Stuck on a convergence argument in $H_0^1(\Omega)$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:47:25Z http://mathoverflow.net/feeds/question/38309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38309/stuck-on-a-convergence-argument-in-h-01-omega Stuck on a convergence argument in $H_0^1(\Omega)$. Dorian 2010-09-10T14:04:41Z 2011-04-29T04:14:05Z <p>I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.</p> <p>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.</p> <p><strong>Question:</strong> If $u_k \to u$ in $L^{p+1}$ for $p + 1 &lt; 2^*=\frac{2n}{n-2}$ 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 $H^{-1}(\Omega)$.</p> <p><em>My idea:</em> 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 <code>$||w||_{H^{-1}} \leq ||w||_{L^q}$</code> 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 <em>if</em> 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 $2^*$. </p> <p>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.</p> http://mathoverflow.net/questions/38309/stuck-on-a-convergence-argument-in-h-01-omega/63339#63339 Answer by Otared Kavian for Stuck on a convergence argument in $H_0^1(\Omega)$. Otared Kavian 2011-04-28T21:00:13Z 2011-04-29T04:14:05Z <p>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 $1 \leq q &lt; \infty$, 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.).</p> <p>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 $(n-2)p &lt; n+2$) you can conclude that $(-\Delta)^{-1} |u_k|^{p-1}u_k \to (-\Delta)^{-1}|u|^{p-1}u$ in $H^1_0$.</p>