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On the multiplicity of the solution set of some nonlinear boundary value problems. II. (English) Zbl 0615.35033

The authors study the nonlinear elliptic equation \(\Delta u+f(x,u)=h(x)\) in some bounded domain \(\Omega \subset {\mathbb{R}}^ n\) with \(u|_{\partial \Omega}=0\). Assuming that \(a_{\pm}(x)=\lim_{u\to \pm \infty} (\partial f/\partial u)(x,u)\) is höldercontinuous on \({\bar \Omega}\) with \(a_-(x)<\lambda_ 1\), \(a_+(x)\in (\lambda_ n,\lambda_{n+1})\) on \({\bar \Omega}\) for some \(n\in {\mathbb{N}}\), where \(\{\lambda_ k\}\) are the eigenvalues of the Laplacian, it is shown, that the equation has at least three (two) solutions for n even (odd), provided \(h=s\psi_ 1\) with s large and \(\psi_ 1\) is the first eigenfunction of \(\Delta u+a_+(x)u\). In the case of \(a_{\pm}=const.\), stronger results are known, see e.g. H. Hofer [Math. Ann. 261, 493- 514 (1982; Zbl 0488.47034)].
Reviewer: M.Wiegner

MSC:

35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0488.47034
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