An elliptic equation with no monotonicity condition on the nonlinearity

Spradlin, Gregory S.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 786-794.

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

Classification : 35J20, 35J60
Mots clés : mountain-pass theorem, variational methods, Nehari manifold, Brouwer degree, concentration-compactness

[1] F. Alessio and P. Montecchiari, Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity. Ann. Instit. Henri Poincaré 16 (1999) 107-135. | Numdam | Zbl

[2] A. Bahri and Y.-Y. Li, On a Min-Max Procedure for the Existence of a Positive Solution for a Certain Scalar Field Equation in ${ℝ}^N$. Revista Iberoamericana 6 (1990) 1-17. | Zbl

[3] P. Caldiroli, A New Proof of the Existence of Homoclinic Orbits for a Class of Autonomous Second Order Hamiltonian Systems in ${ℝ}^N$. Math. Nachr. 187 (1997) 19-27. | Zbl

[4] P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Comm. Appl. Nonlinear Anal. 1 (1994) 97-129. | Zbl

[5] V. Coti Zelati, P. Montecchiari and M. Nolasco, Multibump solutions for a class of second order, almost periodic Hamiltonian systems. Nonlinear Ord. Differ. Equ. Appl. 4 (1997) 77-99. | Zbl

[6] V. Coti Zelati and P. Rabinowitz, Homoclinic Orbits for Second Order Hamiltonian Systems Possessing Superquadratic Potentials. J. Amer. Math. Soc. 4 (1991) 693-627. | Zbl

[7] K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, New York (1985). | MR | Zbl

[8] M. Estaban and P.-L. Lions, Existence and non existence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh 93 (1982) 1-14. | Zbl

[9] B. Franchi, E. Lanconelli and J. Serrin, Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations in ${𝐑}^N$. Adv. Math. 118 (1996) 177-243. | Zbl

[10] L. Jeanjean and K. Tanaka, A Note on a Mountain Pass Characterization of Least Energy Solutions. Adv. Nonlinear Stud. 3 (2003) 445-455. | Zbl

[11] L. Jeanjean and K. Tanaka, A remark on least energy solutions in ${ℝ}^N$. Proc. Amer. Math. Soc. 131 (2003) 2399-2408. | Zbl

[12] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. Instit. Henri Poincaré 1 (1984) 102-145 and 223-283. | Numdam | Zbl

[13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York (1989). | MR | Zbl

[14] P. Rabinowitz, Homoclinic Orbits for a class of Hamiltonian Systems. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 33-38. | Zbl

[15] 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). | MR | Zbl

[16] 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).

[17] G. Spradlin, Existence of Solutions to a Hamiltonian System without Convexity Condition on the Nonlinearity. Electronic J. Differ. Equ. 2004 (2004) 1-13. | Zbl

[18] G. Spradlin, A Perturbation of a Periodic Hamiltonian System. Nonlinear Anal. Theory Methods Appl. 38 (1999) 1003-1022. | Zbl

[19] G. Spradlin, Interacting Near-Solutions of a Hamiltonian System. Calc. Var. PDE 22 (2005) 447-464. | Zbl

[20] E. Serra, M. Tarallo and S. Terracini, On the existence of homoclinic solutions to almost periodic second order systems. Ann. Instit. Henri Poincaré 13 (1996) 783-812. | Numdam | Zbl

[21] G. Whyburn, Topological Analysis. Princeton University Press (1964). | MR | Zbl