Vtyjkyuk

Problem:

Consider the BVP:
beginequation
labeleq1
begincases
fracpartial^2Psipartialrho^2 + frac1rholeft(frac1-h^2rho^21+h^2rho^2right)fracpartialPsipartialrho+frac1+h^2rho^2rho^2fracpartial^2Psipartialzeta^2 = frac-2hF_01+h^2rho^2,, & texton Omega,,\
Psi(rho,zeta) = 0,, & texton partialOmega,.
endcases
endequation
where $Omega$ is the ellipse $(x,y)inmathbbR^2mid (x-R_0)^2/a^2 + y^2/b^2leq 1$ with $R_0-a>0$ and $rho,zeta$ are polar coordinates. I am trying to determine the number of critical points the solution to the above equation has, and the nature of these critical points.

My Attempts:

Numerical Solutions using Mathematica:

The numerical solution to the PDE using NDSolveValue has a single critical point on the line $zeta= 0$. Moreover, along this line we find that $fracpartialPsipartialrho$ seems to be monotonically decreasing. Another point to note is that as $b/a$ increases past $1$, the location of the critical point shifts to the right of $R_0$. Shown below is the solution for $a=1,b=3$.

Analysis

The above question is equivalent to finding the zeroes of $Y =(fracpartialPsipartialrho,fracpartialPsipartialzeta)= (Y_1,Y_2)$ satisfying
beginequation
labeleq2
left[left(fracpartialpartialrho,frac1+h^2rho^2rho^2fracpartialpartialzetaright) + frac1rholeft(frac1-h^2rho^21+h^2rho^2right)left(1,0right)right]cdotleft(Y_1,Y_2right) = frac-2hF_01+h^2rho^2
endequation

By the symmetry of the PDE, we have that $Psi(rho, -zeta) = Psi(rho,zeta)$. Thus $Y_2(rho, -zeta) = -Y_2(rho, zeta)$ so $Y_2(rho,0) = 0$. The boundary condition $Psi = 0$ on $partialOmega$ implies that $nablaPsi$ points inwards on the boundary, so $Y_1(R_0-a,0)$ and $Y_1(R_0+a,0)$ are both non-zero and have opposite sign. Therefore at some point on the line $zeta=0$ we have $Y_1(rho,zeta)=0$ and so there is at least one critical point on the line $zeta = 0$.

Looking for solutions along $zeta=0$

points along $zeta = 0$:}
Along $zeta=0$, we can consider $Y_1$ as a function of only $rho$. So the above equation becomes
$$
Y_1'(rho) + frac1rholeft(frac1-h^2rho^21+h^2rho^2right)Y_1(rho) = frac-2hF_01+h^2rho^2 - frac1+h^2rho^2rho^2fracpartial Y_2partialzeta(rho,0)
$$
which implies,
$$
(IY_1)' = frac-2hrho F_0(1+h^2rho^2)^2 - frac1rhofracpartial Y_2partialzeta(rho,0)
$$
where $I = expleft(int_1^rhofrac1rholeft(frac1-h^2rho^21+h^2rho^2right),dsright) = fracrho1+h^2rho^2$. Since our numerical results suggest that $Y_1$ is monotonically decreasing, one approach would be to show that $(IY_1)' < 0$ everywhere on $zeta=0$. Unfortunately, $fracpartial Y_2partialzeta(rho,0)$ depends on boundary information not just on the line $zeta = 0$.

Does anyone have any advice for how I can further approach this problem?