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Combined effects of concave and convex nonlinearities in some elliptic problems. (English) Zbl 0805.35028
Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^ N$$ and consider the semilinear elliptic problem $-\Delta u = f_ \lambda (x,u) \quad \text{ in } \Omega, \quad u = 0 \quad \text{ on } \partial \Omega \tag{1}$ where $$f_ \lambda : \Omega \times \mathbb{R} \to \mathbb{R}$$ and $$\lambda$$ is a real parameter. When $$f_ \lambda$$ is sublinear, for example, $$f_ \lambda = \lambda u^ q$$, $$0<q<1$$, it is known that (1) has a unique positive solution for all $$\lambda>0$$. On the other hand, if $$f_ \lambda = \lambda | u |^{q-1}u$$, the problem (1) admits infinitely many solutions. The purpose of the present note is to study (1) when $$f_ \lambda$$ is the sum of a sublinear and superlinear term. It is shown that the combined effects of these two nonlinearities change considerably the structure of the solution set.
Reviewer: V.Mustonen (Oulu)

MSC:
 35J60 Nonlinear elliptic equations
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