Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue (Q1272327)

In this paper are given some results concerning the existence of multiple solutions for the following equation in a bounded domain \(\Omega\) of \(\mathbb{R}^{N}\) with smooth boundary \[ -\Delta u=p(x,u) \] with \(u=0\) on the boundary , by assuming \[ p_{0}=\lim_{u\rightarrow 0}\frac { p(x,t)}{u}, \] \[ p_{\infty }=\lim_{|u|\rightarrow +\infty }\frac{ p(x,u)}{u}. \] The authors, using the Morse theory, prove the existence of at least two solutions when the nonlinearity crosses at least one eigenvalue between \(0\) and infinity. Let \(0 \leq \lambda _{1}\leq \lambda _{2}\leq\ldots \) be the eigenvalues of \( (-\Delta)\), with Dirichlet boundary conditions, they prove that there are two nontrivial solutions when \(p_{0} = p_{\infty }=\lambda _{j+1}\). The two point boundary value problem is also studied.