Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.

*(English)*Zbl 0541.35029Semilinear elliptic equations involving critical Sobolev exponents were considered being hard to attack because of the lack of compactness. Indeed the well known nonexistence results of Pokhožaev asserts that, for a starshaped domain, there is no nontrivial solution for the BVP with critical Sobolev power function as nonlinear term. Surprisingly, it is proved in this paper that the lower term can reverse this situation.

The method used here is essentially close to that employed in Yamabe’s problem by Th. Aubin [J. Math. Pures Appl., IX. Sér. 55, 269–296 (1976; Zbl 0336.53033)]. Namely, a version of the mountain pass theorem without the Palais-Smale condition is applied. The decisive device in order to overcome this lack of compactness is to estimate the mountain pass value by a number associated with the best Sobolev constant. The following typical example is discussed in this paper: \((*)\quad -\Delta u=u^ p+\mu \quad u^ q\) on \(\Omega\), \(u>0\) on \(\Omega\), \(u=0\) on \(\partial \Omega\), \(n=\dim \Omega\), where \(p=(n+2)/(n-2)\), \(1<q<p\) and \(\mu>0\) is a constant. When \(n\geq 4\), \((*)\) has a solution for every \(\mu>0\). When \(n=3\), (a) if \(3<q<5\), \((*)\) has a solution for every \(\mu>0\); (b) if \(1<q\leq 3\), \((*)\) possesses a solution only for \(\mu \geq\mu_0\) for some \(\mu_ 0>0\). However, in case \(1<q\leq 3\), the problem is left open for \(\mu<\mu_ 0\).

The method used here is essentially close to that employed in Yamabe’s problem by Th. Aubin [J. Math. Pures Appl., IX. Sér. 55, 269–296 (1976; Zbl 0336.53033)]. Namely, a version of the mountain pass theorem without the Palais-Smale condition is applied. The decisive device in order to overcome this lack of compactness is to estimate the mountain pass value by a number associated with the best Sobolev constant. The following typical example is discussed in this paper: \((*)\quad -\Delta u=u^ p+\mu \quad u^ q\) on \(\Omega\), \(u>0\) on \(\Omega\), \(u=0\) on \(\partial \Omega\), \(n=\dim \Omega\), where \(p=(n+2)/(n-2)\), \(1<q<p\) and \(\mu>0\) is a constant. When \(n\geq 4\), \((*)\) has a solution for every \(\mu>0\). When \(n=3\), (a) if \(3<q<5\), \((*)\) has a solution for every \(\mu>0\); (b) if \(1<q\leq 3\), \((*)\) possesses a solution only for \(\mu \geq\mu_0\) for some \(\mu_ 0>0\). However, in case \(1<q\leq 3\), the problem is left open for \(\mu<\mu_ 0\).

Reviewer: K.Chang

##### MSC:

35J60 | Nonlinear elliptic equations |

35J20 | Variational methods for second-order elliptic equations |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

##### Keywords:

positive solutions; best Sobolev constant; isoperimetric inequality; limiting Sobolev exponent; Semilinear elliptic equations; critical Sobolev exponents; mountain pass theorem
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\textit{H. Brézis} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029)

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##### References:

[1] | Ambrosetti, J. Funct. Anal. 14 pp 349– (1973) |

[2] | Aubin, J. Diff. Geom. 11 pp 573– (1976) |

[3] | Aubin, J. Math. Pures et Appl. 55 pp 269– (1976) |

[4] | Bliss, J. London Math. Soc. 5 pp 40– (1930) |

[5] | Brezis, Comm. Pure Appl. Math. |

[6] | Brezis, Comm. Pure Appl. Math. 33 pp 667– (1980) |

[7] | Brezis, J. Math. Pures et Appl. 58 pp 137– (1979) |

[8] | Brezis, Proc. Amer. Math. Soc. |

[9] | Coffman, Arch. Rat. Mech. Anal. 46 pp 81– (1972) |

[10] | A non-linear boundary value problem with many positive solutions, to appear. |

[11] | Crandall, Arch. Rat. Mech. Anal. 58 pp 207– (1975) |

[12] | Ekeland, J. Math. Anal. Appl. 47 pp 324– (1974) |

[13] | Gidas, Comm. Math. Phys. 68 pp 209– (1979) |

[14] | An isoperimetric inequality for functions analytic in multiply connected domains, Report Mittag-Leffler Institut, 1970. |

[15] | Joseph, Arch. Rat. Mech. Anal. 49 pp 241– (1973) |

[16] | Kazdan, Comm. Pure Appl. Math. 28 pp 567– (1975) |

[17] | Keener, J. Diff. Eq. 16 pp 103– (1974) |

[18] | Lieb, Studies in Appl. Math. 57 pp 93– (1977) · Zbl 0369.35022 · doi:10.1002/sapm197757293 |

[19] | Lieb, Annals of Math. |

[20] | Lions, SIAM Review 24 pp 441– (1982) |

[21] | The method of concentration–compactness and applications to the best constants, in preparation. |

[22] | McLeod, Proc. Nat. Acad. Sc. USA 78 pp 6592– (1981) |

[23] | Peletier, R. Arch. Rat. Mech. Anal. 81 pp 181– (1983) |

[24] | Pohozaev, Soviet Math. Doklady 6 pp 1408– (1965) |

[25] | Russian Dokl. Akad. Nauk SSSR 165 pp 33– (1965) |

[26] | Rabinowitz, J. Funct. Anal. 7 pp 487– (1971) |

[27] | Rabinowitz, Indiana Univ. Math. J. 23 pp 729– (1974) |

[28] | An example for the plasma problem with infinitely many solutions, unpublished note. |

[29] | Talenti, Annali di Mat. 110 pp 353– (1976) |

[30] | The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on R3, to appear. |

[31] | Trudinger, Ann. Sc. Norm. Sup. Pisa 22 pp 265– (1968) |

[32] | Variational problems for gauge fields, Seminar on Differential Geometry, Editor, Princeton University Press, 1982, pp. 455–464. |

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