Composition operators from Sobolev spaces into Lebesgue spaces
Keywords:
Sobolev embeddings, composition operators, eigenvalues, p–Laplacian, Multiplicity of solutions
Abstract
In this paper, we shall obtain a compactness of weighted Sobolev embeddings and use it to get a composition operators from Sobolev spaces into Lebesgue spaces. Applying these results we shall study the multiplicity for singular asymptotically linear p–Laplacian problems.
References
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Z. W. Birnbaum and W. Orlicz. Über die verallgemeinerung des berriffes der zueinander konjugierten potenzen. Stu. Math., 30 (1968), 21–42.
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.
A. Anane, Etude des valeurs propres et de la résonnance pour l’opérateur p-laplacien. C. R. Ac. Sc. Paris, 305 (1987), 725–728.
J. Appell and P. Zabreiko, Nonlinear superposition operators. Cambridge University Press, 2008.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
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G. Dinca, P. Jebelean and J. Mawhin. Variational and topological methods for Dirichlet problems with p–laplacian. Portugaliae Mathematica 58 (2001), 339–378.
T. K. Donaldson and N. S. Trudinger. Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8 (1971), 52–75.
D. M. Duc and N. Q. Huy. Non-uniformly asymptotically linear p–Laplacian problems. Nonlinear Analysis 92 (2013), 183–197.
D. M. Duc. Nonlinear singular elliptic equations. J. London Math. Soc. 40(2) (1989), 420–440.
I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474.
D. G. De Figueiredo. Lectures on the Ekeland variational principle with applications and detours. Tata Institute of Fundational research, Bombay, 1989.
D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, Springer, Berlin, 2001.
B. J. Jaye, V. G. Mazya and I. E. Verbitsky. Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights, Advances in Mathematics 232 (2013), 513–542.
M. A. Kranosel’skii. Topological methods in the theory of nonlinear integral equations. Macmillan, New York, 1964.
A. Kufner. Weighted Sobolev spaces. Wiley, New York, 1985.
A. Kufner, O. John and S. Fucik. Function spaces. Noordhoff, Leyden 1977.
S. Fucik and A. Kufner. Nonlinear Differential Equations. Vol. 2, Elsevier, 1980.
M. Marcus and V. J. Mizel. Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217–229.
M. Marcus and V. J. Mizel. Complete characterization of functions which act, via superposition, on Sobolev spaces. Transactions of AMS 251 (1979), 187–218.
J. Matkwoski. Functional equation and Nemytskii operators. Funkc. Ekvac. 25 (1982) 127–132.
V. G. Mazja. Sobolev spaces, Springer, Berlin, 1985.
B. Opic and A. Kufner. Remark on compactness of imbeddings in weighted spaces. Math. Nachr. 133 (1987), 63–70.
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Series: De Gruyter Series in Nonlinear Analysis and Applications 3, 2011.
M. Struve. Variational methods. Springer, Berlin, 2008.
E. Zeidler. Nonlinear functional analysis and its applications III: Variational methods and optimization. Springer, Berlı́n, 1985.
Z. W. Birnbaum and W. Orlicz. Über die verallgemeinerung des berriffes der zueinander konjugierten potenzen. Stu. Math., 30 (1968), 21–42.
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.
A. Anane, Etude des valeurs propres et de la résonnance pour l’opérateur p-laplacien. C. R. Ac. Sc. Paris, 305 (1987), 725–728.
J. Appell and P. Zabreiko, Nonlinear superposition operators. Cambridge University Press, 2008.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
M. Cuesta. Eigenvalue problems for the p–laplacian with indefinite weights. Electronic Journal of Differential Equations. 2001(33) (2001), 1–9.
G. Dinca, P. Jebelean and J. Mawhin. Variational and topological methods for Dirichlet problems with p–laplacian. Portugaliae Mathematica 58 (2001), 339–378.
T. K. Donaldson and N. S. Trudinger. Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8 (1971), 52–75.
D. M. Duc and N. Q. Huy. Non-uniformly asymptotically linear p–Laplacian problems. Nonlinear Analysis 92 (2013), 183–197.
D. M. Duc. Nonlinear singular elliptic equations. J. London Math. Soc. 40(2) (1989), 420–440.
I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474.
D. G. De Figueiredo. Lectures on the Ekeland variational principle with applications and detours. Tata Institute of Fundational research, Bombay, 1989.
D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, Springer, Berlin, 2001.
B. J. Jaye, V. G. Mazya and I. E. Verbitsky. Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights, Advances in Mathematics 232 (2013), 513–542.
M. A. Kranosel’skii. Topological methods in the theory of nonlinear integral equations. Macmillan, New York, 1964.
A. Kufner. Weighted Sobolev spaces. Wiley, New York, 1985.
A. Kufner, O. John and S. Fucik. Function spaces. Noordhoff, Leyden 1977.
S. Fucik and A. Kufner. Nonlinear Differential Equations. Vol. 2, Elsevier, 1980.
M. Marcus and V. J. Mizel. Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217–229.
M. Marcus and V. J. Mizel. Complete characterization of functions which act, via superposition, on Sobolev spaces. Transactions of AMS 251 (1979), 187–218.
J. Matkwoski. Functional equation and Nemytskii operators. Funkc. Ekvac. 25 (1982) 127–132.
V. G. Mazja. Sobolev spaces, Springer, Berlin, 1985.
B. Opic and A. Kufner. Remark on compactness of imbeddings in weighted spaces. Math. Nachr. 133 (1987), 63–70.
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Series: De Gruyter Series in Nonlinear Analysis and Applications 3, 2011.
M. Struve. Variational methods. Springer, Berlin, 2008.
E. Zeidler. Nonlinear functional analysis and its applications III: Variational methods and optimization. Springer, Berlı́n, 1985.
Published
2019-12-29
How to Cite
Aziz, W. (2019). Composition operators from Sobolev spaces into Lebesgue spaces. Divulgaciones Matemáticas, 20(2), 45-62. Retrieved from https://mail.produccioncientificaluz.org/index.php/divulgaciones/article/view/36630
Section
Research papers
