A global curve of stable, positive solutions for a $p$-Laplacian problem

Rynne, Bryan P.

Electronic Journal of Differential Equations, Tome 2010 (2010).

Résumé

Summary: We consider the boundary-value problem $$\displaylines{ - \phi_p (u'(x))' = \lambda f(x,u(x)) , \quad x \in (0,1),\cr u(0) = u(1) = 0, }$$ where $$\displaylines{ f(x,\xi) > 0, \quad (x,\xi) \in [0,1] \times \mathbb{R} ,\cr (p-1)f(x,\xi) \ge f_\xi(x,\xi) \xi , \quad (x,\xi) \in [0,1] \times (0,\infty) . }$$ These assumptions on $f$ imply that the trivial solution $(\lambda,u)=(0,0)$ is the only solution with $\lambda=0$ or $u=0$, and if $\lambda > 0$ then any solution $u$ is $\em $positive, that is, $u > 0$ on $(0,1)$.

Détail

Classification : 34B15
Keywords: ordinary differential equations, p-Laplacian, nonlinear boundary value problems, positive solutions, stability