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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: \[ -\Delta u=u^p+\lambda f(x)u^q\text{ in } \Omega \quad 0\leq u\in H_0^1(\Omega),\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(0<q<1<p<2^*\); \(2^*=\frac{N+2}{N-2}\) if \(N\geq 3\), \(2^*=\infty\) 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 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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