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A multiplicity result including a sign-changing solution for an inhomogeneous Neumann problem with critical exponent

Published online by Cambridge University Press:  26 March 2007

Norimichi Hirano  and
Naoki Shioji
Norimichi Hirano
Affiliation:
Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan (hirano@math.sci.ynu.ac.jp; shioji@math.sci.ynu.ac.jp)
Naoki Shioji
Affiliation:
Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan (hirano@math.sci.ynu.ac.jp; shioji@math.sci.ynu.ac.jp)
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Abstract

We study the existence of multiple solutions for the problem

and we show that at least one of them is sign changing. Here Ω is a bounded domain in ℝN with N ⩾ 5 whose boundary is of class C2, ∂/∂ν is the outward normal derivative, and f ∈ LN/2(Ω) whose L2N/(N+2)(Ω) norm is sufficiently small.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 333 - 347
Copyright
Copyright © Royal Society of Edinburgh 2007

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References

1 Adimurthi, and Mancini, G.. The Neumann problem for elliptic equation with critical non linearity. In Nonlinear analysis: a tribute in honor of G. Prodi, Quaderni di Scuola Normale Superiore, Pisa (ed. Ambrosetti, A. and Marino, A.), pp. 9–25 (Pisa: Scuola Normale Superiore, 1991).Google Scholar
2 Brćzis, H. and Kato, T.. Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. 58 (1979), 137–151.Google Scholar
3 Brézis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
4 Bréis, H. and Nirenberg, L.. A minimization problem with critical exponent and nonzero data. In Symmetry in nature, Quaderni di Scuola Normale Superiore, Pisa, pp. 129–140 (Pisa: Scuola Normale Superiore, 1989).Google Scholar
5 Cao, D. and Noussair, E. S.. Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents. Indiana Univ. Math. J. 44 (1995), 1249–1271.CrossRefGoogle Scholar
6 Cao, D. and Noussair, E. Z.. Multiplicity of positive and nodal solutions for nonlinear elliptic problems in ℝ N . Annls Inst. H. Poincaré Analyse Non Linéaire 13 (1996), 567–588.CrossRefGoogle Scholar
7 Castro, A., Cossio, J., and Neuberger, J. M.. A sign-changing solution for a superlinear Dirichlct problem. Rocky Mt. J. Math. 27 (1997), 1041–1053.CrossRefGoogle Scholar
8 Chabrowski, J.. Weak convergence methods for semilinear elliptic equations (World Scientific, 1999).CrossRefGoogle Scholar
9 Cherrier, P.. Meilleures constantes dans des inégalités relatives aux espaces de Sobolev. Bull. Set. Math. 108 (1984), 225–262. (In French, with English summary.)Google Scholar
10 Clapp, M. and Weth, T.. Multiple solutions of nonlinear scalar field equations. Comrnun. PDEs 29 (2004), 1533–1554.CrossRefGoogle Scholar
11 Comte, M. and Knaap, M. C.. Existence of solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary condition in general domains. Diff. Integ. Eqns 4 (1991), 1133–1146.Google Scholar
12 Comte, M. and Tarantello, G.. A Neumann problem with critical Sobolev exponent. Houston J. Math. 18 (1992), 279–294.Google Scholar
13 Devillanova, G. and Solimini, S.. A multiplicity result for elliptic equations at critical growth in low dimension. Commun. Contemp. Math. 5 (2003), 171–177.CrossRefGoogle Scholar
14 Hirano, N., Saccon, C. and Shioji, N.. Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Diff. Eqns 9 (2004), 197–220.Google Scholar
15 Neuberger, J. M.. A sign-changing solution for a superlinear Dirichlet problem with a reaction term nonzero at zero. Nonlin. Analysis 33 (1998), 427–441.CrossRefGoogle Scholar
16 Struwe, M.. Infinitely many solutions of superlinear boundary value problems with rotational symmetry. Arch. Math. 36 (1981), 360–369.CrossRefGoogle Scholar
17 Struwe, M.. Superlinear elliptic boundary value problems with rotational symmetry. Arch. Math. 39 (1982), 233–240.CrossRefGoogle Scholar
18 Struwe, M.. Variational methods. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, vol. 34 (Springer, 1996).CrossRefGoogle Scholar
19 Tarantello, G.. Nodal solutions of semilincar elliptic equations with critical exponent. Diff. Integ. Eqns 5 (1992), 25–42.Google Scholar
20 Tarantello, G.. Multiplicity results for an inhomogeneous Neumann problem with critical exponent. Manuscr. Math. 81 (1993), 57–78.CrossRefGoogle Scholar
21 Wang, X. J.. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Diff. Eqns 93 (1991), 283–310.CrossRefGoogle Scholar
22 Willeni, M.. Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (Birkhäuser, 1996).CrossRefGoogle Scholar
23 Xie, Z.. Multiplicity of positive and nodal solutions for nonlinear elliptic problems in ℝ N . Proc. R. Soc. Edinb. A 128 (1998), 1069–1097.CrossRefGoogle Scholar