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 $(f_{k_j})_j$ 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 $H^1_0$.