# American Institute of Mathematical Sciences

September  2012, 17(6): 1729-1750. doi: 10.3934/dcdsb.2012.17.1729

## Existence and compactness for weak solutions to Bellman systems with critical growth

 1 Ashbel Smith Professor, The University of Texas at Dallas, Chair Professor of Risk and Decision Analysis, The Hong Kong Polytechnic University, WCU Distinguished Professor, Ajou University, 800 W. Campbell Rd, SM30, Richardson,TX 75080-3021, United States 2 Mathematical Institute, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic 3 Institute for Applied Mathematics, Department of Applied Analysis, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received  May 2011 Revised  August 2011 Published  May 2012

We deal with nonlinear elliptic and parabolic systems that are the Bellman systems associated to stochastic differential games as a main motivation. We establish the existence of weak solutions in any dimension for an arbitrary number of equations ("players"). The method is based on using a renormalized sub- and super-solution technique. The main novelty consists in the new structure conditions on the critical growth terms with allow us to show weak solvability for Bellman systems to certain classes of stochastic differential games.
Citation: Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729
##### References:
 [1] A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23-67. [2] A. Bensoussan and J. Frehse, $C^\alpha$-regularity results for quasilinear parabolic systems, Comment. Math. Univ. Carolin., 31 (1990), 453-474. [3] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002. [4] A. Bensoussan and J. Frehse, Smooth solutions of systems of quasilinear parabolic equations, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 169-193 (electronic). [5] A. Bensoussan and J. Frehse, Systems of Bellman equations to stochastic differential games with discount control, Boll. Unione Mat. Ital. (9), 1 (2008), 663-681. [6] A. Bensoussan and J. Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling, Rend. Mat. Appl. (7), 29 (2009), 1-16. [7] A. Bensoussan, J. Frehse and J. Vogelgesang, On a class of nonlinear elliptic systems with applications to Stackelberg and Nash differential games, Chin. Ann. Math., to appear, 2010. [8] A. Bensoussan, J. Frehse and J. Vogelgesang, Systems of Bellman equations to stochastic differential games with non-compact coupling, Discrete Contin. Dyn. Syst., 27 (2010), 1375-1389. doi: 10.3934/dcds.2010.27.1375. [9] A. Bensoussan and J.-L. Lions, "Impulse Control and Quasivariational Inequalities," $\mu$, Gauthier-Villars, Montrouge, Heyden & Son, Inc., Philadelphia, PA, 1984. [10] L. Boccardo, The Fatou lemma approach to the existence in quasilinear elliptic equations with natural growth terms, Complex Var. Elliptic Equ., 55 (2010), 445-453. doi: 10.1080/17476930903276241. [11] M. Bulíček and J. Frehse, On nonlinear elliptic Bellman systems for a class of stochastic differential games in arbitrary dimension, Math. Models Methods Appl. Sci., 21 (2011), 215-240. doi: 10.1142/S0218202511005027. [12] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. [13] J. Frehse, A discontinuous solution of a mildly nonlinear elliptic system, Math. Z., 134 (1973), 229-230. doi: 10.1007/BF01214096. [14] J. Frehse, Existence and perturbation theorems for nonlinear elliptic systems, in "Nonlinear Partial Differential Equations and their Applications," Collège de France Seminar, Vol. IV (Paris, 1981/1982), Res. Notes in Math., 84, Pitman, Boston, MA, (1983), 87-111. [15] J. Frehse, A refinement of Rellich's theorem, Rend. Mat. (7), 5 (1985), 229-242. [16] J. Frehse, Remarks on diagonal elliptic systems, in "Partial Differential Equations and Calculus of Variations," Lecture Notes in Math., 1357, Springer, Berlin, (1988), 198-210. [17] J. Frehse, Bellman systems of stochastic differential games with three players, in "Optimal Control and Partial Differential Equation," Conference, (2001), 3-22. [18] A. Friedman, "Stochastic Differential Equations and Applications," Vol. 2, Probability and Mathematical Statistics, Vol. 28, Academic Press [Harcourt Brace Jovanovich Publishers], New York-London, 1976. [19] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [20] S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings, in "Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations," Vol. 1, 2, 3 (Beijing, 1980), Science Press, Beijing, (1982), 481-615. [21] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. [22] O. A. Ladyžhenskaya and N. N. Ural'ceva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. [23] R. Landes, On the existence of weak solutions of perturbated systems with critical growth, J. Reine Angew. Math., 393 (1989), 21-38. [24] F. Murat, L'injection du cône positif de $H^-1$ dans $W^{-1,q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. [25] W. von Wahl and M. Wiegner, Über die Hölderstetigkeit schwacher Lösungen semilinearer elliptischer Systeme mit einseitiger Bedingung, Manuscripta Math., 19 (1976), 385-399. doi: 10.1007/BF01278926. [26] M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme, Math. Z., 147 (1976), 21-28. [27] M. Wiegner, "Das Existenz- und Regularitätsproblem bei Systemen nichtlinearer elliptischer Differentialgleichungen," Habilitation thesis, University of Bochum, 1977.

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##### References:
 [1] A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory, J. Reine Angew. Math., 350 (1984), 23-67. [2] A. Bensoussan and J. Frehse, $C^\alpha$-regularity results for quasilinear parabolic systems, Comment. Math. Univ. Carolin., 31 (1990), 453-474. [3] A. Bensoussan and J. Frehse, "Regularity Results for Nonlinear Elliptic Systems and Applications," Applied Mathematical Sciences, 151, Springer-Verlag, Berlin, 2002. [4] A. Bensoussan and J. Frehse, Smooth solutions of systems of quasilinear parabolic equations, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 169-193 (electronic). [5] A. Bensoussan and J. Frehse, Systems of Bellman equations to stochastic differential games with discount control, Boll. Unione Mat. Ital. (9), 1 (2008), 663-681. [6] A. Bensoussan and J. Frehse, Diagonal elliptic Bellman systems to stochastic differential games with discount control and noncompact coupling, Rend. Mat. Appl. (7), 29 (2009), 1-16. [7] A. Bensoussan, J. Frehse and J. Vogelgesang, On a class of nonlinear elliptic systems with applications to Stackelberg and Nash differential games, Chin. Ann. Math., to appear, 2010. [8] A. Bensoussan, J. Frehse and J. Vogelgesang, Systems of Bellman equations to stochastic differential games with non-compact coupling, Discrete Contin. Dyn. Syst., 27 (2010), 1375-1389. doi: 10.3934/dcds.2010.27.1375. [9] A. Bensoussan and J.-L. Lions, "Impulse Control and Quasivariational Inequalities," $\mu$, Gauthier-Villars, Montrouge, Heyden & Son, Inc., Philadelphia, PA, 1984. [10] L. Boccardo, The Fatou lemma approach to the existence in quasilinear elliptic equations with natural growth terms, Complex Var. Elliptic Equ., 55 (2010), 445-453. doi: 10.1080/17476930903276241. [11] M. Bulíček and J. Frehse, On nonlinear elliptic Bellman systems for a class of stochastic differential games in arbitrary dimension, Math. Models Methods Appl. Sci., 21 (2011), 215-240. doi: 10.1142/S0218202511005027. [12] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. [13] J. Frehse, A discontinuous solution of a mildly nonlinear elliptic system, Math. Z., 134 (1973), 229-230. doi: 10.1007/BF01214096. [14] J. Frehse, Existence and perturbation theorems for nonlinear elliptic systems, in "Nonlinear Partial Differential Equations and their Applications," Collège de France Seminar, Vol. IV (Paris, 1981/1982), Res. Notes in Math., 84, Pitman, Boston, MA, (1983), 87-111. [15] J. Frehse, A refinement of Rellich's theorem, Rend. Mat. (7), 5 (1985), 229-242. [16] J. Frehse, Remarks on diagonal elliptic systems, in "Partial Differential Equations and Calculus of Variations," Lecture Notes in Math., 1357, Springer, Berlin, (1988), 198-210. [17] J. Frehse, Bellman systems of stochastic differential games with three players, in "Optimal Control and Partial Differential Equation," Conference, (2001), 3-22. [18] A. Friedman, "Stochastic Differential Equations and Applications," Vol. 2, Probability and Mathematical Statistics, Vol. 28, Academic Press [Harcourt Brace Jovanovich Publishers], New York-London, 1976. [19] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [20] S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings, in "Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations," Vol. 1, 2, 3 (Beijing, 1980), Science Press, Beijing, (1982), 481-615. [21] O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. [22] O. A. Ladyžhenskaya and N. N. Ural'ceva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. [23] R. Landes, On the existence of weak solutions of perturbated systems with critical growth, J. Reine Angew. Math., 393 (1989), 21-38. [24] F. Murat, L'injection du cône positif de $H^-1$ dans $W^{-1,q}$ est compacte pour tout $q < 2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. [25] W. von Wahl and M. Wiegner, Über die Hölderstetigkeit schwacher Lösungen semilinearer elliptischer Systeme mit einseitiger Bedingung, Manuscripta Math., 19 (1976), 385-399. doi: 10.1007/BF01278926. [26] M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme, Math. Z., 147 (1976), 21-28. [27] M. Wiegner, "Das Existenz- und Regularitätsproblem bei Systemen nichtlinearer elliptischer Differentialgleichungen," Habilitation thesis, University of Bochum, 1977.
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