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Hi I'm stuck with the proof of a concentration-compactness lemma. We have the following equation in $\mathbb{R}^N, N \ge 3$: $$ -\Delta u +u=|u|^{p-2}u, $$ where $2 < p < 2^{*}$. The functional assoc …

Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$. Suppose the restriction of $f$ on $Y$ has a critical point $x_0 \in Y$. Q: Is $x_0$ a critical point of $f$?

Consider a second order Hamiltonian system of the type $$ \ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N. $$ Under very `natural assumptions' it is possible to prove the existence of a non constant $T$-pe …

Let $\Gamma \subset \mathbb{R}^2$ be a closed simple $C^1$ curve. For every $x \in \mathbb{R}^2\setminus\Gamma$ there exists some $p(x) \in \Gamma$ such that $$ (H) \quad \text{ dist}(x,\Gamma)=|x-p( …

Consider the following equation in $\mathbb{R}^N, N \ge 3$: $$ (E) \quad -\Delta u +u=|u|^{p-2}u, $$ where $2 < p < 2^{*} =2N/(N-2)$. Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the functional th …