Asymptotically radial solutions in expanding annular domains. (English) Zbl 1276.35083

Summary: In this paper, we consider the problem \[ \begin{cases} -\Delta u=u^{p}\quad&\text{in }\Omega_R,\\ u=0 \quad&\text{on }\partial\Omega_R,\end{cases}\tag{0.1} \] where \(p>1\) and \(\Omega_R\) is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as \(R \longrightarrow \infty\). The main goal of the paper is to prove, for large \(R\), the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.


35J61 Semilinear elliptic equations
35B09 Positive solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[1] Ackermann, N., Clapp, M., Pacella, F.: Alternating sign multibump solutions of semilinear elliptic equations in expanding tubular domains (Preprint) · Zbl 1273.35132
[2] Adams R.A.: Sobolev Spaces. Academic Press, New York (1978) · Zbl 0347.46040
[3] Bahri A., Coron J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41, 253–294 (1988) · Zbl 0649.35033
[4] Byeon J.: Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Differ. Equ. 136, 136–165 (1997) · Zbl 0878.35043
[5] Catrina F., Wang Z.Q.: Nonlinear elliptic equations on expanding symmetric domains. J. Differ. Equ. 156, 153–181 (1999) · Zbl 0944.35026
[6] Clapp M., Musso M., Pistoia A.: Multipeak solutions to the Bahri–Coron problem in domains with a shrinking hole. J. Funct. Anal. 256, 275–306 (2009) · Zbl 1236.35044
[7] Clapp M., Pacella F.: Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size. Math. Z. 259, 575–589 (2008) · Zbl 1143.35052
[8] Coffman C.: A nonlinear boundary value problem with many positive solutions. J. Differ. Equ. 54, 429–437 (1984) · Zbl 0569.35033
[9] Coron J.M.: Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Sér. I Math. 299, 209–212 (1984) · Zbl 0569.35032
[10] Dancer E.N., Yan S.: Multibump solutions for an elliptic problem in expanding domains. Comm. Partial Differ. Equ. 27, 23–55 (2002) · Zbl 1011.35059
[11] del Pino M., Felmer P., Musso M.: Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differ. Equ. 182, 511–540 (2002) · Zbl 1014.35028
[12] del Pino M., Wei J.: Supercritical elliptic problems in domains with small holes. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 507–520 (2007) · Zbl 1387.35291
[13] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (1983) · Zbl 0562.35001
[14] Grossi M., Pacella F., Yadava S.L.: Symmetry results for perturbed problems and related questions. Topol. Methods Nonlinear Anal. 21, 211–226 (2003) · Zbl 1274.35021
[15] Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer-Verlag, Berlin (1995) · Zbl 0836.47009
[16] Kazdan J., Warner F.W.: Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28, 567–597 (1975) · Zbl 0325.35038
[17] Li G., Yan S., Yang J.: An elliptic problem with critical growth in domains with shrinking holes. J. Differ. Equ. 198, 275–300 (2004) · Zbl 1086.35046
[18] Li Y.Y.: Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differ. Equ. 83, 348–367 (1990) · Zbl 0748.35013
[19] Lin S.S.: Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus. J. Differ. Equ. 103, 338–349 (1993) · Zbl 0803.35053
[20] Lin S.S.: Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli. J. Differ. Equ. 120, 255–288 (1995) · Zbl 0839.35039
[21] Ni W.M., Nussbaum R.: Uniqueness and nonuniqueness for positive radial solutions of {\(\Delta\)}u + f(u, r) = 0. Comm. Pure Appl. Math. 38, 67–108 (1985) · Zbl 0581.35021
[22] Pacard F.: Radial and non-radial solutions of u = {\(\lambda\)}f(u) on an annulus of $${\(\backslash\)mathbb{R}\^n,\(\backslash\),n\(\backslash\)geq3}$$ . J. Differ. Equ. 102, 103–138 (1993) · Zbl 0799.35089
[23] Pohozaev S.: Eigenfunctions of the equation {\(\Delta\)}u + {\(\lambda\)}f(u) = 0. Sov. Math. Dokl. 6, 1408–1411 (1965) · Zbl 0141.30202
[24] Rey O.: Sur un problème variationnel non compact: l’effet de petits trous dans le domaine. C. R. Acad. Sci. Paris Sér. I Math. 308, 349–352 (1989) · Zbl 0686.35047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.