Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^2(S^{N-1})$ normalized. So $ \phi_0=1$ and $ \lambda_0=0$; $ \lambda_1 = \lambda_2= ... = \lambda_N=N-1$ and $ \lambda_{N+1} =2N$.

Aside. The classical maximum principle shows that if $-\Delta u(x)=f(x)$ in $B$ with zero BC (all boundary conditions in post are zero Dirichlet) we have $\| u\|_{L^\infty} \le \frac{ \|f\|_{L^\infty}}{2N}$. If $f$ has no $k=0$ mode it then appears we have the slightly better estimate $\| u\|_{L^\infty} \le \frac{ \|f\|_{L^\infty}}{4(N+1)}$ (or at least that appears to be the estimate on gets for the $k=1$ mode).

Question. Suppose $f=f(r)>0$ and suppose $g(x)$ has no $k=0$ mode. Additionally we assume $ |g| \le f$. Now suppose $-\Delta v_0 =f$ and $-\Delta v_1=2g$.

Can we say anything about $ v_0-v_1$? In particular can we say $ v_0-v_1 \ge 0$?

Thanks