How to prove this Poincare Inequality

Question

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.

Answer 1

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.