Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$:
$$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$
Providing reasons specify if the $\inf J$ over $X$ is attained of not
(1)- $a>0, b \geq 0, X =\{ u \in C^1([0,1]); u(0)=0, u(1)=0 \}$
(2)- $a>0, b > 0, X =\{ u \in AC([0,1]); u(0)=0, u(1)=0 \}$
(3)- $a<0, b \geq 0, X =\{ u \in AC([0,1]); u(0)=0, u(1)=1 \}$
My proof
I am trying to apply the following theorem to find the answer.
Special version of Tonelli’s theorem Assume that the function $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{R},\,\, g(x, \xi): [a,b] \times \mathbb{R} \to \mathbb{R}$ are continuous, $f$ is bounded below, $g$ is convex in $\xi$ and satisfies
$$\exists r>1,\, \exists C>0\,\, \text{such that}\,\, g(x,\xi) \ge C| \xi|^r,\,\, \forall (x, \xi) \in [a,b] \times \mathbb{R}.$$
Then there exists a minimizer of the functional $$J[u] = \displaystyle\int_a^b (f(x,u(x)) + g(x,u'(x))) dx$$ in the space $X= \{ u \in AC([a,b]); u(a)=\alpha, u(b)= \beta \}.$
Here I considered $f(x,u(x)) = b \ln (1+u^2(x))$ and $g(x,u'(x))= (u'(x)^2 -a)^2$. Am I in the right path? I am not sure what should I de next?