## On the number of positive solutions of some weakly nonlinear equations on annular regions.(English)Zbl 0705.35043

We discuss the number of positive solutions of $-\Delta u=u^{\alpha}\text{ in } D;\quad u=0\text{ on } \partial D,$ where $$1<\alpha <(m+2)/(m-2)$$ $$(\alpha >1$$ if $$m=2)$$ and D is a domain in $$R^ m$$. In particular, we prove that, under reasonable hypotheses, the number of positive solutions is unaffected by the introduction of small holes into D. In particular our result implies the uniqueness of the positive solution if D is a true annulus with a small central hole. We also produce a number of counter examples showing how complicated these problems are. We include a general result on the existence of large solutions.
Reviewer: E.N.Dancer

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations
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### References:

 [1] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure App. Math.36, 437–444 (1983) · Zbl 0541.35029 [2] Cafarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behaviour of semilinear elliptic equations with Sobolev exponent. Comm. Pure App. Math.42, 271–297 (1989) · Zbl 0702.35085 [3] Coffman, C.V.: A nonlinear boundary value problem with many positive solutions. J. Diff. Equations54, 429–437 (1984) · Zbl 0569.35033 [4] Dancer, E.N.: The effect of domain shape on the number of positive solutions of certain nonlinear equations. J. Diff. Equations74, 120–156 (1988) · Zbl 0662.34025 [5] Dancer, E.N.: On the influence of domain shape on the existence of large solutions of some superlinear problems. Math. Ann.285, 647–669 (1990) · Zbl 0699.35103 [6] Dancer, E.N.: A note on an equation with critical exponent. Bull. Lond. Math. Soc.20, 600–602 (1988) · Zbl 0646.35027 [7] Dancer, E.N.: Certain weakly nonlinear partial differential equations on annuli. Nonlinear Anal.14, 441–442 (1990) · Zbl 0695.35077 [8] Dancer, E.N.: A new degree forS 1-invariant gradient mappings and applications. Analyse Nonlineaire2, 329–370 (1985) · Zbl 0579.58022 [9] Figueredo, de, D.G., Lions, P.L., Nussbaum, R.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pure et Appliquées61, 41–63 (1982) · Zbl 0452.35030 [10] Gidas, B., Ni, W., Nirenberg, L.: Symmetry and related properties by the maximum principle. Commun. Math. Phys.68, 209–243 (1979) · Zbl 0425.35020 [11] Gidas, B., Spruck, J.: Global and local behaviour of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math.34, 525–598 (1981) · Zbl 0465.35003 [12] Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equations6, 883–901 (1981) · Zbl 0462.35041 [13] Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003 [14] Li, Yan Yan: Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differ. Equations83, 348–367 (1990) · Zbl 0748.35013 [15] Lin, S.S.: On non radially symmetric bifurcation in the annulus. J. Differ. Equations80, 251–279 (1989) · Zbl 0688.35005 [16] Ni, W., Nussbaum, R.: Uniqueness and nonuniqueness for positive radial solutions of {$$\Delta$$}u+f(u,r)=0. Commun. Pure Appl. Math.38, 67–108 (1985) · Zbl 0581.35021
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