Exact multiplicity results for a p-Laplacian problem with concave–convex–concave nonlinearities

✍ Scribed by Idris Addou; Shin-Hwa Wang

• Book ID: 104330475
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 278 KB
• Volume: 53
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(02)00298-5

✦ Synopsis

We study the exact number of positive solutions of a two-point Dirichlet boundary-value problem involving the p-Laplacian operator. We consider the case p = 2 as well as the case p ¿ 1, when the nonlinearity f satisÿes f(0) = 0 and has two distinct simple positive zeros and such that f changes sign exactly twice on (0; ∞). Note that we may allow that f changes sign more than twice on (0; ∞). Some interesting examples of quartic polynomials are given. In particular, for f(u) = -u 2 (u -1)(u -2), we study the evolution of the bifurcation curves of the p-Laplacian problem as p increases from 1 to inÿnity, and hence are able to determine the exact multiplicity of positive solutions for each p ¿ 1.