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

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## 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## Preview

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