Let B = {(x, y) ∈ ℝ2 : x2 + y2 < 1} be the open unit disc in ℝ2, ∂B = {(x, y) ∈ ℝ2 : x2 + y2 = 1} be its boundary and B̅ = B ∪ ∂B. For λ ∈ (0, ∞), let Sλ be the set of twice continuously differentiable functions in B, that are
continuous on B and satisfy
\(\left(\frac{∂ u}{∂ x}\right)^2+\lambda\left(\frac{∂ u}{∂ y}\right)^2=1\), in B
u(x, y) = 0 on ∂B.
Then which of the following statements are true?
1
S1 = ø
2
S2 = ø
3
S1 has exactly one element and S2 has exactly two elements.
4
S1 and S2 are both infinite.