CUBO A Mathematical Journal Vol.14, N^o02, (175-182). June 2012

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On a Condition for the Nonexistence of W-Solutions of Nonlinear High-Order Equations with L¹ -Data

 

Alexander A. Kovalevsky* and Francesco Nicolosi**

Institute of Applied Mathematics and Mechanics, Rosa Luxemburg St. 74, 83114 Donetsk, Ukraine email: alexkvl@iamm.ac.donetsk. ua

Department ofMathematics and Informatics, University of Catania, 95125 Catania, Italy email: fnicolosi@dmi.unict.it

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ABSTRACT

In a bounded open set of we consider the Dirichlet problem for nonlinear order equations in divergence form with right-hand sides. It is supposed that , and the coefficients of the equations admit the growth of rate with respect to the derivatives of order m of unknown function. We establish that under the condition for some L¹ -data the corresponding Dirichlet problem does not have W-solutions.

Keywords and Phrases: Nonlinear high-order equations in divergence form, data, Dirichlet problem, W-solution, nonexistence of W-solutions.

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RESUMEN

En un conjunto abierto y acotado de consideramos el problema de Dirichlet para ecuaciones no lineales de orden en la forma divergente con lados L¹ -right-hand. Se supone que , y los coeficientes de las ecuaciones admiten el radio de crecimiento con respecto a las derivadas de orden m de la función desconocida. Establecemos que bajo la condición para algn data el problema de Dirichlet correspondiente no tiene W-soluciones.

 

2010 AMS Mathematics Subject Classification: 35G30, 35J40, 35J60.

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References

[1] Ph. Benilán, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J.L. Vazquez, An L¹ -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995) 241-273.         [ Links ]

[2] N. Dunford and J.T. Schwartz, Linear operators. Part I. General theory, John Wiley & Sons, Inc., New York, 1988.         [ Links ]

[3] A.A. Kovalevskii, Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in L¹, Izv. Math. 65 (2001) 231-283.         [ Links ]

[4] A. Kovalevsky, Entropy solutions of Dirichlet problem for a class of nonlinear elliptic high-order equations with data, Nelinejnye granichnye zadachi 12 (2002) 119-127.         [ Links ]

[5] A. Kovalevsky and F. Nicolosi, Solvability of Dirichlet problem for a class of degenerate nonlinear high-order equations with data, Nonlinear Anal. 47 (2001) 435-446.         [ Links ]

[6] A. Kovalevsky and F. Nicolosi, Existence of solutions of some degenerate nonlinear elliptic fourth-order equations with data, Appl. Anal. 81 (2002) 905-914.         [ Links ]

[7] J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969.         [ Links ]

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Received: March 2011. Revised: December 2011.