A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

Gabrielle Nornberg; Delia Schiera; Boyan Sirakov

doi:10.3934/dcds.2020128

Previous Article
Convergence and structure theorems for order-preserving dynamical systems with mass conservation
DCDS Home
This Issue
Next Article
Indefinite nonlinear diffusion problem in population genetics

June 2020, 40(6): 3857-3881. doi: 10.3934/dcds.2020128

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

Gabrielle Nornberg ^1, , Delia Schiera ^2, and Boyan Sirakov ^3,,

1.	Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense 400, 13566-590 São Carlos - SP, Brazil
2.	Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy
3.	Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, 22451-900 Gávea - Rio de Janeiro, Brazil

^* Corresponding author

Dedicated to Professor Wei-Ming Ni with admiration

Received May 2019 Revised October 2019 Published February 2020

Fund Project: G. Nornberg was supported by Fapesp grant 2018/04000-9, São Paulo Research Foundation

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as

$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle = \lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $

for

$ i = 1, \cdots, n $

, in a bounded

$ C^{1, 1} $

domain

$ \Omega\subset \mathbb{R}^N $

with Dirichlet boundary conditions; here

$ n\geq 1 $

$ \lambda \in \mathbb{R} $

$ c_{ij}, \, h_i \in L^\infty(\Omega) $

$ c_{ij}\geq 0 $

$ M_i $

satisfies

$ 0<\mu_1 I\leq M_i\leq \mu_2 I $

, and

$ F_i $

is an uniformly elliptic Isaacs operator.

We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

Keywords: A priori estimates, elliptic system, multiplicity, existence and nonexistence, fully nonlinear operators.

Mathematics Subject Classification: Primary: 35J47, 35J60, 35J66; Secondary: 35A01, 35A16, 35P30.

Citation: Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

References:

[1]	D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka, Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335. doi: 10.1016/j.jfa.2015.01.014. Google Scholar
[2]	S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987. doi: 10.1016/j.jde.2008.10.026. Google Scholar
[3]	C. Bandle and W. Reichel, Solutions of quasilinear second-order elliptic boundary value problems via degree theory, in Handbook of Differential Equations (eds. M. Chipot and P. Quittner), Stationary Partial Differential Equations, vol.1. Elsevier, NorthHolland, Amsterdam, (2004), 1–70. Google Scholar
[4]	G. Barles, A. Blanc, C. Georgelin and M. Kobylanski, Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Sc. Norm. Sup. Pisa, 28 (1999), 381-404. Google Scholar
[5]	G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rat. Mech. Anal., 133 (1995), 77-101. doi: 10.1007/BF00375351. Google Scholar
[6]	H. Berestycki, L. Nirenberg and S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105. Google Scholar
[7]	L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa, 11 (1984), 213-235. Google Scholar
[8]	L. Boccardo, F. Murat and J.P. Puel, Existence de solutions faibles des équations elliptiques quasi-lineaires à croissance quadratique, in: Nonlinear P.D.E. and Their Applications (eds. H. Brézis and J.L. Lions), Collège de France Seminar, vol. IV, Research Notes in Mathematics, Pitman, London, 84 (1983), 19–73. Google Scholar
[9]	J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. I. H. Poincaré, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001. Google Scholar
[10]	L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. Google Scholar
[11]	L. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. Google Scholar
[12]	C. De Coster and L. Jeanjean, Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient, J. Differential Equations, 262 (2017), 5231-5270. doi: 10.1016/j.jde.2017.01.022. Google Scholar
[13]	M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar
[14]	P. Felmer, A. Quaas and B. Sirakov, Resonance phenomena for second-order stochastic control equations, SIAM Journal on Mathematical Analysis, 42 (2010), 997-1024. doi: 10.1137/080744268. Google Scholar
[15]	V. Ferone and F. Murat, Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonl. Anal., 42 (2000), 1309-1326. doi: 10.1016/S0362-546X(99)00165-0. Google Scholar
[16]	L. Jeanjean and B. Sirakov, Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264. doi: 10.1080/03605302.2012.738754. Google Scholar
[17]	J. L. Kazdan and R. J. Kramer, Invariant criteria for existence of solutions to second-order quasi-linear elliptic equations, Comm. Pure Appl. Math., 31 (1978), 619-645. doi: 10.1002/cpa.3160310505. Google Scholar
[18]	S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. Google Scholar
[19]	S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan., 61 (2009), 723-755. doi: 10.2969/jmsj/06130723. Google Scholar
[20]	S. Koike and A. Świech, Local maximum principle for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Commun. Pure Appl. Anal., 11 (2012), 1897-1910. doi: 10.3934/cpaa.2012.11.1897. Google Scholar
[21]	G. Nornberg and B. Sirakov, A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852. doi: 10.1016/j.jfa.2018.06.017. Google Scholar
[22]	G. Nornberg, $C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures et Appl., 128 (2019), 297-329. doi: 10.1016/j.matpur.2019.06.008. Google Scholar
[23]	G. Nornberg, Methods of the Regularity Theory in the Study of Partial Differential Equations with Natural Growth in the Gradient, Ph.D. thesis, PUC-Rio, 2018. doi: 10.17771/PUCRio.acad.36015. Google Scholar
[24]	A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002. Google Scholar
[25]	B. Sirakov, Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE, International Mathematics Research Notices, 2018 (2018), 7457-7482. doi: 10.1093/imrn/rnx107. Google Scholar
[26]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Archive for Rational Mechanics and Analysis, 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9. Google Scholar
[27]	B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. Google Scholar
[28]	B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. Google Scholar
[29]	P. Souplet, A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient, Nonlinear Anal., 121 (2015), 412-423. doi: 10.1016/j.na.2015.02.005. Google Scholar
[30]	N. Winter, $W^{2, p}$ and $W^{1, p}$ estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377. Google Scholar

show all references

References:

[1]	D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka, Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335. doi: 10.1016/j.jfa.2015.01.014. Google Scholar
[2]	S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations, 246 (2009), 2958-2987. doi: 10.1016/j.jde.2008.10.026. Google Scholar
[3]	C. Bandle and W. Reichel, Solutions of quasilinear second-order elliptic boundary value problems via degree theory, in Handbook of Differential Equations (eds. M. Chipot and P. Quittner), Stationary Partial Differential Equations, vol.1. Elsevier, NorthHolland, Amsterdam, (2004), 1–70. Google Scholar
[4]	G. Barles, A. Blanc, C. Georgelin and M. Kobylanski, Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Sc. Norm. Sup. Pisa, 28 (1999), 381-404. Google Scholar
[5]	G. Barles and F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rat. Mech. Anal., 133 (1995), 77-101. doi: 10.1007/BF00375351. Google Scholar
[6]	H. Berestycki, L. Nirenberg and S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105. Google Scholar
[7]	L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Sup. Pisa, 11 (1984), 213-235. Google Scholar
[8]	L. Boccardo, F. Murat and J.P. Puel, Existence de solutions faibles des équations elliptiques quasi-lineaires à croissance quadratique, in: Nonlinear P.D.E. and Their Applications (eds. H. Brézis and J.L. Lions), Collège de France Seminar, vol. IV, Research Notes in Mathematics, Pitman, London, 84 (1983), 19–73. Google Scholar
[9]	J. Busca and B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. I. H. Poincaré, 21 (2004), 543-590. doi: 10.1016/j.anihpc.2003.06.001. Google Scholar
[10]	L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. Google Scholar
[11]	L. Caffarelli, M. G. Crandall, M. Kocan and A. Świech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. Google Scholar
[12]	C. De Coster and L. Jeanjean, Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient, J. Differential Equations, 262 (2017), 5231-5270. doi: 10.1016/j.jde.2017.01.022. Google Scholar
[13]	M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar
[14]	P. Felmer, A. Quaas and B. Sirakov, Resonance phenomena for second-order stochastic control equations, SIAM Journal on Mathematical Analysis, 42 (2010), 997-1024. doi: 10.1137/080744268. Google Scholar
[15]	V. Ferone and F. Murat, Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonl. Anal., 42 (2000), 1309-1326. doi: 10.1016/S0362-546X(99)00165-0. Google Scholar
[16]	L. Jeanjean and B. Sirakov, Existence and multiplicity for elliptic problems with quadratic growth in the gradient, Comm. Part. Diff. Eq., 38 (2013), 244-264. doi: 10.1080/03605302.2012.738754. Google Scholar
[17]	J. L. Kazdan and R. J. Kramer, Invariant criteria for existence of solutions to second-order quasi-linear elliptic equations, Comm. Pure Appl. Math., 31 (1978), 619-645. doi: 10.1002/cpa.3160310505. Google Scholar
[18]	S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. Google Scholar
[19]	S. Koike and A. Świech, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan., 61 (2009), 723-755. doi: 10.2969/jmsj/06130723. Google Scholar
[20]	S. Koike and A. Świech, Local maximum principle for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Commun. Pure Appl. Anal., 11 (2012), 1897-1910. doi: 10.3934/cpaa.2012.11.1897. Google Scholar
[21]	G. Nornberg and B. Sirakov, A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852. doi: 10.1016/j.jfa.2018.06.017. Google Scholar
[22]	G. Nornberg, $C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures et Appl., 128 (2019), 297-329. doi: 10.1016/j.matpur.2019.06.008. Google Scholar
[23]	G. Nornberg, Methods of the Regularity Theory in the Study of Partial Differential Equations with Natural Growth in the Gradient, Ph.D. thesis, PUC-Rio, 2018. doi: 10.17771/PUCRio.acad.36015. Google Scholar
[24]	A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135. doi: 10.1016/j.aim.2007.12.002. Google Scholar
[25]	B. Sirakov, Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE, International Mathematics Research Notices, 2018 (2018), 7457-7482. doi: 10.1093/imrn/rnx107. Google Scholar
[26]	B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Archive for Rational Mechanics and Analysis, 195 (2010), 579-607. doi: 10.1007/s00205-009-0218-9. Google Scholar
[27]	B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. Google Scholar
[28]	B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. Google Scholar
[29]	P. Souplet, A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient, Nonlinear Anal., 121 (2015), 412-423. doi: 10.1016/j.na.2015.02.005. Google Scholar
[30]	N. Winter, $W^{2, p}$ and $W^{1, p}$ estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377. Google Scholar

Figure 1. Illustration of Theorem 2.4

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Illustration of Theorem 2.5 for $ \mu_2 h\gneqq 0 $ small in $ L^p $-norm

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158
[2]	Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393
[3]	Yayun Li, Yutian Lei. On existence and nonexistence of positive solutions of an elliptic system with coupled terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1749-1764. doi: 10.3934/cpaa.2018083
[4]	Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[5]	Monica Lazzo. Existence and multiplicity results for a class of nonlinear elliptic problems in $\mathbb(R)^N$. Conference Publications, 2003, 2003 (Special) : 526-535. doi: 10.3934/proc.2003.2003.526
[6]	Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[7]	Qianzhong Ou. Nonexistence results for a fully nonlinear evolution inequality. Electronic Research Announcements, 2016, 23: 19-24. doi: 10.3934/era.2016.23.003
[8]	Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[9]	Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166
[10]	D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[11]	Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[12]	Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148
[13]	Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[14]	Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020436
[15]	Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152
[16]	Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845
[17]	Isabeau Birindelli, Francoise Demengel. The dirichlet problem for singluar fully nonlinear operators. Conference Publications, 2007, 2007 (Special) : 110-121. doi: 10.3934/proc.2007.2007.110
[18]	Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
[19]	Xiaohui Yu. Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros. Communications on Pure & Applied Analysis, 2013, 12 (1) : 451-459. doi: 10.3934/cpaa.2013.12.451
[20]	Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167

2020 Impact Factor: 1.392

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.392

Tools

Figures and Tables

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors