Multiplicity result for semilinear dissipative hyperbolic equations (Q1282995)

The paper concerns the problem \[ \partial^2_t u+\beta\partial_tu- \Delta u- \lambda_1u+ g(u)= {s\phi_1\over \sqrt{2\pi}}+ h(t,x),\quad 0< t<2\pi,\quad x\in\Omega, \] \[ u= 0,\quad 0<t<2\pi,\quad x\in\partial\Omega, \] \[ u(0,x)= u(2\pi, x),\quad x\in\Omega. \] Here \(g:\mathbb{R}\to\mathbb{R}\) is a linearly growing function such that \[ \liminf_{|u|\to\infty} f(u)= +\infty\quad\text{and} \quad \limsup_{u\to-\infty} {g(u)\over u}< \lambda_2- \lambda_1. \] Further, \(\Omega\subset \mathbb{R}^n\) is a bounded domain with smooth boundary, \(\lambda_1\) and \(\lambda_2\) are the first eigenvalues of \(-\Delta\) on \(\Omega\) with zero Dirichlet boundary conditions, \(\phi_1\) is the eigenfunction corresponding to \(\lambda_1\) normalized by \(\int_\Omega \phi_1(x)^2 dx=1\), and \(h\in L^2((0, 2\pi)\times \Omega)\) is such that \(\int^{2\pi}_0 \int_\Omega h(t,x) \phi_1(x) dx dt= 0\). The author proves the following Ambrosetti-Prodi type result. Let \(s_0:= \sqrt{2\pi} \int_\Omega \phi(x) dx\). Then there exists a real number \(s_1\geq s_0\) such that for \(s>s_1\) there exist at least two solutions, for \(s= s_1\) there exists at least one solution, and for \(s<s_0\) there is no solution.