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Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN

Published online by Cambridge University Press:  14 November 2011

Ziqing Xie
Ziqing Xie
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha 410081, P.R. China
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Abstract

We consider the following problem:

where

for x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 5 , 1998 , pp. 1069 - 1097
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Bahri, A. and Li, Y. Y.. On a min-max procedure for the existence of a positive solution for certain scalar field equations in RN. Rev. Mat. Iberoamericana 6 (1990), 1–15.CrossRefGoogle Scholar
2A. Bahri and Lions, P. L.. On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincare Anal. Non linéaire. (to appear).Google Scholar
3Benci, V. and Cerami, G.. The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Rational Mech. Anal 114 (1991), 79–93.CrossRefGoogle Scholar
4Brezis, H. and Lieb, E.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 486–90.CrossRefGoogle Scholar
5Cao, D. M.. Positive solutions and bifurcation from the essential spectrum of a semilinear elliptic equation in RN. Nonlinear Anal. 15 (1990), 1045–52.CrossRefGoogle Scholar
6Cao, D. M., Li, G. B. and Zhong, X.. A note on the number of the positive solutions of some nonlinear elliptic problems. Nonlinear Anal, (to appear).Google Scholar
7Cao, D. M. and Noussair, E. S.. Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN. Ann. Inst. H. Poincaré Anal. Non linéaire 13 (1996), 567–88.CrossRefGoogle Scholar
8Cerami, G. and Passaseo, D.. Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with ‘rich’ topology. Nonlinear Anal. 18 (1992), 109–19.CrossRefGoogle Scholar
9Ekeland, I.. On the variational principle. J. Math. Anal. Appl. 47 (1974), 324–53.CrossRefGoogle Scholar
10Kwong, M. K.. Uniqueness of positive solutions of …u — u + up = 0 in RN. Arch. Rational Mech. Anal. 105 (1989), 243–66.CrossRefGoogle Scholar
11Li, Yi. Remarks on a semilinear elliptic equation on RN. J. Differential Equations 74 (1988), 34–49.CrossRefGoogle Scholar
12Lions, P. L.. The concentration-compactness principle in the calculation of variations. The locally compact case. Ann. Inst. H. Poincare, Anal. Non Lineaire 1 (1984), 102–45; 223–83.Google Scholar
13Lions, P. L.. On positive solutions of semilinear elliptic equations in unbounded domains. In Nonlinear Diffusion Equations and their Equilibrium States (New York: Springer, 1988).Google Scholar
14Noussair, E. S. and Cao, D. M.. Multiplicity results for an inhomogeneous nonlinear elliptic problem. Dig. Integ. Eqns. (to appear).Google Scholar
15Passaseo, D.. Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math. 65 (1989), 147–66.CrossRefGoogle Scholar
16Tarantello, G.. On nonhomogenous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Lineaire 9 (1992), 281–304.CrossRefGoogle Scholar
17Zhu, X. P.. Multiple entire solutions of semilinear elliptic equation. Nonlinear Anal. 12 (1988), 1297–316.CrossRefGoogle Scholar
18Zhu, X. P. and Cao, D. M.. The concentration-compactness principle in nonlinear elliptic equation. Ada Math. Sci. 9 (1989), 307–28.Google Scholar