# American Institute of Mathematical Sciences

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

 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.
Citation: Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. [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. [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. [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. [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. [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. [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. [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. [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. [27] B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. [28] B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] S. Koike, Perron's method for Lp-viscosity solutions, Saitama Math. J., 23 (2005), 9-28. [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. [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. [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. [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. [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. [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. [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. [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. [27] B. Sirakov, Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient, preprint, arXiv: 1509.04495. [28] B. Sirakov, A new method of proving a priori bounds for superlinear elliptic PDE, preprint, arXiv: 1904.03245. [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. [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.
Illustration of Theorem 2.4
Illustration of Theorem 2.5 for $\mu_2 h\gneqq 0$ small in $L^p$-norm
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