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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 |
$ -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), $ |
$ i = 1, \cdots, n $ |
$ C^{1, 1} $ |
$ \Omega\subset \mathbb{R}^N $ |
$ n\geq 1 $ |
$ \lambda \in \mathbb{R} $ |
$ c_{ij}, \, h_i \in L^\infty(\Omega) $ |
$ c_{ij}\geq 0 $ |
$ M_i $ |
$ 0<\mu_1 I\leq M_i\leq \mu_2 I $ |
$ F_i $ |
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. |
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