Mathematische Zeitschrift

, Volume 228, Issue 4, pp 723–744

# 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
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