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.


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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