A Kirchhoff‐type equation involving critical exponent and sign‐changing weight functions in dimension four

Abstract

In the paper, we study the existence and multiplicity of positive solutions for the following Kirchhoff equation involving concave‐convex nonlinearities: {centerarray−a∫normalΩ∇u2dx+bΔu=λg(x)uq−2u+h(x)up−2uinΩ,arrayu∈H01(Ω). We obtain the existence and multiplicity of solutions of by variational methods and concentration compactness principle.

Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave–convex nonlinearities

“…The equation was first proposed by Kirchhoff [1] as an extension of the classical D' Alembert's wave equation to describe free vibrations of elastic strings. Several existence results for equation (E 1 ) have been obtained in recent years; see [2][3][4][5][6][7] and references therein. Moreover, other similar arguments are also obtained; see [8][9][10][11][12].…”