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On a superlinear elliptic equation. (English) Zbl 0733.35043
The author deals with the following problem: \[ -\Delta u=f(u)\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with regular boundary, assuming that
(f1) \(f\in C^ 1(R,R)\), \(f(0)=f'(0)=0;\)
(f2) There are constants \(C_ 1,C_ 2\) such that \[ | f(t)| \leq C_ 1+C_ 2| t|^{\alpha},\quad 1<\alpha <(n+2)/(n-2) \] (f3) There are constants \(\mu >2\), \(M>0\) such that \[ 0<\mu F(t)\leq tf(t),\quad | t| \geq M,\text{ where } F(t)=\int^{t}_{0}f(r)dr. \] The main result is
Theorem. If f satisfies (f1)(f2)(f3), then the problem above possesses at least three nontrivial solutions.
In a classical paper, Ambrosetti and Rabinowitz obtained two nontrivial solutions, and infinitely many in the case of odd nonlinearities f. Infinitely many solutions can be obtained in case \(n=1\). The author establishes existence of multiple solutions in case \(n\geq 2\) without assuming any symmetry.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
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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