國立清華大學 數學系

In this talk, we study the bifurcation of positive solutions for two problems with the one-dimensional $(p,q)$-Laplace equation $$ (|u'|^{p-2}u')' + (|u'|^{q-2}u')' + \lambda(|u|^{p-2}u+|u|^{q-2}u)=0, \quad u(-L)=u(L)=0 \leqno{(1)} $$ and $$ (|u'|^{p-2}u')' + (|u'|^{q-2}u')' + \lambda u^{r-1}=0, \quad u(-1)=u(1)=0, \leqno{(2)} $$ where $p>q>1$, $r>1$, $L>0$ and $\lambda>0$ is a parameter. First we prove that there exist five different types of bifurcation diagrams for (1) and that the structure of positive solutions to (1) changes depending on $L$. Next we consider (2). Then there are five types of order relations for $(p,q,r)$. We prove that there are two types of bifurcation curves that are increasing, two types that are decreasing, and one that is not monotone and turns exactly once. As a consequence, we prove the existence of multiple positive solutions and show the uniqueness of positive solutions for a bifurcation parameter in a certain range. This is a joint work with Ryuji Kajikiya (Saga University, Japan), Inbo Sim (University of Ulsan, Korea) and Mieko Tanaka (Tokyo University of Science, Japan).