Li, Shujie; Zhang, Zhitao Sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities. (English) Zbl 0951.35050 Acta Math. Sin., Engl. Ser. 16, No. 1, 113-122 (2000). Summary: We use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solution theorems. At last we get up to six nontrivial solutions. Cited in 15 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 47J30 Variational methods involving nonlinear operators 35A15 Variational methods applied to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:sign-changing solutions; Dirichlet problems; multiple solutions; jumping nonlinearities Software:conley PDFBibTeX XMLCite \textit{S. Li} and \textit{Z. Zhang}, Acta Math. Sin., Engl. Ser. 16, No. 1, 113--122 (2000; Zbl 0951.35050) Full Text: DOI References: [1] N Dancer, Yihong Du. Existence of changing sign solutions for semilinear problems with jumping non-linearities at zero. Proceedings of the Royal Society of Edinburgh, 1994, 124A, 1165-1176 · Zbl 0819.35054 [2] E N Dancer, Yihong Du. Multiple solutions of some semilinear elliptic equations via generalized conley index. Journal of Mathematical Analysis and Applications, 1995, 189: 848-871 · Zbl 0834.35049 [3] Jingxian Sun, Zhaoli Liu. Calculus of variations and super- and sub-solution in reverse order. Acta Mathematica Sinica, 1994, 37(4): 512-514 (In Chinese) · Zbl 0810.47059 [4] Kungching Chang. A variant mountain pass lemma. Science in China, (A), 1983, (4) (In Chinese) [5] M H A Newman. Elements of the Topology of Plane Sets of Points. London: Cambridge University Press, 1939 · Zbl 0021.06704 [6] Kungching Chang. Infinite Dimensional Morse Theory and Multiple Solutions Problems. Birkhäuser, 1993 [7] Shujie Li, Zhitao Zhang. Sign-changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems. (preprint) · Zbl 0951.35050 [8] H Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach Spaces. SIAM Review, 1976, 18(4): 620-709 · Zbl 0345.47044 [9] H Amann. A note on theory for gradient mapping. Proceedings of the American Mathematical Society, 1982, 85(4): 591-595 · Zbl 0501.58012 [10] H Hofer. Variational and topological methods in partially ordered Hilbert spaces. Math Ann, 1982, 261: 493-514 · Zbl 0488.47034 [11] Kungching Chang, Shujie Li, Jiaquan Liu. Remarks on multiple solutions for asymptotically linear elliptic boundry value problems. Topological methods for Nonlinear Analysis, Journal of the Juliusz Schauder Center, 1994, 3: 179-187 · Zbl 0812.35031 [12] Dajun Guo, Jingxian Sun. Ordinary differential equations in abstract spaces. The Publishing House of Shandong Technology and Science, 1989 (In Chinese) [13] Zhaoli Liu. Multiple solutions of differential equations. Doctorial thesis of Shandong University, 1992 · Zbl 0764.45004 [14] Alfonso Castro, Jorge Cossio. Multiple solutions for a nonlinear Dirichlet problem. SIAM J Math Anal, 1994, 25(6): 1554-1561 · Zbl 0807.35039 [15] Alfonso Castro, Jorge Cossio, John M Neuberger. A sign-changing solution for a superlinear Dirichlet problem. to appear in Rocky Mountain J M (1997) · Zbl 0907.35050 [16] Alfonso Castro, Jorge Cossio, John M Neuberger. A minimax principle, index of the critical point, and existence of sign-changing solutions to Elliptic boundary value problems. (preprint) · Zbl 0901.35028 [17] Thomas Bartsch, Zhiqiang Wang. On the existence of sign-changing solutions for semilinear Dirichlet problems. (preprint) · Zbl 0903.58004 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.