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**Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. II.**
*(English)*
Zbl 0989.35053

Summary: The author considers the problem:
\[
\begin{aligned} \Delta u+hu+ f(u)=0 & \text{ in }\Omega_R\\ u=0 & \text{ on }\partial \Omega_R\\ u>0 & \text{ in }\Omega_R, \end{aligned}
\]
where \(\Omega_R\equiv \{x\in\mathbb{R}^N \mid R-1<|x |< R+1\}\) and the function \(f\) and the constant \(h\) satisfy suitable assumptions. This problem is invariant under the orthogonal coordinate transformations, in other words, \(O(N)\)-symmetric. Let \(G\) be an infinite closed subgroup of \(O(N)\). He investigates how the symmetry subgroup \(G\) affects the structure of positive solutions. Considering a natural \(G\) group action on a sphere \(S^{N-1}\), we give a partial order on the space of \(G\)-orbits \(\{xG\mid x\in S^{N-1}\}\). In a previons paper [J. Byeon, J. Differ. Equations 163, No. 2, 429-474 (2000; Zbl 0952.35054)], the author has studied the effect of symmetry on the structure of positive solutions when the number of elements of \(xG\) is finite for some \(x\in S^{N-1}\). In this paper, he studies the effect when \(xG\) is an infinite set for any \(x\in S^{N-1}\). In fact, in view of the partial order, a critical (locally minimal) orbital set will be defined. Then it is shown that when \(R\to\infty\), a critical orbital set produces a solution of our problem whose energy goes to \(\infty\) and is concentrated around the scaled critical orbital set.

### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

### Citations:

Zbl 0952.35054
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\textit{J. Byeon}, J. Differ. Equations 173, No. 2, 321--355 (2001; Zbl 0989.35053)

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### References:

[1] | Berestycki, H.; Nirenberg, L., Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, (), 115-164 · Zbl 0705.35004 |

[2] | Berestycki, B.; Nirenberg, L.; Varadhan, S.R.S., The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. pure appl. math., 57, 47-92, (1994) · Zbl 0806.35129 |

[3] | Borel, A., Seminar on transformation group, Ann. math. stud., 46, (1960) |

[4] | Bredon, G.E., Transformation groups with orbits of uniform dimension, Michigan math. J., 8, 139-147, (1961) · Zbl 0121.39501 |

[5] | Byeon, J., Existence of many nonequivalent non-radial positive solutions of semilinear elliptic equations on three dimensional annuli, J. differential equations, 136, 136-165, (1997) · Zbl 0878.35043 |

[6] | Byeon, J., Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. partial differential equations, 22, 1731-1769, (1997) · Zbl 0883.35040 |

[7] | Byeon, J., Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, J. differential equations, 163, 429-474, (2000) · Zbl 0952.35054 |

[8] | Coffman, C.V., A nonlinear boundary value problem with many positive solutions, J. differential equations, 54, 429-437, (1984) · Zbl 0569.35033 |

[9] | Conner, P.E., Orbits of uniform dimension, Michigan math. J., 6, 25-32, (1959) · Zbl 0087.37901 |

[10] | Curtis, M.L., Matrix groups, (1979), Springer-Verlag New York · Zbl 0425.22013 |

[11] | Ding, W., On a conformally invariant elliptic equation on \(R\)^{n}, Comm. math. phys., 107, 331-335, (1986) · Zbl 0608.35017 |

[12] | Esteban, M.J.; Lions, P.L., A compactness lemma, Nonlinear anal., 7, 381-385, (1983) · Zbl 0512.46035 |

[13] | Gidas, B.; Ni, W.N.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020 |

[14] | Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041 |

[15] | Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehren 224, (1983), Springer-Verlag Berlin/New York · Zbl 0691.35001 |

[16] | Hsiang, W.Y.; Lawson, B., Minimal submanifolds of low cohomogeneity, J. differential geom., 5, 1-38, (1971) · Zbl 0219.53045 |

[17] | Li, C.M., Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. partial differential equations, 16, 585-615, (1991) · Zbl 0741.35014 |

[18] | Li, Y.Y., Existence of many positive solutions of semilinear elliptic equations in annulus, J. differential equations, 83, 348-367, (1990) · Zbl 0748.35013 |

[19] | Lieb, E.H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Inv. math., 74, 441-448, (1983) · Zbl 0538.35058 |

[20] | Lin, S.S., Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. differential equations, 120, 255-288, (1995) · Zbl 0839.35039 |

[21] | Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case. part 1, Ann. inst. H. Poincaré, 1, 109-145, (1984) · Zbl 0541.49009 |

[22] | Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case. part 2, Ann. inst. H. Poincaré, 1, 223-283, (1984) · Zbl 0704.49004 |

[23] | Ladyzhenskaya, O.A.; Ural’tseva, N.N., Linear elliptic and quasilinear elliptic equations, (1968), Academic Press New York/London · Zbl 0164.13002 |

[24] | Montgomery, D.; Samelson, H.; Yang, C.T., Exceptional orbits of highest dimension, Ann. of math., 64, 131-141, (1956) · Zbl 0074.26001 |

[25] | Montgomery, D.; Samelson, H.; Zippin, L., Singular points of a compact transformation group, Ann. of math., 63, 1-9, (1956) · Zbl 0074.25905 |

[26] | Montgomery, D.; Yang, C.T., The existence of a slice, Ann. of math., 65, 108-116, (1957) · Zbl 0078.16202 |

[27] | Palais, R.S., The principle of symmetric criticality, Comm. math. phys., 69, 19-30, (1979) · Zbl 0417.58007 |

[28] | Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1984), Springer-Verlag New York/Berlin · Zbl 0153.13602 |

[29] | Rabinowitz, P.H., Minimax methods in critical point theory with application to differential equations, CBMS regional conference series math., 65, (1986), Amer. Math. Soc Providence |

[30] | Struwe, M., Variational methods; application to nonlinear partial differential equations and Hamiltonian systems, (1990), Springer-Verlag Berlin/New York |

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