Uniqueness of entire solutions to quasilinear equations of $ p $-Laplace type

Nguyen Cong Phuc; Igor E. Verbitsky; Nguyen Cong Phuc; Igor E. Verbitsky

doi:10.3934/mine.2023068

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-33. doi: 10.3934/mine.2023068

Previous Article Next Article

Research article Special Issues

Uniqueness of entire solutions to quasilinear equations of $ p $-Laplace type

Nguyen Cong Phuc ^{1
,
,},
Igor E. Verbitsky ²

1.
Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA
2.
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received: 25 August 2022 Revised: 22 November 2022 Accepted: 24 November 2022 Published: 08 December 2022

We prove the uniqueness property for a class of entire solutions to the equation

$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $

where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,

$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $

in the sub-natural growth case $ 0 < q < p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.
- quasilinear equations,
- p-Laplace,
- entire solutions,
- uniqueness of solutions,
- sub-natural growth
Citation: Nguyen Cong Phuc, Igor E. Verbitsky. Uniqueness of entire solutions to quasilinear equations of $ p $-Laplace type[J]. Mathematics in Engineering, 2023, 5(3): 1-33. doi: 10.3934/mine.2023068

Related Papers:

Abstract

We prove the uniqueness property for a class of entire solutions to the equation

$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $

where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,

$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $

in the sub-natural growth case $ 0 < q < p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.

References

[1]	D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Berlin, Heidelberg: Springer, 1996. https://doi.org/10.1007/978-3-662-03282-4
[2]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vázquez, An $L^1$ theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 22 (1995), 241–273.
[3]	M.-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation with absorption or source term and measure data, Adv. Nonlinear Stud., 3 (2003), 25–63. https://doi.org/10.1515/ans-2003-0102 doi: 10.1515/ans-2003-0102
[4]	L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149–169. https://doi.org/10.1016/0022-1236(89)90005-0 doi: 10.1016/0022-1236(89)90005-0
[5]	L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right hand side measures, Commun. Part. Diff. Eq., 17 (1992), 641–655. https://doi.org/10.1080/03605309208820857 doi: 10.1080/03605309208820857
[6]	L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539–551. https://doi.org/10.1016/S0294-1449(16)30113-5 doi: 10.1016/S0294-1449(16)30113-5
[7]	H. Brezis, S. Kamin, Sublinear elliptic equations in $\mathbb{R}^n$, Manuscripta Math., 74 (1992), 87–106. https://doi.org/10.1007/BF02567660 doi: 10.1007/BF02567660
[8]	D. T. Cao, I. E. Verbitsky, Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms, Calc. Var., 52 (2015), 529–546. https://doi.org/10.1007/s00526-014-0722-0 doi: 10.1007/s00526-014-0722-0
[9]	D. T. Cao, I. E. Verbitsky, Nonlinear elliptic equations and intrinsic potentials of Wolff type, J. Funct. Anal., 272 (2017), 112–165. https://doi.org/10.1016/j.jfa.2016.10.010 doi: 10.1016/j.jfa.2016.10.010
[10]	G. Dal Maso, A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 25 (1997), 375–396.
[11]	A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the H-convergence of quasi-linear parabolic equations, Annali di Matematica Pura ed Applicata, 170 (1996), 207–240. https://doi.org/10.1007/BF01758989 doi: 10.1007/BF01758989
[12]	G. Dal Maso, F. Murat, A. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 28 (1999), 741–808.
[13]	H. Dong, H. Zhu, Gradient estimates for singular $p$-Laplace type equations with measure data, arXiv: 2102.08584.
[14]	F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961–2998. https://doi.org/10.1016/j.jfa.2010.08.006 doi: 10.1016/j.jfa.2010.08.006
[15]	F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093–1149. https://doi.org/10.1353/ajm.2011.0023 doi: 10.1353/ajm.2011.0023
[16]	E. Giusti, Direct methods in the calculus of variations, River Edge, NJ: World Scientific, 2003. https://doi.org/10.1142/5002
[17]	L. I. Hedberg, T. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier, 33 (1983), 161–187. https://doi.org/10.5802/aif.944 doi: 10.5802/aif.944
[18]	J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford: Oxford Univ. Press, 1993.
[19]	T. Kilpeläinen, T. Kuusi, A. Tuhola-Kujanpää, Superharmonic functions are locally renormalized solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 775–795. https://doi.org/10.1016/j.anihpc.2011.03.004 doi: 10.1016/j.anihpc.2011.03.004
[20]	T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 19 (1992), 591–613.
[21]	T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137–161. https://doi.org/10.1007/BF02392793 doi: 10.1007/BF02392793
[22]	T. Kilpeläinen, X. Xu, On the uniqueness problem for quasilinear elliptic equations involving measures, Rev. Mat. Iberoam., 12 (1996), 461–475. https://doi.org/10.4171/RMI/204 doi: 10.4171/RMI/204
[23]	M. A. Krasnoselskii, Positive solutions of operator equations, Groningen: P. Noordhoff, 1964.
[24]	T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., 207 (2013), 215–246. https://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
[25]	A. Malusa, M. M. Porzio, Renormalized solutions to elliptic equations with measure data in unbounded domains, Nonlinear Anal., 67 (2007), 2370–2389. https://doi.org/10.1016/j.na.2006.09.007 doi: 10.1016/j.na.2006.09.007
[26]	V. G. Maz'ya, Sobolev spaces, with applications to elliptic partial differential equations, 2 Eds., Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-15564-2
[27]	G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (5), 6 (2007), 195–261.
[28]	Q.-H. Nguyen, N. C. Phuc, Good-$\lambda$ and Muckenhoupt-Wheeden type bounds in quasilinear measure datum problems, with applications, Math. Ann., 374 (2019), 67–98. https://doi.org/10.1007/s00208-018-1744-2 doi: 10.1007/s00208-018-1744-2
[29]	Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equation with measure data, J. Funct. Anal., 278 (2020), 108391. https://doi.org/10.1016/j.jfa.2019.108391 doi: 10.1016/j.jfa.2019.108391
[30]	Q.-H. Nguyen, N. C. Phuc, Existence and regularity estimates for quasilinear equations with measure data: the case $1 < p \leq \frac{ 3n-2}{2n-1}$, Analysis & PDE, in press.
[31]	Q.-H. Nguyen, N. C. Phuc, A comparison estimate for singular p-Laplace equations and its consequences, arXiv: 2202.11318.
[32]	N. C. Phuc, A sublinear Sobolev inequality for $p$-superharmonic functions, Proc. Amer. Math. Soc., 145 (2017), 327–334. https://doi.org/10.1090/proc/13322 doi: 10.1090/proc/13322
[33]	N. C. Phuc, I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. Math., 168 (2008), 859–914. https://doi.org/10.4007/annals.2008.168.859 doi: 10.4007/annals.2008.168.859
[34]	N. C. Phuc, I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., 256 (2009), 1875–1906. https://doi.org/10.1016/j.jfa.2009.01.012 doi: 10.1016/j.jfa.2009.01.012
[35]	A. Seesanea, I. E. Verbitsky, Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms, Adv. Calc. Var., 13 (2020), 53–74. https://doi.org/10.1515/acv-2017-0035 doi: 10.1515/acv-2017-0035
[36]	N. S. Trudinger, X. J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369–410. https://doi.org/10.1353/ajm.2002.0012 doi: 10.1353/ajm.2002.0012
[37]	I. E. Verbitsky, Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms, Adv. Calc. Var., in press. https://doi.org/10.1515/acv-2021-0004
[38]	I. E. Verbitsky, Global pointwise estimates of positive solutions to sublinear equations, St. Petersburg Math. J., in press.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)