92a:42024
42.: J. L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Harmonic Analysis, Proc., Minneapolis, 1981; , Lecture Notes in Math. 908 (1982), 86-101. MR 83h:42025
43.: E. T. Sawyer, Two weight norm inequalities for fractional and Poisson integrals, Harmonic Analysis, Proc., Minneapolis, 1981; Lecture Notes in Math. 908 (1982), 102-127.MR 83k:42020b
44.: E. T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, Trans. Amer. Math. Soc. 308 (1988), 533-545.MR 89d:26009
45.: E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. MR 94i:42024
46.: E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Analysis 5 (1996), 523-580. MR 98g:42026
47.: M. Schechter, Superlinear elliptic boundary value problems, Manuscripta Math. 86 (1995), 253-267. MR 96e:35052
48.: J. Serrin, On the Harnack inequality for linear elliptic equations, J. d'Analyse Math. 4 (1956), 292-308. MR 18:398f
49.: G. Sweers, Positivity for a strongly coupled elliptic system, J. Geom. Anal. 4 (1994), 121-142. MR 95g:35052
50.: S. Treil and A. Volberg, Weighted imbedding and weighted norm inequalities for the Hilbert transform and maximal operators, St. Petersburg Math. J. 7 (1996), 1017-1032. MR 97c:42017
51.: B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Linköping Studies in Sci. and Techn., Dissert. 387, Linköping University, 1995.
52.: I. E. Verbitsky and R. L. Wheeden, Weighted inequalities for fractional integrals and applications to semilinear equations, J. Funct. Anal. 129 (1995), 221-241. MR 95m:42025
53.: I. E. Verbitsky and R. L. Wheeden, Weighted norm inequalities for integral operators, Trans. Amer. Math. Soc. 350 (1998), 3371-3391. MR 98k:42020
54.: R. L. Wheeden and S. Zhao, Weak type estimates for operators of potential type, Studia Math. 119 (1996), 149-160. MR 97d:42013
55.: K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradients of solutions of elliptic differential equations, Math. Scand. 21 (1967), 13-67. MR 39:621
56.: Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), 309-334. MR 88f:60142
57.: W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. MR 91e:46046

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35J60, 42B25, 47H15, 31B15

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: 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
Posted: 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.
Copyright of article: Copyright 1999, American Mathematical Society

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