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.

MSC:

 35J61 Semilinear elliptic equations 35B09 Positive solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs
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References:

 [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
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