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We prove that a class of superlinear indefinite problems with homogeneousNeumann boundary conditions admits an arbitrarily high number of positivesolutions, provided that the parameters of the problem are adequately chosen.The sign-changing weight in front of the nonlinearity is taken to be piecewiseconstant, which allows to perform a sharp phase-plane analysis, firstly tostudy the sets of points reached at the end of the regions where the weight isnegative, and then to connect such sets through the flow in the positive part.Moreover, we study how the number of solutions depends on the amplitude of theregion in which the weight is positive, using the latter as the mainbifurcation parameter and constructing the corresponding global bifurcationdiagrams.

High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions