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Superlinear problems without Ambrosetti and Rabinowitz growth condition. (English) Zbl 1158.35400
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.

##### MSC:
 35P30 Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory 35J20 Second order elliptic equations, variational methods 35B38 Critical points in solutions of PDE 35B60 Continuation of solutions of PDE 35D05 Existence of generalized solutions of PDE (MSC2000)
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##### 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 · doi:10.1016/0022-1236(73)90051-7 [2] Costa, D. G.; Magalhães, C. A.: Variational elliptic problems which are nonquadratic at infinity, Nonlinear anal. 23, 1401-1412 (1994) · Zbl 0820.35059 · doi:10.1016/0362-546X(94)90135-X [3] Jeanjean, L.: On the existence of bounded palais -- Smale sequences and application to a landesman -- lazer type problem set on RN, Proc. roy. Soc. Edinburgh sect. A 129, 787-809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147 [4] Palais, R. S.: Critical point theory and the minimax principle, Proc. sympos. Pure math. 15, 185-212 (1968) · Zbl 0212.28902 [5] Schechter, M.; Zou, W.: Superlinear problems, Pacific J. Math. 214, 145-160 (2004) · Zbl 1134.35346 · doi:10.2140/pjm.2004.214.145 [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 · doi:10.1006/jfan.2001.3798 [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 · doi:10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D [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 · doi:10.1512/iumj.2003.52.2273 [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 · doi:10.1007/s000330050128