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**Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.**
*(English)*
Zbl 0541.35029

Semilinear 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### Citations:

Zbl 0336.53033
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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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