# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${\psi }_{1}$, a strict super solution ${\phi }_{1}$, a strict sub solution ${\psi }_{2}$, and a super solution ${\phi }_{2}$ for

$\left(1·1\right)\phantom{\rule{1.em}{0ex}}Lu\left(x\right)=-{\Delta }u\left(x\right)+qu\left(x\right)=f\left(x,u\left(x\right)\right);\phantom{\rule{1.em}{0ex}}x\in {\Omega },$
$\left(1·2\right)\phantom{\rule{1.em}{0ex}}Bu\left(x\right)=u\left(x\right)=0;\phantom{\rule{1.em}{0ex}}x\in \partial {\Omega },$

such that ${\psi }_{1}<{\phi }_{1}<{\phi }_{2}$, ${\psi }_{1}<{\psi }_{2}<{\phi }_{2}$ and ${\psi }_{2}\nleqq {\phi }_{1}$, then (1.1)-(1.2) has at least three distinct solutions ${u}_{s}$ $\left(s=1,2,3\right)$ such that ${\psi }_{1}\le {u}_{1}<{u}_{2}<{u}_{3}\le {\phi }_{2}$. (Here we write ${z}_{1}<{z}_{2}$ when ${z}_{1}\le {z}_{2}$ and ${z}_{1}\ne {z}_{2}·\right)$ This extends the work by H. Amann [SIAM Review 18, 620- 709 (1976; Zbl 0345.47044)] where the case ${\psi }_{1}<{\phi }_{1}<{\psi }_{2}<{\phi }_{2}$ was studied.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B35 Stability of solutions of PDE
##### Keywords:
existence; multiplicity; sub solution; super solution