How does the author easily conclude that we can construct such a sequence? On page 8 of the book I am reading (available here, I added a screenshot below), the author constructs a sequence of functions in the unit disk having some nice properties. In particular, each element of the sequence is continuous, even piece-wise smooth and satisfies


*

*$\phi_n = 0$ on the boundary of the disk,

*$\int_{\mathbb{D}} \lvert \nabla \phi_n \rvert^2 \to 0$
Now, notice that $\phi_n(0)\to\infty$. The author claims that, by the same construction, we can find a sequence $\phi_n$ satisfying the aforementioned properties and such that $\phi_n$ diverges in a dense subset of the unit disk.
I don't see how this could possibly be true. If anyone could clarify what the author means I would appreciate it!

 A: I will assume the existence your function. Further I will assume it is $≥0$ by the definition in your image. Let $x_k$ be a dense enumeration of the interior of the unit ball. Let $d_k$ be the distance of $x_k$ to the exterior of the ball. Extend $\phi_n$ to be zero outside the unit ball. Now
$$f_{n,k}:=\phi_n\left(\frac{x-x_k}{d_k}\right) \cdot d_k^{1+\frac N2}$$
Here $N$ is the dimension of the ball you are looking at. This is a $≥0$ function supported entirely within the unit ball and $f_{n,k}(x_k)\to\infty$ as $n\to\infty$. Further because of the multiplication with $d_k^{1+\frac N2}$ the integral of $\|\nabla f_{n,k}\|^2$ over the unit ball is independent of $k$. Now we have our function:
$$\varphi_n(x) := \sum_{k=1}^n 2^{-2k}\ f_{n,k}(x).$$
Verify that $\int\|\nabla \varphi_n\|^2≤\int \|\phi_n\|^2$ and thus your condition $2$ is satisfied $(*)$. Everything is supported inside the unit ball by construction, so condition $1$ also is satisfied.
Now your functions are positive everywhere and $f_{n,k}(x_k)\to \infty$. For that reason also $\varphi_n(x_k)$ will go to infinity, as the explosion of $f_{n,k}$ at this point cannot be compensated by negative terms.
$(*)$ for this point you may need $(a+b)^2≤2a^2+2b^2$. This will turn $\left(\sum_{k=1}^n a_k\right)^2 ≤\sum_{k=1}^n 2^ka_k^2$. In the sum above the $2^k$ terms will be eaten by the suppressing factor. 
