Existence and multiplicity results for classes of elliptic resonant problems. (Q1856804)

In this paper, the author establishes existence and multiplicity results for elliptic resonant problems \[ \begin{aligned} -&\Delta u = \lambda_k u + f(u) \text{ in } \Omega,\\ &u=0 \text{ on }\partial\Omega; \end{aligned} \] where \(\lambda_k\) is an eigenvalue of \(-\Delta\), and \[ \liminf_{v \in E_k, \| v\| \to \infty} \frac{\pm 1}{\| v\| }\int_\Omega \int_0^{v(x)}f(s)\,ds\,dx \geq c \tag{\(f^{\pm}\)} \] for an appropriate constant \(c\), where \(E_k\) is the eigenspace associated to \(\lambda_k\). Let \(f_0(u) = (\lambda_k-\lambda_m)u + f(u)\) and \(F_0(u) = \int_0^u f_0(s)\,ds\), and consider the conditions \[ \pm F_0(u) > 0 \text{ for \(| u| \) small}. \tag{\(f_0^{\pm}\)} \] For \(k=1\) and \(2\), under the conditions \((f^+)\) or \((f^-)\) with \((f_0^+)\) or \((f_0^-)\), and suitable values of \(m\) (with extra assumptions), the author obtain multiplicity results. The proofs rely on the Morse theory and new observations on the critical groups of degenerate critical points of a local linking type.