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Mathematische Zeitschrift

, Volume 228, Issue 4, pp 723–744 | Cite as

Liouville–type theorems for semilinear elliptic equations involving the Sobolev exponent

  • Chang-Shou Lin

Abstract.

In this article, we prove there are no positive smooth solutions of \( \Delta u +K(x) u^{\frac{n+2}{n-2}} =0 \quad \text{ in } \mathbb{R}^n , \leqno(0.1) \) where \(K(x)\in C^1(\mathbb{R}^n)\) satisfies one of the following conditions: (i) K is a subharmonic function in \({\mathbb{R}}^n\) with \(K(\infty) =\lim\limits_{|x|\to +\infty}K(x) >0\), and the derivative \(|\nabla K(x)|\) satisfies \( c_1|x|^{-(l+1)}\leq |\nabla K(x)|\leq c_2|x|^{-(l+1)} , \) where \(l>\frac{1}{2}\) for \(n=3\), \(l>1\) for \(4\leq n\leq 6\) and \(l\geq \frac{n-2}{2}\) for \(n\geq 7\). (ii) \(K(x) \neq \text{constant}\) is nondecreasing along each ray \(\{ t\xi | t\geq 0\}\) for any unit vector \(\xi\) in \(\mathbb{R}^n\) and \(\lim\limits_{|x|\to +\infty}K(x)=K(\infty) >0\). (iii) \(K(x_1, \cdots , x_n) \equiv K(x_1)\) is nondecreasing in \(x_1\), \(K\equiv K_1\) for \(x_1\leq b\) and \(K\equiv K_2>0\) for \(x_1\geq a\), where \(K_1,K_2,a\) and b are constants. Various generalizations to a more general class of nonlinearities are also considered.

Keywords

Unit Vector Elliptic Equation General Class Smooth Solution Type Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Chang-Shou Lin
    • 1
  1. 1. Department of Mathematics, National Chung Cheng University, Ming Hsiung, Chia Yi, Taiwan (E-mail: cslin@math.ccu.edu.tw) TW

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