Existence for a singular Sturm-Liouville "eigenvalue" problem with non-homogeneous boundary condition

Question

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

Answer 1

Write your differential equation as $Ph = \lambda h$, the differential operator is unbounded and symmetric on the Hilbert space $\mathcal{L}_N$, with an appropriate domain. As analyzed by Amadori-Gladiali in the reference you pointed out, with the boundary condition $h(1)=0$ it is also self-adjoint and has a discrete spectrum, meaning that the resolvent $(P-\lambda)^{-1}$ exists as a bounded operator for all $\lambda \not\in \sigma(P)$ not belonging to the spectrum.

Now comes a standard idea. Choosing some sufficiently regular $\tilde{h}$ satisfying $\tilde{h}(1) = \alpha$ and setting $h=\tilde{h}+f$, your homogeneous equation with inhomogenous boundary conditions becomes an inhomogeneous equation with homogeneous boundary conditions $(P-\lambda)f = -(P-\lambda)\tilde{h}$ (when $\tilde{h}$ is sufficiently regular, the right-hand side is in $\mathcal{L}_N$). The solution $f = (P-\lambda)^{-1} [-(P-\lambda)\tilde{h}]$ will be in $\mathcal{L}_N$ and unique as long as $\lambda\not\in\sigma(P)$. Since the spectrum only has eigenvalues, it's a matter simple linear algebra to figure out what happens when $\lambda \in \sigma(P)$.