I have a class of PDEs of the form $$ -\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0 $$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and here). Then, I consider their Green function, being these linear equations, and I want to solve $$ -\Box G(x)+\lambda\phi_0^2(x)G(x)=\delta^4(x). $$ Using linearity, I can apply a Fourier transform and I get $$ p^2\tilde G(p)+\lambda\sum_{n=-\infty}^\infty b_n\tilde G(p_n-p)=1. $$ Is there any reasonable approach to get a solution out of this equation?

I have a class of PDEs of the form $$ -\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0 $$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and here). Then, I consider their Green's functions, being these linear equations, and I want to solve $$ -\Box G(x)+\lambda\phi_0^2(x)G(x)=\delta^4(x). $$ Using linearity, I can apply a Fourier transform and I get $$ p^2\tilde G(p)+\lambda\sum_{n=-\infty}^\infty b_n\tilde G(p_n-p)=1. $$ Is there any reasonable approach to get a solution out of this equation?