Consider the following dual integral equations which has been obtained while solving a mixed boundary value problem: $$ \int_0^\infty \psi_k(\lambda) J_k (\lambda r) \mathrm{d}\lambda = F(r) \delta_{1k} \qquad (0 < r < R) $$
$$ \int_0^\infty \lambda \psi_k(\lambda) J_k (\lambda r) \mathrm{d}\lambda = 0 \qquad (r>R) $$ where $k \in \mathbb{N}$ and $\psi_k(\lambda)$ are the unknown functions. In addition, $F(r)$ is a known (radial) function.
The solution for $k=1$ can readily be obtained following the standard recipes by Sneddon. Accordingly, the order of Bessel function can be reduced to zero by multiplying by $r$ and differentiating by $r$ to obtain $(k=1)$ $$ \int_0^\infty \lambda \psi_1(\lambda) J_0 (\lambda r) \mathrm{d}\lambda = \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d} r} \left( rF(r) \right) 0 := G(r) \qquad (0 < r < R) $$
$$ \int_0^\infty \lambda^2 \psi_1(\lambda) J_0 (\lambda r) \mathrm{d}\lambda = 0 \qquad (r>R) $$
By making use of the identity $$ \int_0^\infty J_0(\lambda r) \sin(\lambda t) \mathrm{d}\lambda = \frac{H(t-r)}{\left(t^2-r^2\right)^{1/2}} $$ a solution of the outer problem above $(r>R)$ can be chosen of the form $$ \psi_1(\lambda) = \frac{1}{\lambda^2} \int_0^R \chi_1 (t) \sin(\lambda t) \mathrm{d} t $$ where $\chi_1(t)$ is now the new unknown function. Upon substitution of this solution into the inner problem equation, the problem can eventually be transformed into a Fredholm integral.
My question is:
For $k \ne 1$, $\psi_k(\lambda) = 0$ are trivial solutions of the problem. Are these unique solutions or other solutions also can exist?
Any help or advice would be highly appreciated.
Thank you.