\begin{align} & f(x)^2 \frac d {dx}\, \frac{g(x)}{f(x)} = f(x)g'(x) - f'(x)g(x). \tag 0 \\[10pt] & g(x)^2 \frac d {dx}\, \frac{f(x)}{g(x)} = g(x)f'(x) - g'(x)f(x). \tag 0 \\[10pt] \text{Therefore } & f(x)^2 \frac d {dx}\, \frac{g(x)}{f(x)} + g(x)^2 \frac d {dx}\, \frac{f(x)}{g(x)} = 0. \tag 1 \en...

Consider the $(d+1)$-Laplacian denoted as $\Delta_{d+1}$, ($p$-laplacian with $p=d+1$) and its eigenfunctions in $\mathbb{R}^d$. I'd like to know whether we can say that any linear combination of eigenfunctions belonging to the same eigenvalue is also an eigenfunction with the same eigenvalue? ...