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On semilinear elliptic equations involving concave--convex nonlinearities and sign-changing weight function. (English) Zbl 1153.35036
The author studies the multiplicity results of positive solutions of the following elliptic equation: $$-\Delta u=u^p+\lambda f(x)u^q\text{ in } \Omega \quad 0\le u\in H_0^1(\Omega),\tag 1$$ where $\Omega$ is a bounded domain in $\bbfR^N$, $0<q<1<p<2^*$; $2^*=\frac{N+2}{N-2}$ if $N\ge 3$, $2^*=\infty$ if $N=2$, and $f$ is a given function, $\lambda>0$. The author under suitable assumptions on the data (1) proves that (1) possesses at least two positive solutions for $\lambda$ is sufficiently small.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Second order elliptic equations, variational methods 35J25 Second order elliptic equations, boundary value problems 47J30 Variational methods (nonlinear operator equations)
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##### References:
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