Why should the map $-\Delta^{-1}$ continuous?

Question

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for solutions in $E=\tilde{H}^1_0$, which means this functions are also radially simmetric and in $H^1_0$. The scalar product is $$\langle u,v \rangle_E = \int_\Omega \nabla u \cdot \nabla v $$

The last step of the proof is based on the definition of a functional $T:E\to E$ such that $$\langle Tu,v \rangle_E = \int_\Omega |x|^\lambda |u|^\tau v.$$ How do I know such an opetator is well-defined?

Moreover, it says it can be written in the form $Tu = -\Delta^{-1} (|x|^\lambda |u|^\tau),$ and it goes on saying that the map $T_1 : H^{-1} \to E$ such that $|x|^\lambda|u|^\tau \mapsto -\Delta^{-1} (|x|^\lambda |u|^\tau)$ is continuous. I can't get why this should be well-defined either.

Answer 1

Let $2*=2\,n/(n-2)$. Then $H_0\subset L^{2^*}$. But more can be said for radial functions. By a result of Strauss, if $u\in\tilde H_0(\Omega)$, then $u$ is equal a.e. to a continuous function on $\Omega\setminus\{0\}$ and $$ |u(x)\le C\,|x|^{-(n-2)/2}\,\|\nabla u\|_2. $$ This and related results (some by Ni) can be found in

PABLO L. DE NA ́POLI AND IRENE DRELICHMAN, ELEMENTARY PROOFS OF EMBEDDING THEOREMS FOR POTENTIAL SPACES OF RADIAL FUNCTIONS, arXiv 1404.7468.

This implies that $|x|^\lambda|u|^\tau$ is in some $L^P$ space, and can be integrated against $v\in\tilde H^1_0\subset L^{2*}$. I have not gone through the calculations, but I assume that this will happen precisely when $\lambda$ and $\tau$ satisfy the conditions given in the paper. This defines $|x|^\lambda|u|^\tau$ as an element in the dual of $\tilde H^1_0$, and $\Delta^{-1}\colon H^{-1}\to H^1_0$ is continuous.