The third solution of semilinear elliptic boundary value problems and applications to singular perturbation problems (Q1206393)

The author studies an elliptic problem of the type \[ \Delta u(x)+f(x,u(x))=0,\;x\in\Omega, \qquad p(x)u(x)+q(x) {{\partial u} \over {\partial \nu}}(x)= \varphi(x),\;x\in\partial\Omega. \tag{P} \] Following the method of \textit{K. Schmitt} [Nonlinear Anal., Theory, Methods Appl. 2, 263-309 (1978; Zbl 0378.35022)], he introduces the notion of quasi- supersolution and quasi-subsolution of (P). In the main result he proves that, if \(\overline{w}_ 1\), \(\overline {w}_ 2\) are quasi-subsolutions and \(\widehat {w}_ 1\), \(\widehat {w}_ 2\) are quasi-supersolutions of (P) such that \[ \forall x\in \overline {\Omega}:\;i\leq j \Longrightarrow \overline{w}_ i(x)\leq \widehat {w}_ j(x), \qquad \exists x_ 0\in \overline{\Omega}:\;\overline{w}_ 2(x_ 0)> \widehat {w}_ 1(x_ 0) \] and if \(\widehat{w}_ 1\) and \(\overline{w}_ 2\) are not solutions of (P), then (P) possesses at least three distinct solutions. This result is then applied to a singular perturbation problem of the form \[ \varepsilon^ 2 \Delta u(x)+ f(x,u(x))=0,\;x\in\Omega, \qquad u(x)=0,\;x\in\partial\Omega. \]