Some remarks on the number of solutions of some nonlinear elliptic problems (Q1070109)

This paper contains several (sharp) multiplicity results for the problem \[ -\Delta u=f(u)+t\phi \quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with a smooth boundary \(\partial \Omega\), \(f\in C^ 1(R)\) is asymptotically linear, \(\phi\) is a (normalized) eigenfunction of the Laplacian with homogeneous Dirichlet boundary conditions, and \(t\in R\) is a parameter. These results depend on the nature of the interaction of f with the first three eigenvalues of the Laplacian. They improve upon previous results of several authors, including \textit{A. C. Lazer} and \textit{P. J. McKenna} [J. Math. Anal. Appl. 84, 282-294 (1981; Zbl 0496.35039) as well as subsequent papers], and \textit{H. Hofer} [Math. Ann. 261, 493-514 (1982; Zbl 0488.47034)].