Let $\mathbb{R}^n$ be the $n$-dimensional real vector space with Cartesian coordinates $x=(x^1,\ldots, x^n)\in \mathbb{R}^n$. I'm searching for a non-trivial example of a function $A:\mathbb{R}^n \rightarrow \mathbb{R}$, which is continuously differentiable on $\mathbb{R}^n\backslash \lbrace 0\rbrace$ (i.e. $A\in C^\infty(\mathbb{R}^n\backslash \lbrace 0\rbrace)$), such that it satisfies \begin{align} A(tx)&=tA(x),\newline \sum^{n}_{i,j=1}\xi^i A(x)\frac{\partial^2 A(x)}{\partial x^i \partial x^j} \xi^j &\le 0, \end{align} for all $t\in \mathbb{R}$, $x, \xi \in \mathbb{R}^n$.

Obviously, if $A$ is chosen to be a linear function, it is a solution to the problem above. But are there any non-linear solutions for dimension $n\ge 3$? Is there some topological argument against the existence of such solutions?