A COMPLETE CLASSIFICATION OF BIFURCATION DIAGRAMS OF A $P$-LAPLACIAN DIRICHLET PROBLEM II. GENERALIZED NONLINEARITIES

Abstract

We study the bifurcation diagrams of classical positive solutions $u$ with $\lVert u \rVert_{\infty} \in (0,\infty)$ of the $p$-Laplacian Dirichlet problem $$\left\{\begin{array}{ll} \varphi_{p}(u'(x)))' + \lambda f_{q,r}(u(x)) = 0,\ \ \ -1 \lt x \lt 1, \\ u(-1) = 0 = u(1), \end{array}\right.$$ where $p \gt 1$, $\varphi_{p}(y) = |y|_{p-2}$, $(\varphi(u'))'$ is the one dimensional $p$-Laplacian, $\lambda \gt 0$ is a bifurcation parameter, and $$f_{q,r}(u) = \left\{\begin{array}{ll} \mid 1-u \mid^q,\ \ \ \textrm{if } 0 \lt u \leq 1, \\ \mid 1-u \mid^r,\ \ \ \textrm{if } u \gt 0, \end{array}\right.$$ with positive constants $q$ and $r$. We give explicit formulas of bifurcation curves of classical positive solutions on the $\left( \lambda ,\left\Vert u\right\Vert _{\infty }\right) $-plane. More importantly, for different $(p,q,r)$, we give a complete classification of all bifurcation diagrams. Hence we are able to determine the (exact) multiplicity of classical positive solutions for each $(p,q,r,\lambda )$. Our results generalize the results of Lee et,al [J. Math. Anal. Appl., 330 (2007), 276-290] with nonlinearity $f_{q,r}$ generalized from $q = r \gt 0$ to $q,r \gt 0$.