# 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 & Continuous Dynamical Systems - A, 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
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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