## An elliptic problem with jumping nonlinearities.(English)Zbl 1161.35395

Summary: It is proved that the elliptic problem $-\Delta u=f(x,u)\;\text{in}\;\Omega,\quad u=0\;\text{on}\;\partial\Omega,$ on a bounded domain $$\Omega\subset\mathbb R^ N$$ with smooth boundary has two sign-changing solutions and one positive solution if $$f\in\mathbb{C}^ 1(\overline\Omega\times\mathbb R,\mathbb R)$$ satisfies $$\mu_ i<f_ u'(x,0)<\mu_ {i+1}$$ for some $$i\geq 2,$$ $\limsup_ {u\to+\infty}f(x,u)/u<\mu_ 1<\liminf_ {u\to-\infty}f(x,u)/u,$ and $\limsup_ {u\to-\infty}f(x,u)/u<\infty,$ where $$\mu_ 1<\mu_ 2\leq\mu_ 3\leq\cdots$$ are the eigenvalues of $$-\Delta$$ with Dirichlet zero boundary conditions on $$\Omega$$ counting their multiplicity.

### MSC:

 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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### References:

 [1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044 [2] Bartsch, T.; Chang, K.-C.; Wang, Z.-Q., On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233, 655-677, (2000) · Zbl 0946.35023 [3] Bartsch, T.; Liu, Z.L., Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. differential equations, 9, 645-676, (2004) · Zbl 1100.58005 [4] Bartsch, T.; Wang, Z.-Q., On the existence of sign changing solutions for semilinear Dirichlet problems, Topology methods nonlinear anal., 7, 115-131, (1996) · Zbl 0903.58004 [5] Chang, K.C., Infinite-dimensional Morse theory and multiple solution problems, (1993), Birkhäuser Boston [6] Chang, K.C., $$H^1$$ versus $$C^1$$ isolated critical points, C. R. acad. sci. Paris Sér. I math., 319, 441-446, (1994) · Zbl 0810.35025 [7] Chang, K.C.; Li, S.J.; Liu, J.Q., Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topology methods nonlinear anal., 3, 179-187, (1994) · Zbl 0812.35031 [8] Dancer, E.N.; Du, Y.H., The generalized Conley index and multiple solutions of semilinear elliptic problems, Abstracts appl. anal., 1, 103-135, (1996) · Zbl 0933.35069 [9] Dancer, E.N.; Du, Y.H., Yihong A note on multiple solutions of some semilinear elliptic problems, J. math. anal. appl., 211, 626-640, (1997) · Zbl 0880.35046 [10] Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. ann., 261, 493-514, (1982) · Zbl 0488.47034 [11] Hofer, H., A note on the topological degree at a critical point of mountainpass-type, Proc. amer. math. soc., 90, 309-315, (1984) · Zbl 0545.58015 [12] Hofer, H., A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. London math. soc., 31, 566-570, (1985) · Zbl 0573.58007 [13] Li, S.J.; Wang, Z.-Q., Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. d’analyse math., 81, 373-396, (2000) · Zbl 0962.35065 [14] Li, S.J.; Wang, Z.-Q., Ljusternik – schnirelman theory in partially ordered Hilbert spaces, Trans. amer. math. soc., 354, 3207-3227, (2002) · Zbl 1219.35067 [15] Liu, Z.L., Positive solutions of superlinear elliptic equations, J. funct. anal., 167, 370-398, (1999) · Zbl 0951.35051 [16] Liu, Z.L.; Li, S.J., Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications, Nodea/nonlinear differential equations appl., 10, 133-170, (2003) · Zbl 1290.35052 [17] Liu, Z.L.; Sun, J.X., Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. differential equations, 172, 257-299, (2001) · Zbl 0995.58006 [18] Liu, Z.L.; Sun, J.X., Four versus two solutions of semilinear elliptic boundary value problems, Calc. var. partial differential equations, 14, 319-327, (2002) · Zbl 0996.35017 [19] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer New York · Zbl 0676.58017 [20] Tian, G., On the mountain pass lemma, Kexue tongbao, 29, 1150-1154, (1984), (English version) · Zbl 0588.58012
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