A result on multiplicity of solutions of semilinear boundary value problems (Q1113346)

We prove the existence of (at least) two solutions for the one- dimensional nonlinear boundary value problem \[ (P_ t)\quad - u''=g(u)+h(x),\quad u(0)=u(\pi)=0, \] where the nonlinearity g(u) is supposed to be asymptotically linear for \(u\to -\infty\) and divergent for \(u\to a^->0\). We use a variational approach: the solutions of \((P_ t)\) are thought to be critical points of a functional defined on an open set of \(H^ 1_ 0((0,p))\). We prove that this functional has a local minimum and then, using a deformation lemma of V. Benci, that there is a ``mountain-pass'' solution.