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On pairs of positive solutions for a class of quasilinear elliptic problems. (English) Zbl 1240.35206

Summary: We prove, by using bifurcation theory, the existence of at least two positive solutions for the quasilinear problem \(-\Delta _{p}u = f(x,u)\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(N>p>1\) and \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^{N}\), \(N\geq 2\), and the non-linearity \(f\) is a locally Lipschitz continuous function, among other assumptions.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35B32 Bifurcations in context of PDEs
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