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 (and continuous).

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 symmetric 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 operator 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 (and continuous).