Existence results for singular anisotropic elliptic boundary-value problems

Kim, Eun Heui

Electronic Journal of Differential Equations, Tome 2000 (2000).

Résumé

Summary: We establish the existence of a positive solution for anisotropic singular quasilinear elliptic boundary-value problems. As an example of the problems studied we have $$ u^au_{xx}+u^bu_{yy}+\lambda(u+1)^{a+r}=0 $$ with zero Dirichlet boundary condition, on a bounded convex domain in ${\Bbb R}^2$. Here $0\leq b\leq a$, and $\lambda, r$ are positive constants. When 0 r 1 (sublinear case), for each positive $\lambda$ there exists a positive solution. On the other hand when $r>1$ (superlinear case), there exists a positive constant $\lambda^*$ such that for $\lambda$ in $(0,\lambda^*)$ there exists a positive solution, and for $\lambda^* \lambda$ there is no positive solution.

Détail

Classification : 35J65, 35J70
Keywords: anisotropic, singular, sublinear, superlinear, elliptic boundary-value problems