An Existence Theorem for the Neumann Problem Involving the p-Laplacian

Giovanni Anello; Giuseppe Cordaro

doi:10.68381/jca1009

Abstract

We deal with the existence of weak solutions for a Neumann problem involving the p-Laplacian with asymmetric nonlinearity. We show that our result gives new conditions on the right hand side which assure the existence and localization of at least one weak solution. In particular, we make a comparison with a result obtained in a recent paper by S. Villegas where he studies the same problem using a different approach.

Author information

Contact details are reproduced from the original publication and may be historical.

Giovanni Anello

Dept. of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy

anello@dipmat.unime.it

Giuseppe Cordaro

Dept. of Mathematics, University of Messina, 98166 Sant'Agata-Messina, Italy

cordaro@dipmat.unime.it

Keywords

Variational principle
critical point
weak solution
Neumann problem
p-Laplacian

References

G. Anello, G. Cordaro: Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri, Arch. Math. (Basel), to appear.Google Scholar
D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (1977).https://doi.org/10.1007/978-3-642-96379-7 Google Scholar
V. I. Burenkov, V. A. Gusakov: On precise constants in Sobolev imbedding theorems, Soviet Math. Dokl. 35 (1987) 651–655.Google Scholar
V. I. Burenkov: Sobolev spaces on domains, B. G. Teubner, Stuttgart, Leipzig (1998) 184–185, ISBN 3-8154-2068-7.Google Scholar
B. Ricceri: A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410.https://doi.org/10.1016/S0377-0427(99)00269-1 Google Scholar
B. Ricceri: Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001) 331–340.https://doi.org/10.1017/S0024609301008001 Google Scholar
B. Ricceri: Applications de théorèmes de semi-continuité inférieure, C. R. Acad. Sci. Paris, Sér. I, 295 (1982) 265–267.Google Scholar
S. Villegas: A Neumann problem with asymmetric nonlinearity and a related minimizing problem, J. Differential Equations 145 (1998) 145–155.https://doi.org/10.1006/jdeq.1997.3396 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.

[1] G. Anello, G. Cordaro: Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri, Arch. Math. (Basel), to appear.Google Scholar

[2] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (1977).https://doi.org/10.1007/978-3-642-96379-7 Google Scholar

[3] V. I. Burenkov, V. A. Gusakov: On precise constants in Sobolev imbedding theorems, Soviet Math. Dokl. 35 (1987) 651–655.Google Scholar

[4] V. I. Burenkov: Sobolev spaces on domains, B. G. Teubner, Stuttgart, Leipzig (1998) 184–185, ISBN 3-8154-2068-7.Google Scholar

[5] B. Ricceri: A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410.https://doi.org/10.1016/S0377-0427(99)00269-1 Google Scholar

[6] B. Ricceri: Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001) 331–340.https://doi.org/10.1017/S0024609301008001 Google Scholar

[7] B. Ricceri: Applications de théorèmes de semi-continuité inférieure, C. R. Acad. Sci. Paris, Sér. I, 295 (1982) 265–267.Google Scholar

[8] S. Villegas: A Neumann problem with asymmetric nonlinearity and a related minimizing problem, J. Differential Equations 145 (1998) 145–155.https://doi.org/10.1006/jdeq.1997.3396 Google Scholar