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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.

35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
35J20Second order elliptic equations, variational methods
35B38Critical points in solutions of PDE
35B60Continuation of solutions of PDE
35D05Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI
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[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