First of all, there should be a constraint on the exponent $\tau$ to guarantee certain integrability  conditions. Define

$$n^*:=\frac{2(n+\lambda)}{n-2}, $$

where $n$ denotes the dimension of the ball. Then $E$ embeds continuously in the Banach space $ X:=L^{n^*}(0,1; r^{\lambda+n-1} dr)$. (For a proof of this I refer to this [old paper of mine][1].)

Define $\tau^*:=n^*-1$. (This is the critical Sobolev exponent.) Assume


$$\tau \leq \tau^*. $$

Set $r:=|x|$, $x\in \mathbb{R}^n$. Use  Holder's inequality  for the conjugate exponents $n^*$ and $\frac{n^*}{\tau^*}$  and the Sobolev embedding $E\to X$ to verify that

$$\left\vert\int_\Omega r^\lambda |u|^\tau v dx\right\vert \leq C\Vert u\Vert_E\cdot \Vert v\Vert_E. $$

This proves that $Tu$ defines a linear functional on $E$. The dual of $E$ is $H^{-1}$.

The Laplacian  defines an isomorphism $\Delta E\to H^{-1}$ with inverse  $\Delta^{-1}$. The continuity statement   is proved by observing that the map

$$ E\ni u\mapsto |u|^\tau \in L^{\frac{n^*}{\tau^*}}(0,1, r^{\lambda+n-1} dr). $$

is continuous.





  [1]: http://www3.nd.edu/~lnicolae/Existence_and_regularity.pdf