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Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem

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  • We study the bifurcation diagrams of positive solutions of the $p$-Laplacian Dirichlet problem \begin{eqnarray*} (\varphi_p(u'(x)))'+f_\lambda(u(x))=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray*} where $\varphi_p(y)=|y|^{p-2}y$, $(\varphi_p(u'))'$ is the one-dimensional $p$-Laplacian, $p>1$, the nonlinearity $f_\lambda(u)=\lambda g(u)-h(u),$ $g,h\in C[0,\infty)\cap C^2(0,\infty )$, and $\lambda >0$ is a bifurcation parameter. Under certain hypotheses on functions $g$ and $h$, we give a complete classification of bifurcation diagrams. We prove that, on the $(\lambda, |u|_\infty)$-plane, each bifurcation diagram consists of exactly one curve which has exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions for each $\lambda >0.$ In addition, we show the evolution phenomena of bifurcation diagrams of polynomial nonlinearities with positive coefficients.
    Mathematics Subject Classification: 34B15, 34B18.


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