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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.

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
Full Text: DOI Numdam EuDML
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