Consider the following definitions: Let $\phi: R \rightarrow R$ a $C^1$ function.
*) "$\phi$ satisfies the Palais-Smale compactness condition at the level c, or $(PS)_c$ for short, if every sequence $(u_j)\subset R$ such that $\phi(u_j) \rightarrow c$ and $\phi'(u_j)\rightarrow0$ has a convergent subsequence"
**)"$\phi$ satisfies the Cerami condition at the level c, or $(C)_c$ for short, if every sequence suchthat $\phi(u_j)\rightarrow c$ ,$(1+||u_j||)\phi'(u_j)\rightarrow 0$ has a convergent subsequence"
I know that if $\varphi $ satisfies $(*)$ then satisfies (**). But if satisfies $(* *)$ then satisfies $(*)$ ?
I believe that the conditions are equivalents, but i dont know how to prove this. Someone can give me a help to prove or give a counter example?
Consider for example the Dirichlet problem $$ \begin{cases} -\Delta u = \alpha u -g(u) &\text{in $\Omega$}\\ u=0 &\text{on $\partial \Omega$} \end{cases} $$ with standard smoothness of $\Omega$ and $\alpha \in \mathbb{R}$. Assume that $\lim_{|t| \to +\infty} g(t)=0$ and $\lim_{|t| \to +\infty} \int_0^t g(s)\, \mathrm{d}s=\beta\in\mathbb{R}$. In general, the Palais-Smale condition for the Euler functional $$ I(u)=\frac{1}{2}\int_\Omega |\nabla u|^2\, \mathrm{d}x -\frac{\alpha}{2}\int_\Omega |u|^2 \, \mathrm{d}x + \int_\Omega G(u)\, \mathrm{d}x $$ (where $G'=g$ and $G(0)=0$) is not satisfied, in general. However the Cerami condition can hold true. You can refer to the paper Abstract critical point theorems and applications... by Bartolo, Benci and Fortunato, published in 1983 on Nonlinear Analysis, volume 7, pages 981-1012.