Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms
HTML articles powered by AMS MathViewer
- by A. Seesanea and I. E. Verbitsky
- St. Petersburg Math. J. 31 (2020), 557-572
- DOI: https://doi.org/10.1090/spmj/1614
- Published electronically: April 30, 2020
- PDF | Request permission
Abstract:
The paper is devoted to the existence problem for positive solutions ${u \in L^{r}(\mathbb {R}^{n})}$, $0<r<\infty$, to the quasilinear elliptic equation \begin{equation*} -\Delta _{p} u = \sigma u^{q} \ \text { in } \ \mathbb {R}^n \end{equation*} in the subnatural growth case $0<q< p-1$, where $\Delta _{p}u = \mathrm {div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $\sigma$ is a nonnegative measurable function (or measure) on $\mathbb {R}^n$.
The techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of $\Delta _{p}$ such as the $\mathcal {A}$-Laplacian $\mathrm {div} \mathcal {A}(x,\nabla u)$, or the fractional Laplacian $(-\Delta )^{\alpha }$ on $\mathbb {R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\mathrm {div} (\mathcal {A} \nabla u)$ on an arbitrary domain $\Omega \subseteq \mathbb {R}^n$ with a positive Green function.
References
- David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
- Alano Ancona, Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators, Nagoya Math. J. 165 (2002), 123–158. MR 1892102, DOI 10.1017/S0027763000008187
- L. Boccardo and L. Orsina, Sublinear equations in $L^s$, Houston J. Math. 20 (1994), no. 1, 99–114. MR 1272564
- M. Brelot, Lectures on potential theory, Lectures on Mathematics, vol. 19, Tata Institute of Fundamental Research, Bombay, 1960. Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy. MR 0118980
- Haïm Brezis and Shoshana Kamin, Sublinear elliptic equations in $\textbf {R}^n$, Manuscripta Math. 74 (1992), no. 1, 87–106. MR 1141779, DOI 10.1007/BF02567660
- Cao Tien Dat and Igor E. Verbitsky, Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 529–546. MR 3311903, DOI 10.1007/s00526-014-0722-0
- Dat Cao and Igor Verbitsky, Nonlinear elliptic equations and intrinsic potentials of Wolff type, J. Funct. Anal. 272 (2017), no. 1, 112–165. MR 3567503, DOI 10.1016/j.jfa.2016.10.010
- Dat T. Cao and Igor E. Verbitsky, Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations, Nonlinear Anal. 146 (2016), 1–19. MR 3556326, DOI 10.1016/j.na.2016.08.008
- Carme Cascante, Joaquin M. Ortega, and Igor E. Verbitsky, On $L^p$-$L^q$ trace inequalities, J. London Math. Soc. (2) 74 (2006), no. 2, 497–511. MR 2269591, DOI 10.1112/S0024610706023064
- Alexander Grigor′yan and Wolfhard Hansen, Lower estimates for a perturbed Green function, J. Anal. Math. 104 (2008), 25–58. MR 2403428, DOI 10.1007/s11854-008-0015-7
- A. Grigor′yan and I. E. Verbitsky, Pointwise estimates of solutions to nonlinear equations for nonlocal operators, Ann. Sc. Norm. Super. Pisa CL(5) 20 (2020), no. 2.
- V. G. Maz′ja and V. P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27 (1972), no. 6, 67–138. MR 0409858
- L. I. Hedberg and Th. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 4, 161–187. MR 727526
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR 2305115
- Petr Honzík and Benjamin J. Jaye, On the good-$\lambda$ inequality for nonlinear potentials, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4167–4180. MR 2957206, DOI 10.1090/S0002-9939-2012-11352-8
- B. Jawerth, C. Perez, and G. Welland, The positive cone in Triebel-Lizorkin spaces and the relation among potential and maximal operators, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988) Contemp. Math., vol. 107, Amer. Math. Soc., Providence, RI, 1990, pp. 71–91. MR 1066471, DOI 10.1090/conm/107/1066471
- Tero Kilpeläinen, Tuomo Kuusi, and Anna Tuhola-Kujanpää, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 6, 775–795. MR 2859927, DOI 10.1016/j.anihpc.2011.03.004
- Tero Kilpeläinen and Jan Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994), no. 1, 137–161. MR 1264000, DOI 10.1007/BF02392793
- Tuomo Kuusi and Giuseppe Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci. 4 (2014), no. 1, 1–82. MR 3174278, DOI 10.1007/s13373-013-0048-9
- Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. MR 1461542, DOI 10.1090/surv/051
- Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
- Stephen Quinn and Igor E. Verbitsky, A sublinear version of Schur’s lemma and elliptic PDE, Anal. PDE 11 (2018), no. 2, 439–466. MR 3724493, DOI 10.2140/apde.2018.11.439
- Adisak Seesanea and Igor E. Verbitsky, Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms, Adv. Calc. Var. 13 (2020), no. 1, 53–74. MR 4048382, DOI 10.1515/acv-2017-0035
- Adisak Seesanea and Igor E. Verbitsky, Solutions to sublinear elliptic equations with finite generalized energy, Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 6, 21. MR 3881877, DOI 10.1007/s00526-018-1448-1
- Neil S. Trudinger and Xu-Jia Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), no. 2, 369–410. MR 1890997
- Igor E. Verbitsky, Sublinear equations and Schur’s test for integral operators, 50 years with Hardy spaces, Oper. Theory Adv. Appl., vol. 261, Birkhäuser/Springer, Cham, 2018, pp. 467–484. MR 3792109
- —, Wolff’s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations, Nonlinear Anal. (in press); doi:10.1016/j.na.2019.04.015.
Bibliographic Information
- A. Seesanea
- Affiliation: Department of Mathematics, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan
- Email: seesanea@math.sci.hokudai.ac.jp
- I. E. Verbitsky
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: verbitskyi@missouri.edu
- Received by editor(s): November 1, 2018
- Published electronically: April 30, 2020
- Additional Notes: A. S. is partially supported by JSPS KAKENHI Grant no. 17H01092
- © Copyright 2020 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 557-572
- MSC (2010): Primary 35J92; Secondary 35J20, 42B37
- DOI: https://doi.org/10.1090/spmj/1614
- MathSciNet review: 3985926
Dedicated: Dedicated to the memory of S. G. Mikhlin