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

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.