# American Institute of Mathematical Sciences

June  2016, 9(3): 869-893. doi: 10.3934/dcdss.2016033

## On nonlinear and quasiliniear elliptic functional differential equations

 1 Central Economics and Mathematical Institute, Russian Academie of Science, Nakhimovskii pr. 47, Moscow, 117418, Russian Federation

Received  March 2015 Revised  October 2015 Published  April 2016

We consider nonlinear elliptic functional differential equations. The corresponding operator has the form of a product of nonlinear elliptic differential mapping and linear difference mapping. It were obtained sufficient conditions for solvability of the Dirichlet problem. A concrete example shows that a nonlinear differential--difference operator may not be strongly elliptic even if the nonlinear differential operator is strongly elliptic and the linear difference operator is positive definite. The analysis is based on the theory of pseudomonotone--type operators and linear theory of elliptic functional differential operators.
Citation: Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033
##### References:
 [1] H. Brésis, Équations et inéquations non linéaires dans les espaces vectoriels en dualitè, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280. [2] F. E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc., 76 (1970), 999-1005. doi: 10.1090/S0002-9904-1970-12530-7. [3] H. Gajewski, K. Gróger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie-Verlag, Berlin, 1974. [4] Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 475-554. doi: 10.1007/BF01089137. [5] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271-310. doi: 10.1007/BF02392210. [6] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964. [7] G. I. Laptev, The first boundary problem for second-order quasilinear elliptic equations with double degeneration, Differential Equations, 30 (1994), 1057-1068. [8] J.-L. Lions, Quelques Methodes de Resolution de Problemes Aux Limities Non Lineaires, Dunod, Paris, 1969. [9] S. I. Pokhozhaev, Solvability of nonlinear equations with odd operators, Funkcional. Anal. i Prilo\v zen, 1 (1967), 66-73. [10] A. V. Razgulin, Rotational multi-petal waves in optical systems with 2-D feedback, Chaos in Optics. Proc. SPIE, ed. R.Roy, 2039 (1993), 342-352. [11] I. V. Skrypnik, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 555-629. [12] A. L. Skubachevskii, The first boundary value problem for strongly elliptic differential-difference equations, J. Differential Equations, 63 (1986), 332-361. doi: 10.1016/0022-0396(86)90060-4. [13] A. L. Skubachevskii., Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, 91, Birkhäuser, Basel-Boston-Berlin, 1997. [14] A. L. Skubachevskii, Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics, Nonlinear Anal., 32 (1998), 261-278. doi: 10.1016/S0362-546X(97)00476-8. [15] O. V. Solonukha, On a class of essentially nonlinear elliptic differential-difference equations, Proc. of the Steklov Inst. of Math., 283 (2013), 226-244. doi: 10.1134/S0081543813080154. [16] M. I. Vishik and O. A. Ladyzhenskaya, Boundary value problem for partial differential equations and certain classes of operator equations, American Mathematical Society Translations, Ser. 2, 10 (1958), 223-281.

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##### References:
 [1] H. Brésis, Équations et inéquations non linéaires dans les espaces vectoriels en dualitè, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175. doi: 10.5802/aif.280. [2] F. E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc., 76 (1970), 999-1005. doi: 10.1090/S0002-9904-1970-12530-7. [3] H. Gajewski, K. Gróger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie-Verlag, Berlin, 1974. [4] Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 475-554. doi: 10.1007/BF01089137. [5] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271-310. doi: 10.1007/BF02392210. [6] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964. [7] G. I. Laptev, The first boundary problem for second-order quasilinear elliptic equations with double degeneration, Differential Equations, 30 (1994), 1057-1068. [8] J.-L. Lions, Quelques Methodes de Resolution de Problemes Aux Limities Non Lineaires, Dunod, Paris, 1969. [9] S. I. Pokhozhaev, Solvability of nonlinear equations with odd operators, Funkcional. Anal. i Prilo\v zen, 1 (1967), 66-73. [10] A. V. Razgulin, Rotational multi-petal waves in optical systems with 2-D feedback, Chaos in Optics. Proc. SPIE, ed. R.Roy, 2039 (1993), 342-352. [11] I. V. Skrypnik, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 555-629. [12] A. L. Skubachevskii, The first boundary value problem for strongly elliptic differential-difference equations, J. Differential Equations, 63 (1986), 332-361. doi: 10.1016/0022-0396(86)90060-4. [13] A. L. Skubachevskii., Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, 91, Birkhäuser, Basel-Boston-Berlin, 1997. [14] A. L. Skubachevskii, Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics, Nonlinear Anal., 32 (1998), 261-278. doi: 10.1016/S0362-546X(97)00476-8. [15] O. V. Solonukha, On a class of essentially nonlinear elliptic differential-difference equations, Proc. of the Steklov Inst. of Math., 283 (2013), 226-244. doi: 10.1134/S0081543813080154. [16] M. I. Vishik and O. A. Ladyzhenskaya, Boundary value problem for partial differential equations and certain classes of operator equations, American Mathematical Society Translations, Ser. 2, 10 (1958), 223-281.
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