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Results tagged with calculus-of-variations
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user 14743
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
2
votes
Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$
If I am not grossly mistaken, with your notations, set $c_k := \inf_{\gamma \in \Gamma_k}\max_{u \in \gamma([0,1])} J_k(u)$ (where $\Gamma_k$ is the set of paths $\gamma : [0,1] \to H^1_0(B_k(0))$ suc …
2
votes
Stuck on a convergence argument in $H_0^1(\Omega)$.
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 …