CSIR NET Maths previous year question papers

Define S := {$y \in C^{1}[0, 1] : y(0) = y(1) = 0$}, ‖y‖$(\infty) := \max(x\in[0,1])$|y(x)| for every $y \in S, B_{0}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) < \varepsilon$}, $B_{1}(0, \varepsilon) :=$ {y ∈ S : ‖y‖$(\infty) +$ ‖y′‖$(\infty) < \varepsilon$}. Consider the functional $J : S \to \mathbb{R}$ given by $J[y] = \int_{0}^{1}[(y')^{2} - 2x(y')^{4}]$dx, then there exists an $\varepsilon > 0$ such that

Define S := {$y \in C^{1}[-1, 1] : y(-1) = -1, y(1) = 3$}. Let $\varphi$ be the extremal of the functional $J : S \to \mathbb{R}$ given by $J[y] = \int$₋$_{1}^{1}[(y')^{3} + (y')^{2}]$dx. Define ‖y‖$(\infty) := \max(x\in[-1,1])$|y(x)| for every y ∈ S and let $B_{0}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) < \varepsilon$}, $B_{1}(\varphi, \varepsilon) :=$ {y ∈ S : ‖$y - \varphi$‖$(\infty) +$ ‖$y' - \varphi'$‖$(\infty) < \varepsilon$}. Then which of the following statements are true?