Concave-convex behavior for a Kirchhoff type equation with degenerate nonautonomous coefficient

Author(s)

Massa, Eugenio ^[1]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat, Campus Sao Carlos, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS; v. 28, n. 6 DEC 2021.
• Web of Science Citations: 0

Abstract

In this paper we study positive solutions for the Kirchhoff type equation -M (x, vertical bar vertical bar u vertical bar vertical bar(2) ) Delta u =lambda f(u) with Dirichlet boundary conditions in a bounded domain Omega, where vertical bar vertical bar center dot vertical bar vertical bar is the norm in H-0(1)(Omega) and f, M are suitable functions. The problem is nonvariational since the nonlocal coefficient M, possibly degenerate, depends on the point x is an element of Omega. We show that these properties of M can produce interesting phenomena, even with simple homogeneous right hand sides, providing existence, nonexistence, and multiplicity results, due to the fact that the rate of growth with respect to u on the left hand side may change in Omega. Several model examples are given, including one where M takes the form of the original Kirchhoff coefficient for the elastic string, but with nonhomogeneous material. (AU)

• FAPESP's process: 19/19699-0 - Concave-convex type problems with local and nonlocal operators
• Grantee: Eugenio Tommaso Massa
• Support Opportunities: Scholarships abroad - Research