Consider the following singular Sturm-Liouville problem: $$ -(r^{N - 1}h')' - r^{N - 1}c(r)h = \lambda r^{N - 3} h \text{ in } (0, 1), \qquad h(1) = \alpha $$ where

- $N \in \mathbb N$, $N \geq 3$;
- $c(r) \in L^\infty(0, 1)$;
- $\alpha > 0$.

We begin by defining the functional setting:

$\displaystyle L_N^2 \:= \left\{v:(0, 1) \to \mathbb R \ : \ \int_0^1 r^{N - 1} v^2 \ dr < + \infty \right\}$;

$\displaystyle H^1_N := \left\{v \in L_N^2 \ : \ v' \in L_N^2 \right\}$ with the norm $\displaystyle \|v\|_N^2 = \int_0^1 r^{N - 1} (v^2 + |v'|^2) \ dr$;

$\displaystyle H_{0, N}^1 := \left\{v \in H_N^1 \ : \ v(1) = 0 \right\}$;

$\displaystyle \mathcal L_N := \left\{ v:(0, 1) \to \mathbb R \ : \ \int_0^1 r^{N - 3} v^2 \ dr < + \infty \right\}$;

$\mathcal H_N := H_N^1 \cap \mathcal L_N$;

$\mathcal H_{0, N} := H_{0, N}^1 \cap \mathcal L_N$;

$K_N := \mathcal H_{0, N} + \{\widetilde h\}$ where $\widetilde h \in \mathcal H_N$ is such that $\widetilde h(1) = \alpha$

Observe that $K_N$, being an affine space, has a Hilbert manifold structure with tangent space $T_vK_N = \mathcal H_{0, N}$ at every $v \in K_N$. Then we can define a weak solution of as $h \in K_N$ such that \begin{equation} \int_0^1 r^{N - 1}(h' v' - c(r)hv) \ dr = \lambda \int_0^1r^{N - 3} h v \ dr \quad \forall v \in \mathcal H_{0, N}. \end{equation}

Question: What can one say about existence of (weak?) solutions $(h, \lambda)$ of this problem such that the eigenvalue $\lambda < 0$ and the eigenfunction $h$ is defined in the whole interval $(0, 1)$? Any hope that there will be a solution $h$ for every $\lambda < 0$?

Remark: As noted by Igor Khavkine in the comments, the usage of "eigen-" is imprecise here. I keep it nonetheless.

Thanks in advance