## Existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli.(English)Zbl 1285.35047

Summary: We consider the nonlinear elliptic problem $-{\Delta}u = u^p \quad \text{in } {\Omega}_R, \quad u > 0 \quad \text{in } {\Omega}_R, \quad u = 0 \quad \text{in } {\Omega}_R$ where $$p > 1$$ and $${\Omega}_R = \{x \in \mathbb R^N : R < |x| < R + 1\}$$ with $$N \geqslant 3$$. It is known that, as $$R \to \infty$$, the number of nonequivalent solutions of the above problem goes to $$\infty$$ when $$p \in (1,(N + 2)/(N - 2))$$, $$N \geqslant 3$$. Here we prove the same phenomenon for any $$p > 1$$ by finding $$O(N - 1)$$-symmetric clustering bump solutions which concentrate near the set $$\{(x_1, \dots, x_N) \in {\Omega}_R : x_N = 0\}$$ for large $$R > 0$$.

### MSC:

 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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### References:

 [1] Ackermann, N.; Clapp, M.; Pacella, F., Self-focusing multibump standing waves in expanding waveguides, Milan J. Math., 79, 221-232, (2011) · Zbl 1229.35285 [2] Ambrosetti, A.; Malchiodi, A.; Ni, W.-M., Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, part I, Comm. Math. Phys., 235, 427-466, (2003) · Zbl 1072.35019 [3] Amick, C.; Toland, J., Nonlinear elliptic eigenvalue problems on an infinite strip - global theory of bifurcation and asymptotic bifurcation, Math. Ann., 262, 313-362, (1982) · Zbl 0489.35067 [4] Byeon, J., Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differential Equations, 136, 136-165, (1997) · Zbl 0878.35043 [5] Byeon, J., Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. Differential Equations, J. Differential Equations, 172, 445-447, (2001), (Erratum) [6] Byeon, J., Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, II, J. Differential Equations, 173, 321-355, (2001) · Zbl 0989.35053 [7] Byeon, J.; Tanaka, K., Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains, Calc. Var. Partial Differential Equations, (2013), in press [8] Catrina, F.; Wang, Z.-Q., Nonlinear elliptic equations on expanding symmetric domains, J. Differential Equations, 156, 1, 153-181, (1999) · Zbl 0944.35026 [9] Coffman, C. V., A nonlinear boundary value problem with many positive solutions, J. Differential Equations, 54, 429-437, (1984) · Zbl 0569.35033 [10] Dancer, E. N., On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285, 647-669, (1989) · Zbl 0699.35103 [11] Dancer, E. N.; Yan, S., Multibump solutions for an elliptic problem in expanding domains, Comm. Partial Differential Equations, 27, 23-55, (2002) · Zbl 1011.35059 [12] Gidas, B.; Ni, Wei Ming; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 3, 209-243, (1979) · Zbl 0425.35020 [13] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, Classics Math., (2001), Springer-Verlag Berlin, reprint of the 1998 edition · Zbl 1042.35002 [14] Kazdan, J.; Warner, F., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28, 567-597, (1975) · Zbl 0325.35038 [15] Li, Y. Y., Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83, 348-367, (1990) · Zbl 0748.35013 [16] Palais, R. S., The principle of symmetric criticality, Comm. Math. Phys., 69, 19-30, (1979) · Zbl 0417.58007
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