The asymptotic behavier of the solution of f″(r)+1rf(r)+a2f(r)=0Consider the Laplace equation with potential term in R2 as Δu+a2u=0 where a&gt;0 is a constant. I am considering the fundamental solution of it, i.e., the solution that is radial symmetric. I denote the fundamental solution as u(x1,x2)=f(r) where r=x12+x22 and I get an ODE,f″(r)+1rf′(r)+a2f(r)=0.I do not know how to solve this ODE and I even do not know the asymptotic behavier on the ∞ and 0. As the fundamental soultion for Laplace equation in R2 is exactly −12πln|x|, I guess that the f here satisfies f∼ln|x|, which means that limx→0|f|ln|x| and limx→∞|f|ln|x| exsit and do not equal to 0. However, I cannot prove it, can you give me some hints or references?

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The asymptotic behavier of the solution of f″(r)+1rf(r)+a2f(r)=0Consider the Laplace equation with potential term in R2 as Δu+a2u=0 where a&amp;gt;0 is a constant. I am considering the fundamental solution of it, i.e., the solution that is radial symmetric. I denote the fundamental solution as u(x1,x2)=f(r) where r=x12+x22 and I get an ODE,f″(r)+1rf′(r)+a2f(r)=0.I do not know how to solve this ODE and I even do not know the asymptotic behavier on the ∞ and 0. As the fundamental soultion for Laplace equation in R2 is exactly −12πln|x|, I guess that the f here satisfies f∼ln|x|, which means that limx→0|f|ln|x| and limx→∞|f|ln|x| exsit and do not equal to 0. However, I cannot prove it, can you give me some hints or references?

delai59qk · Accepted Answer

Step 1Rescale the equation with ρ=arρ2f″+ρf′+ρ2f=0This is the Bessel differential equation with n=0. The homogenous solution isf(ρ)=C1J0(ρ)+C2Y0(ρ)Step 2For a fundamental solution we require thatDf=12πrδ(r)=a22πρδ(ρ)Can you prove from here which C would give you the desired quantity?Hint: near 0, Y0(z)∼2π(γ+log⁡z2) and J0(z)∼1−z24 whereas the homogeneous solutions for the Poisson equation radially were f=C1log⁡r+C2.

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