Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T05:51:55.087Z Has data issue: false hasContentIssue false

An elliptic equation with no monotonicity condition on the nonlinearity

Published online by Cambridge University Press:  11 October 2006

Gregory S. Spradlin*
Affiliation:
Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach, Florida 32114-3900, USA; [email protected]
Get access

Abstract

An elliptic PDE is studied which is a perturbation of an autonomousequation. The existence of a nontrivial solution is proven viavariational methods. The domain of the equation is unbounded, whichimposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. Inan earlier paper with this assumption, a solution was obtained using asimple application of topological (Brouwer) degree. Here, a more subtledegree theory argument must be used.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessio, F. and Montecchiari, P., Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity. Ann. Instit. Henri Poincaré 16 (1999) 107135. CrossRef
Bahri, A. and On, Y.-Y. Li a Min-Max Procedure for the Existence of a Positive Solution for a Certain Scalar Field Equation in $\mathbb{R}^N$ . Revista Iberoamericana 6 (1990) 117.
Caldiroli, P., New Proof, A of the Existence of Homoclinic Orbits for a Class of Autonomous Second Order Hamiltonian Systems in $\mathbb{R}^N$ . Math. Nachr. 187 (1997) 1927. CrossRef
Caldiroli, P. and Montecchiari, P., Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Comm. Appl. Nonlinear Anal. 1 (1994) 97129.
Coti Zelati, V., Montecchiari, P. and Nolasco, M., Multibump solutions for a class of second order, almost periodic Hamiltonian systems. Nonlinear Ord. Differ. Equ. Appl. 4 (1997) 7799. CrossRef
Coti Zelati, V. and Rabinowitz, P., Homoclinic Orbits for Second Order Hamiltonian Systems Possessing Superquadratic Potentials. J. Amer. Math. Soc. 4 (1991) 693627. CrossRef
K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, New York (1985).
Estaban, M. and Lions, P.-L., Existence and non existence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh 93 (1982) 114. CrossRef
Franchi, B., Lanconelli, E. and Serrin, J., Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations in ${\mathbf R}^N$ . Adv. Math. 118 (1996) 177243. CrossRef
Jeanjean, L. and Tanaka, K., Note, A on a Mountain Pass Characterization of Least Energy Solutions. Adv. Nonlinear Stud. 3 (2003) 445455.
Jeanjean, L. and Tanaka, K., A remark on least energy solutions in ${\mathbb R}^N$ . Proc. Amer. Math. Soc. 131 (2003) 23992408.
Lions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. Instit. Henri Poincaré 1 (1984) 102145 and 223–283.
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York (1989).
P. Rabinowitz, Homoclinic Orbits for a class of Hamiltonian Systems. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 33–38. CrossRef
P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Regional Conf. Series in Math., No. 65, Amer. Math. Soc., Providence (1986).
P. Rabinowitz, Théorie du degrée topologique et applications à des problèmes aux limites nonlineaires, University of Paris 6 Lecture notes, with notes by H. Berestycki (1975).
Spradlin, G., Existence of Solutions to a Hamiltonian System without Convexity Condition on the Nonlinearity. Electronic J. Differ. Equ. 2004 (2004) 113.
Spradlin, G., Perturbation, A of a Periodic Hamiltonian System. Nonlinear Anal. Theory Methods Appl. 38 (1999) 10031022. CrossRef
Spradlin, G., Interacting Near-Solutions of a Hamiltonian System. Calc. Var. PDE 22 (2005) 447464. CrossRef
Serra, E., Tarallo, M. and Terracini, S., On the existence of homoclinic solutions to almost periodic second order systems. Ann. Instit. Henri Poincaré 13 (1996) 783812. CrossRef
G. Whyburn, Topological Analysis. Princeton University Press (1964).