Stuck on a convergence argument in $H_0^1(\Omega)$.

Question

I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.

However I've encountered this step along the way which seems clear to me but I'm second guessing whether or not it is true.

Question: If $u_k \to u$ in $L^{p+1}$ for $p + 1 < 2^*=\frac{2n}{n-2}$ then I would like to see that $\Delta^{-1}(|u_k|^{p-1}u_k) \to \Delta^{-1}(|u|^{p-1}u)$ in $H_0^1(\Omega)$. This is of course equivalent to showing that $|u_k|^{p-1}u_k \to |u|^{p-1}u$ in $H^{-1}(\Omega)$.

My idea: Since I have convergence in $L^{p+1}(\Omega)$ it follows that I have convergence in all $L^q(\Omega)$ for $p+1 \geq q \geq 1$. By Sobolev embeddings I believe that it's true that $||w||_{H^{-1}} \leq ||w||_{L^q}$ for any $q$ with $1/q + 1/r = 1$ for $1 \leq r \leq 2^*$. So this should imply the needed $H^{-1}(\Omega)$ convergence if I knew that $|u_k|^{p-1}u_k \to |u|^{p-1}u$ in some $L^q$ within this range. The best however I can say is that I have convergence in $L^{\frac{p+1}{p}}$ since $u_k \to u$ in $L^{p+1}$. But then $1 + 1/p > 1 + \frac{n-2}{n+2} = \frac{2n}{n+2}$ which is the conjugate exponent to $2^*$.

This appears to work but is quite technical and messy and all of the 'proofs' I've seen hint at some "simply energy argument". This doesn't appear simple at all! Therefore I would appreciate any suggestions about a better approach or if someone could point out something wrong with how I've thought about it. I hope this fits within the paramaters of the website.

Answer 1

The answer is indeed yes. First, without loss of generality, one may assume that $u_k \to u$ in $L^{p+1}$ and almost everywhere in $\Omega$ and that there exists $g \in L^{p+1}$ such that $|u_k| \leq g$ a.e. (at this step one uses the « almost reverse » of Lebesgue's dominated convergence: if $f_k \to f $ in $L^q$ with $1 \leq q < \infty$, then there exist a function $g \in L^q$ and a subsequence such that $|f_{k_j}| \leq g$ a.e., $f_{k_j} \to f $ a.e.).

Next, using Lebesgue's dominated convergence theorem one concludes that $|u_k|^{p-1}u_k \to |u|^{p-1}u$ in $L^{(p+1)/p}$. Now, as the imbedding $L^{(p+1)/p} \subset H^{-1}$ is continuous (this is true for $(n-2)p \leq n+2$, but in fact it is compact whenever $(n-2)p < n+2$) you can conclude that $(-\Delta)^{-1} |u_k|^{p-1}u_k \to (-\Delta)^{-1}|u|^{p-1}u$ in $H^1_0$.