I want to ask a question about a statement that I found on the paper: Principal Eigenvalues for Problems With indefinite Weight Function in $R^N$. 


The statement is the following: 


Suppose that $g:\mathbb{R}^2\to\mathbb{R}$ is a $C^{\infty}$ function which changes sign on $\mathbb{R}^2$ and there exist constants $K,R>0$ such that $g(x)\leq-K$ for $|x|>R$. Let $B$ a ball such that 
$$
\int\_{B} g(x) dx <0 
$$  

and $g(x)<0$ if $x\in \mathbb{R}^2\setminus B$. 



>  I would like to know why is true that there exist a positive constant $C\_1>0$ such that 
> $$
> \int\_{B} u^2 dx \leq C\_1\int\_{B}|\nabla u|^2 dx 
> $$
> for all $u\in H^1(B)$ with $\int\_{B}g u^2 dx>0$ ?
> 


The authors give the following reference for this result: 
K.J. Brown, S.S. Lin and A. Terkitas, Existence and noexistence of steady-state solutions for a selection-migration model in popular genetics. J. Math. Biol. 27 (1989), 91-104. 


I can't have access to this paper that's why I am posting this question here. 


I am trying to prove this inequality, but the results I know are based on arguments that requires compact support for $u$ or mean zero. I appreciate if some specialist can give me a hint on how to prove this inequality for the general case (neither compact support or mean zero) or provide an alternative reference where it is proved. Thanks.



How to prove this Poincare Inequality
Just assume that $g$ is a bounded function with $\int\_B g < 0$, and positive somewhere in th ball $B$ . Then the set 
$$\Big \{u\in H^1(B)\, : \|u\|\_2= 1\, ,\, \int\_B g u^2\ge 0 \Big \}$$
is not empty and does not contain constant functions. By weak compactness $\|\nabla u\|\_2$ attains a non-zero minimum on it, and by homogeneity we get the thesis.


