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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.
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
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