Spradlin, Gregory S. An elliptic equation with no monotonicity condition on the nonlinearity. (English) Zbl 1123.35021 ESAIM, Control Optim. Calc. Var. 12, 786-794 (2006). Summary: 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 [cf. the author, Nonlinear Anal., Theory Methods Appl. 38, No. 8 (A), 1003–1022 (1999; Zbl 0939.37032)]. Here, a more subtle degree theory argument must be used. Cited in 1 Document MSC: 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 47H11 Degree theory for nonlinear operators 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:Mountain-pass theorem; variational methods; Nehari manifold; Brouwer degree; concentration-compactness Citations:Zbl 0939.37032 PDFBibTeX XMLCite \textit{G. S. Spradlin}, ESAIM, Control Optim. Calc. Var. 12, 786--794 (2006; Zbl 1123.35021) Full Text: DOI Numdam EuDML References: [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 0919.34044 · Zbl 0919.34044 [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 \(\mathbb{R}^N\) . Revista Iberoamericana 6 ( 1990 ) 1 - 17 . Zbl 0731.35036 · Zbl 0731.35036 [3] P. Caldiroli , A New Proof of the Existence of Homoclinic Orbits for a Class of Autonomous Second Order Hamiltonian Systems in \(\mathbb{R}^N\) . Math. Nachr. 187 ( 1997 ) 19 - 27 . Zbl 0894.58015 · Zbl 0894.58015 [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 0867.70012 · Zbl 0867.70012 [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 0878.34045 · Zbl 0878.34045 [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 0744.34045 · Zbl 0744.34045 [7] K. Deimling , Nonlinear Functional Analysis . Springer-Verlag, New York ( 1985 ). MR 787404 | Zbl 0559.47040 · Zbl 0559.47040 [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 0506.35035 · Zbl 0506.35035 [9] B. Franchi , E. Lanconelli and J. Serrin , Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations in \({\mathbf{R}}^N\) . Adv. Math. 118 ( 1996 ) 177 - 243 . Zbl 0853.35035 · Zbl 0853.35035 [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 1095.35006 · Zbl 1095.35006 [11] L. Jeanjean and K. Tanaka , A remark on least energy solutions in \({\mathbb{R}}^N\) . Proc. Amer. Math. Soc. 131 ( 2003 ) 2399 - 2408 . Zbl 1094.35049 · Zbl 1094.35049 [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 0704.49004 · Zbl 0704.49004 [13] J. Mawhin and M. Willem , Critical Point Theory and Hamiltonian Systems . Springer-Verlag, New York ( 1989 ). MR 982267 | Zbl 0676.58017 · Zbl 0676.58017 [14] P. Rabinowitz , Homoclinic Orbits for a class of Hamiltonian Systems . Proc. Roy. Soc. Edinburgh Sect. A 114 ( 1990 ) 33 - 38 . Zbl 0705.34054 · Zbl 0705.34054 [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 845785 | Zbl 0609.58002 · Zbl 0609.58002 [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 1065.34039 · Zbl 1065.34039 [18] G. Spradlin , A Perturbation of a Periodic Hamiltonian System . Nonlinear Anal. Theory Methods Appl. 38 ( 1999 ) 1003 - 1022 . Zbl 0939.37032 · Zbl 0939.37032 [19] G. Spradlin , Interacting Near-Solutions of a Hamiltonian System . Calc. Var. PDE 22 ( 2005 ) 447 - 464 . Zbl 1063.37053 · Zbl 1063.37053 [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 0873.58032 · Zbl 0873.58032 [21] G. Whyburn , Topological Analysis . Princeton University Press ( 1964 ). MR 165476 | Zbl 0186.55901 · Zbl 0186.55901 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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