MULTIPLE SOLUTIONS FOR DIRICHLET PROBLEMS WHICH ARE SUPERLINEAR AT +∞ AND (SUB-)LINEAR AT−∞

TitleMULTIPLE SOLUTIONS FOR DIRICHLET PROBLEMS WHICH ARE SUPERLINEAR AT +∞ AND (SUB-)LINEAR AT−∞
Publication TypeJournal Article
Year of Publication2009
AuthorsMOTREANU, D, MOTREANU, V, PAPAGEORGIOU, N
Volume13
Issue3
Start Page341
Pagination17
Date Published2009
ISSN1083-2564
AMS35J25, 35J80, 58E05
Abstract

We consider a semilinear Dirichlet elliptic problem with a right-hand side nonlinearity which exhibits an asymmetric growth near +∞ and near −∞. Namely, it is (sub-)linear near −∞ and superlinear near +∞. However, it need not satisfy the Ambrosetti–Rabinowitz condition on the positive semiaxis. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.

URLhttp://www.acadsol.eu/en/articles/13/3/6.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term
asymptotically linear at −∞ and superlinear at +∞, Math. Z. 219 (1995), 499–513.
[2] T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications
to problems with resonance, Nonlinear Anal. 28 (1997), 419–441.
[3] H. Brezis and L. Nirenberg, H1 versus C 1 local minimizers, C. R. Acad. Sci. Paris S´er. I Math. 317 (1993), 465–472.
[4] N. P. C´ac, On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Differential Equations 80 (1989), 379–404.
[5] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhauser,
Boston, 1993.
[6] D. G. Costa and C. A. Magalh˜aes, Variational elliptic problems which are nonquadratic at
infinity, Nonlinear Anal. 23 (1994), 1401–1412.
[7] E. N. Dancer and Z. Zhang, Fuˇc´ık spectrum, sign-changing, and multiple solutions for semilinear
elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl. 250 (2000), 449–464.
[8] G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations 2002, No. 8, 12 pp.
[9] D. G. de Figueiredo and B. Ruf, B. On a superlinear Sturm-Liouville equation and a related
bouncing problem, J. Reine Angew. Math. 421 (1991), 1–22.
[10] L. Gasi´nski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.
[11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
[12] C. A. Magalh˜aes, Multiplicity results for a semilinear elliptic problem with crossing of multiple
eigenvalues, Differential Integral Equations 4 (1991), 129–136.
[13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.

[14] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, A degree theoretic approach for multiple
solutions of constant sign for nonlinear elliptic equations, Manuscripta Math. 124 (2007), 507– 531.
[15] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions
for asymptotically linear, noncoercive elliptic equations, Monatsh. Math., in press.
[16] F. O. de Paiva, Multiple solutions for a class of quasilinear problems, Discrete Contin. Dyn.
Syst. 15 (2006), 669–680.
[17] R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1–16.
[18] K. Perera, Existence and multiplicity results for a Sturm-Liouville equation asymptotically
linear at −∞ and superlinear at +∞, Nonlinear Anal. 39 (2000), 669–684.
[19] K. Perera, Critical groups of critical points produced by local linking with applications, Abstr. Appl. Anal. 3 (1998), 437–446.
[20] M. Schechter, The Fuˇc´ık spectrum, Indiana Univ. Math. J. 43 (1994), 1139–1157.
[21] J. L. V´azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.