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Positive solution for the elliptic problems with sublinear and superlinear nonlinearities. (English) Zbl 1216.35056
Summary: This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities $$-\Delta u=\lambda a(x)u^p+ b(x)u^q$$ in $$\Omega$$, $$u>0$$ in $$\Omega$$, $$u=0$$ on $$\partial\Omega$$, where $$\lambda>0$$ is a real parameter, $$0<p<1<q$$. $$\Omega$$ is a bounded domain in $$\mathbb R^N$$ $$(N\geq 3)$$, and $$a(x)$$ and $$b(x)$$ are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.
##### MSC:
 35J61 Semilinear elliptic equations 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B09 Positive solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35D30 Weak solutions to PDEs 35J20 Variational methods for second-order elliptic equations
##### Keywords:
positive solution; existence
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##### References:
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