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High-energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponents*

Published online by Cambridge University Press:  14 November 2011

Zhi-Qiang Wang
Zhi-Qiang Wang
Affiliation:
Department of Mathematics, Utah State University, Logan, UT 84322, U.S.A
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Extract

We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem −Δu + λu = up in antipodal invariant domains including ball domains with Neumann boundary conditions. Here p is the critical Sobolev exponent (N + 2)/(N − 2). The shape of the solutions and the location of the peaks are also studied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 1003 - 1029
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Adams, R.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2, Adimurthi and Mancini, G.. The Neumann problem for elliptic equations with critical non-linearity. In A tribute in honor of G. Prodi (Pisa: Scuola Normale Superiore, 1991).Google Scholar
3, Adimurthi and Mancini, G.. Geometry and topology of the boundary in the critical Neumann problem (preprint).Google Scholar
4, Adimurthi and Yadava, S. L.. Critical Sobolev exponent problem in RN (n ≧ 4) with Neumann boundary condition. Proc. Indian Acad. Sci. 100 (1990), 275–84.Google Scholar
5, Adimurthi and Yadava, S. L.. Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents. Arch. Rational Mech. Anal. 115 (1991), 275–96.CrossRefGoogle Scholar
6, Adimurthi, Pacella, F and Yadava, S. L.. Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity (preprint).Google Scholar
7Bianchi, G. and Egnell, H.. A note on the Sobolev inequality. J. Fund. Anal. 100 (1991), 18–24.CrossRefGoogle Scholar
8Brezis, H.. Nonlinear elliptic equations involving the critical Sobolev exponent. In Survey and Perspectives in Directions in P.D.E., eds Grandall, , Rabinowitz, and Turner, , 17–36 (New York: Academic Press, 1987).Google Scholar
9Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical exponents. Comm. Pure Appl. Math. 36 (1983), 437–77.CrossRefGoogle Scholar
10Budd, C., Knapp, M. C. and Peletier, L. A.. Asymptotic behavior of solutions of elliptic equations with critical exponents and Neumann boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 225–50.CrossRefGoogle Scholar
11Gidas, B. and Spruck, I.. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6 (1981), 883–901.CrossRefGoogle Scholar
12Gidas, B., Ni, W.-M. and Nirenberg, L.. Symmetry of positive solutions of nonlinear elliptic equations in R n. Adv. Math., Suppl. Stud. 7A (1981), 369–402.Google Scholar
13Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (New York: Springer, 1983).Google Scholar
14Guillemin, V. and Pollack, A.. Differential Topology (Englewood Cliffs, NJ: Prentice-Hall, 1974).Google Scholar
15Keller, E. F. and Segel, L. A.. Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26(1970), 399–415.CrossRefGoogle ScholarPubMed
16Lin, C.-S. and Ni, W.-M.. On the diffusion coefficient of a semilinear Neumann problem. Lecture Notes in Mathematics 1340 (1988), 160–74.CrossRefGoogle Scholar
17Lin, C.-S., Ni, W.-M. and Takagi, I.. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988), 1–27.CrossRefGoogle Scholar
18Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1 and Part 2. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–45; 223–83.CrossRefGoogle Scholar
19Ni, W.-M.. On the positive radial solutions of some semilinear elliptic equations on R n. Appl. Math. Optim. 9(1983), 373–80.CrossRefGoogle Scholar
20Ni, W.-M., Pan, X.-B. and Takagi, I.. Singular behavior of least energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67 (1992), 1–20.CrossRefGoogle Scholar
21Ni, W.-M. and Takagi, I.. On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 45 (1991) 819–51.CrossRefGoogle Scholar
22Ni, W.-M. and Takagi, I.. On the existence and shape of solutions to a semilinear Neumann problem. In Progress in Nonlinear Differential Equations, eds Ni, Lloyd, Peletier and Serrin, 425–436 (Boston: Birkhäuser, 1992).Google Scholar
23Ni, W.-M. and Takagi, I.. Spike-layers in semilinear elliptic singular perturbation problems (preprint).Google Scholar
24Palais, R.. The principle of symmetric criticality. Comm. Math. Phys. 69 (1979), 19–30.CrossRefGoogle Scholar
25Rey, O.. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Fund. Anal. 89 (1990), 1–52.CrossRefGoogle Scholar
26Struwe, M.. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187 (1984), 511–17.CrossRefGoogle Scholar
27Wang, X.-J.. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93 (1991), 283–310.CrossRefGoogle Scholar
28Wang, Z.-Q.. On the existence of multiple, single-peaked solutions of a semilinear Neumann problem. Arch. Rational Mech. Anal. 120 (1992), 375–99.CrossRefGoogle Scholar
29Wang, Z.-Q.. On the existence of positive solutions for semilinear Neumann problems in exterior domains. Comm. Partial Differential Equations 17 (1992), 1309–25.CrossRefGoogle Scholar
30Wang, Z.-Q.. The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents. Differential Integral Equations 8 (1995), 1533–54.CrossRefGoogle Scholar
31Wang, Z.-Q.. Nonlinear Neumann problems with critical exponent in symmetric domains. In Proceedings of the conference on variational methods in nonlinear analysis, Erice-Sicily, Italy, May, 1992 (to appear).Google Scholar
32Wang, Z.-Q.. On the shape of solutions for a nonlinear Neumann problem in symmetric domains. In Exploiting Symmetry in Applied and Numerical Analysis, eds Georg, Allgower and Miranda, , Lectures in Applied Mathematics 29, 433–42 (Providence, RI: American Mathematical Society, 1993).Google Scholar
33Wang, Z.-Q.. Remarks on a nonlinear Neumann problem with critical exponent. Houston J. Math. 20(1994), 671–84.Google Scholar