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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 $\bbfR\sp N$ and consider the semilinear elliptic problem $$-\Delta u = f\sb \lambda (x,u) \quad \text{ in } \Omega, \quad u = 0 \quad \text{ on } \partial \Omega \tag 1$$ where $f\sb \lambda : \Omega \times \bbfR \to \bbfR$ and $\lambda$ is a real parameter. When $f\sb \lambda$ is sublinear, for example, $f\sb \lambda = \lambda u\sp q$, $0<q<1$, it is known that (1) has a unique positive solution for all $\lambda>0$. On the other hand, if $f\sb \lambda = \lambda \vert u \vert\sp{q-1}u$, the problem (1) admits infinitely many solutions. The purpose of the present note is to study (1) when $f\sb \lambda$ 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.

35J60Nonlinear elliptic equations
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