Palais-smale condition - Open Access .click

Nonlinear Analysis: Theory, Methods & Applications, 1991
Let \(\phi: X\to\mathbb R\) be a given functional on a Banach space \(X\). \(\phi\) is said to be coercive if \(\phi(u)\to +\infty\) as \(\| u\| \to \infty\). This is equivalent to saying that for each \(d\in\mathbb R\), the set \(\Phi^ d=\{u\in X: \phi(u)\leq d\}\) is bounded. A differentiable functional (in the sense of Fréchet) \(\phi: X\to\mathbb R\
Costa, David G., Silva, Elves Alves de Barros e +1 more
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A Necessary and Sufficient Condition for Palais--Smale Conditions

SIAM Journal on Mathematical Analysis, 1999
Let \(\Omega\) be an arbitrary domain in \({\mathbb{R}}^N\), \(N\geq 2\). Denote \(2^*= +\infty\) if \(N=2\), and \(2^*=2N/(N-2)\) for \(N\geq 3\). The paper establishes a necessary and sufficient condition for the Palais-Smale property related to the boundary value problem \(-\Delta u+u=|u|^{p-2}u\) in \(\Omega\), subject to the Dirichlet condition ...
Chen, Kuan-ju, Wang, Hwai-chiuan
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A weak nonsmooth palais-smale condition and coercivity

Rendiconti del Circolo Matematico di Palermo, 2000
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Kourogenis, Nikolas C., Papageorgiou, Nikolaos S. +1 more
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mathematics
variational methods
fos: mathematics

coercivity
analysis
critical points

critical point
ekeland's principle
variational methods involving nonlinear operators