Miyagaki, O. H.; Souto, M. A. S. Superlinear problems without Ambrosetti and Rabinowitz growth condition. (English) Zbl 1158.35400 J. Differ. Equations 245, No. 12, 3628-3638 (2008). Summary: Superlinear elliptic boundary value problems without Ambrosetti and Rabinowitz growth condition are considered. Existence of nontrivial solution is established by combining some arguments used by Struwe and Tarantello and Schechter and Zou (also by Wang and Wei). Firstly, by using the mountain pass theorem due to Ambrosetti and Rabinowitz solution is constructed for almost every parameter \(\lambda \) by varying the parameter \(\lambda \). Then, the continuation of the solutions is considered. Cited in 5 ReviewsCited in 127 Documents MSC: 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B60 Continuation and prolongation of solutions to PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:variational methods; critical points; superlinear problems; elliptic equations PDFBibTeX XMLCite \textit{O. H. Miyagaki} and \textit{M. A. S. Souto}, J. Differ. Equations 245, No. 12, 3628--3638 (2008; Zbl 1158.35400) Full Text: DOI Link References: [1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 [2] Costa, D. G.; Magalhães, C. A., Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23, 1401-1412 (1994) · Zbl 0820.35059 [3] Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on \(R^N\), Proc. Roy. Soc. Edinburgh Sect. A, 129, 787-809 (1999) · Zbl 0935.35044 [4] Palais, R. S., Critical point theory and the minimax principle, (Global Analysis. Global Analysis, Proc. Sympos. Pure Math., vol. 15 (1968)), 185-212 · Zbl 0212.28902 [5] Schechter, M.; Zou, W., Superlinear problems, Pacific J. Math., 214, 145-160 (2004) · Zbl 1134.35346 [6] Struwe, M.; Tarantello, G., On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B (8), 1, 109-121 (1998) · Zbl 0912.58046 [7] Szulkin, A.; Zou, W., Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187, 25-41 (2001) · Zbl 0984.37072 [8] Wang, G.; Wei, J., Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 233-234, 221-236 (2002) · Zbl 1002.35049 [9] Willem, M.; Zou, W., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52, 109-132 (2003) · Zbl 1030.35068 [10] Zhou, H. S., Positive solution for a semilinear elliptic equations which is almost linear at infinity, Z. Angew. Math. Phys., 49, 896-906 (1998) · Zbl 0916.35036 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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.