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On the superlinear Ambrosetti-Prodi problem. (English) Zbl 0554.35045

The author considers the problem \(-\Delta u=g(x,u)+t\phi_ 1+h\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Omega\) is a smooth bounded domain in \(\mathbb R^ N\), \(\liminf_{s\to \infty} s^{-1}g(x,0)>\lambda_ 1\limsup_{s\to -\infty} g'\!_ 2(x,s)<\lambda_ 1\) and \(\phi_ 1\) is the positive eigenfunction of -\(\Delta\). If \(g\) grows more slowly than the Sobolev exponent at \(+\infty\) and if a technical condition is satisfied, the author shows that the problem has at least two solutions for \(t\) large negative. This partially generalizes a result of the reviewer and Amann and Hess. [More recently, the author and S. Solimini [MRC report 2568 (Wisconsin)] have improved the main result.] (Note that the properness question he raises can be proved if \((\ln s)^ Bs^{- \alpha}g(s)>C>0\) as \(s\to \infty\) where \(C\) is finite and \(\alpha \in [1,(N-2)^{-1}(N+2))\) by a blowing up argument. Here, for simplicity, we assume that \(g\) is independent of \(x\).)
Reviewer: E. N. Dancer

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A15 Variational methods applied to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

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