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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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