Rlaplace[w_] := 
      1/r*D[r*D[w, r], r] + 1/r^2*D[w, ϕ, ϕ]

yht = Expand[Rlaplace[Rlaplace[w[r]]]]

dsol = DSolve[yht == f, w, r]

ww = dsol[[1, 1, 2]]

w1[r_] = 
      ww[r] /. {C[3] -> 0, C[1] -> 0}
w2[r_] = 
      ww[r] /. {C[1] -> D1, C[2] -> D2, C[3] -> D3, 
          C[4] -> D4, f -> 0}

ratk = 
      Solve[{w2[R] == 0, w2'[R] == 
            0, 
w1[a] == 
            w2[a], w1'[a] == w2'[
              a], 
w1''[a] == w2''[
              a], w1'''[a] == w2'''[a]}, {C[2], C[4], D1, D2, 
          D3, D4}]

sol1[r_] = w1[r] /. ratk[[1]]
sol2[r_] = w2[r] /. ratk[[1]]

aa = 1/2;
RR = 1;
ff = 1/π/aa^2;
kuva1 = 
        Plot[sol1[r] /. {f -> ff, a -> aa, R -> RR}, {r, 0, 
            aa}];
kuva2 = 
        Plot[sol2[r] /. {f -> ff, a -> aa, R -> RR}, {r, aa, 
            RR}];
Show[kuva1, kuva2]

lsol1[r_] = 
      Limit[sol1[r] /. {f -> 1/π/a^2}, 
        a -> 0]
lsol2[r_] = 
      Limit[sol2[r] /. {f -> 1/π/a^2}, a -> 0]

Plot[lsol2[r] /. {R -> 1}, {r, 0, 1}]

ratk = 
      Solve[{w2[R] == 0, w2''[R] == 
            0, 
w1[a] == 
            w2[a], w1'[a] == w2'[
              a], 
w1''[a] == w2''[
              a], w1'''[a] == w2'''[a]}, {C[2], C[4], D1, D2, 
          D3, D4}]

sol1[r_] = w1[r] /. ratk[[1]]
sol2[r_] = w2[r] /. ratk[[1]]

aa = 1/2;
RR = 1;
ff = 1/π/aa^2;
kuva1 = 
        Plot[sol1[r] /. {f -> ff, a -> aa, R -> RR}, {r, 0, 
            aa}];
kuva2 = 
        Plot[sol2[r] /. {f -> ff, a -> aa, R -> RR}, {r, aa, 
            RR}];
Show[kuva1, kuva2]

lsol1[r_] = 
      Limit[sol1[r] /. {f -> 1/π/a^2}, 
        a -> 0]
lsol2[r_] = 
      Limit[sol2[r] /. {f -> 1/π/a^2}, a -> 0]

Plot[lsol2[r] /. {R -> 1}, {r, 0, 1}]