The answet is "Yes", in the sense that a solution will exist in terms of a Fourier-Bessel series. However in practice solving the problem numerically is probably more efficient than trying to solve the Sturm-Liouville problem for the radial basis functions or calculating the coefficients.
The boundary condition on \Gamma_2 is not a self-adjoint condition, but you can solve that by taking \phi = \psi + \frac{K_2}{K_1} so that \psi satisfies Laplace's equation together with \psi = V_1 - \frac{K_2}{K_1} on \Gamma_1, \psi = V_2 - \frac{K_2}{K_1} on \Gamma_3, \partial \psi/\partial r = 0 on \Gamma_4 and \partial \psi /\partial r + K_1 \psi = 0 on \Gamma_2.
Since r = 0 is not in the domain, this is one of those rare instances where we will need to use the Bessel function of the second kind Y_0 as well as J_0, and the radial dependence must be \rho_n(r) = \cos \alpha_n J_0(k_nr) + \sin \alpha_nY_0(k_nr) where the eigenvalues k_n, n = 0, 1, \cdots, satisfy <br />
\left| \begin{array}{cc} k_nJ_0'(k_nR_1) & k_nY_0'(k_nR_1) \\ k_nJ_0'(k_nR_2) + K_1 J_0(k_nR_2) & k_nY_0'(k_nR_2) + K_1Y_0(k_nR_2)<br />
\end{array} \right| = 0 and <br />
\tan \alpha_n = -\frac{J_0'(k_nR_1)}{Y_0'(k_nR_1)}.<br /> The \rho_n are orthogonal with respect to the inner product
\langle f, g \rangle = \int_{R_1}^{R_2} f(r) g(r) r\,dr.
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