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A remark on the existence of three solutions via sub-super solutions. (English) Zbl 0647.35031
Nonlinear analysis and applications, Proc. 7th Int. Conf., Arlington/Tex. 1986, Lect. Notes Pure Appl. Math. 109, 561-566 (1987).
Summary: [For the entire collection see Zbl 0632.00014.] We prove that if there exists a sub solution $\psi\sb 1$, a strict super solution $\phi\sb 1$, a strict sub solution $\psi\sb 2$, and a super solution $\phi\sb 2$ for $$ (1.1)\quad Lu(x)=-\Delta u(x)+qu(x)=f(x,u(x));\quad x\in \Omega, $$ $$ (1.2)\quad Bu(x)=u(x)=0;\quad x\in \partial \Omega, $$ such that $\psi\sb 1<\phi\sb 1<\phi\sb 2$, $\psi\sb 1<\psi\sb 2<\phi\sb 2$ and $\psi\sb 2\nleq \phi\sb 1$, then (1.1)-(1.2) has at least three distinct solutions $u\sb s$ $(s=1,2,3)$ such that $\psi\sb 1\le u\sb 1<u\sb 2<u\sb 3\le \phi\sb 2$. (Here we write $z\sb 1<z\sb 2$ when $z\sb 1\le z\sb 2$ and $z\sb 1\ne z\sb 2.)$ This extends the work by {\it H. Amann} [SIAM Review 18, 620- 709 (1976; Zbl 0345.47044)] where the case $\psi\sb 1<\phi\sb 1<\psi\sb 2<\phi\sb 2$ was studied.

35J65Nonlinear boundary value problems for linear elliptic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B35Stability of solutions of PDE