Some hints to prove it, because I want to know whether I can use the Sobolev embedding theorem for this domain $\Omega$ = $[0,\infty)$ or not.
(The Cone Condition) : $\Omega$ satisfies the cone condition if there exists a finite cone C such that each x $\in \Omega $ is the vertex of a finite cone $C_{x}$ contained in $\Omega$ and congruent to C. Note that $C_{x}$ need not be obtained from C by parallel translation, but simply by rigid motion.
Sobolev Imbedding Theorem I want to use :
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ an Let $0\leq j$ and $1\leq m$ be integers and let $1\leq p\leq \infty$. Suppose that $\Omega$ satisfies the cone condition then : if either $n<mp$ or $m=n$ and $p=1$ then, : $$W^{j+m,p}(\Omega) \rightarrow C^{j}_{B}(\Omega)$$
Here $W^{j+m,p}(\Omega)$ is a Sobolev space and $C^{j}_{B}$ is the space of j-times continuously differentiable and bounded functions on $\Omega$
In My case , $\Omega$ = $[0,\infty)$ , j=0 , p=1 and m=1 i.e the sobolev space $H^{1}([0,\infty))