In my book in the chapter about the saddle point theorem there has the following exercise:
Let $\Omega \subset R^n$ $(N \geq 1)$ be a bounded , smooth domain and consider the Dirichlet problem
$$ \left\{ \begin{array}{ccccccc} \Delta u = \lambda_1 + g(u) - h(x) , \ in \ \Omega \\ \\ \hspace{-0.7cm}u = 0 \ on \ \partial \Omega \\ \end{array} \right. $$
where $h: \overline{\Omega} \rightarrow R$ and $g : R \rightarrow R$ are continuous functions with $g$ incresing and such that
$$ g(-\infty) < g(s) < g(+ \infty) , \forall s \in R$$
where $g(+ \infty) = lim_{s \rightarrow + \infty} g(s)$ and $g(- \infty) = lim_{s \rightarrow - \infty} g(s)$ and $\lambda_1$ is the first eigen value of the Laplacian. Show that if $\phi_1 >0$ is a $\lambda_1$ - eigen function and the following Landesman - Lazer condition
$$ g(- \infty) \int_{\Omega} \phi_1 \ dx < \int_{\Omega} h(x) \phi_1 \ dx < g(+\infty) \int_{\Omega} \phi_1 \ dx \ (LL)$$
is satisfied then the problem above has a weak solution. Verify that the second condition (LL) is also necessary for existence of a solution under the other given conditions.
the second part i did. But i dont know how to start the first part .... someone can give me a help with the first part or indicate some reference?
Any help will be apreciated.
thanks in advance