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

On nonlinear and quasiliniear elliptic functional differential equations

Olesya V. Solonukha 1,

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.
Keywords: ellipticity condition, Pseudomonotone operator, strongly ellipticity algebraic condition., strongly elliptic operator.
Mathematics Subject Classification: Primary: 34K10, 35J25; Secondary: 35J4.
Citation: Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 475-554. doi: 10.1007/BF01089137.  Google Scholar

[5]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271-310. doi: 10.1007/BF02392210.  Google Scholar

[6]

M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964.  Google Scholar

[7]

G. I. Laptev, The first boundary problem for second-order quasilinear elliptic equations with double degeneration, Differential Equations, 30 (1994), 1057-1068.  Google Scholar

[8]

J.-L. Lions, Quelques Methodes de Resolution de Problemes Aux Limities Non Lineaires, Dunod, Paris, 1969. Google Scholar

[9]

S. I. Pokhozhaev, Solvability of nonlinear equations with odd operators, Funkcional. Anal. i Prilo\v zen, 1 (1967), 66-73.  Google Scholar

[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. Google Scholar

[11]

I. V. Skrypnik, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 555-629. Google Scholar

[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.  Google Scholar

[13]

A. L. Skubachevskii., Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, 91, Birkhäuser, Basel-Boston-Berlin, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 475-554. doi: 10.1007/BF01089137.  Google Scholar

[5]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966), 271-310. doi: 10.1007/BF02392210.  Google Scholar

[6]

M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964.  Google Scholar

[7]

G. I. Laptev, The first boundary problem for second-order quasilinear elliptic equations with double degeneration, Differential Equations, 30 (1994), 1057-1068.  Google Scholar

[8]

J.-L. Lions, Quelques Methodes de Resolution de Problemes Aux Limities Non Lineaires, Dunod, Paris, 1969. Google Scholar

[9]

S. I. Pokhozhaev, Solvability of nonlinear equations with odd operators, Funkcional. Anal. i Prilo\v zen, 1 (1967), 66-73.  Google Scholar

[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. Google Scholar

[11]

I. V. Skrypnik, Nonlinear elliptic and parabolic equations, J. Sov. Math., 12 (1979), 555-629. Google Scholar

[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.  Google Scholar

[13]

A. L. Skubachevskii., Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, 91, Birkhäuser, Basel-Boston-Berlin, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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