The equation is from the paper "Point defect sink efficiency of low-angle tilt grain boundaries". I get confused that the author get a solution without firstly giving the boundary condition.
the equation is $$\Delta c=\delta(\Gamma)$$ and $\delta(\Gamma)$ is the Dirac delta function of $\Gamma$ $$\delta(\Gamma) = \sum_{j=-\infty}^{+\infty}\delta(x)\delta\left(y-jp\right)$$
they have one solution periodic in y , $$c(x,y)=\frac{1}{4\pi}\ln(2\cosh\frac{2\pi x}{p}-2\cos\frac{2\pi y}{p})+C$$ C is a constant. I can't get the solution while the author didn't put the proof on appendix. I've try WKB theory or boundary layer theory but no useful