Electron. J. Diff. Eqns.,Vol. 2004(2004), No. 132, pp. 1-15.

Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator

Carlos Aranda, Tomas Godoy

Abstract:
Consider the problem
$$ -\Delta_{p}u=g(u) +\lambda h(u)\quad\hbox{in }\Omega $$
with $u=0$ on the boundary, where $\lambda\in(0,\infty)$, $\Omega$ is a strictly convex bounded and $C^{2}$ domain in $\mathbb{R}^{N}$ with $N\geq2$, and 1 less than $p\leq2$. Under suitable assumptions on $g$ and $h$ that allow a singularity of $g$ at the origin, we show that for $\lambda$ positive and small enough the above problem has at least two positive solutions in $C(\overline{\Omega})\cap C^{1}(\Omega)$ and that $\lambda=0$ is a bifurcation point from infinity. The existence of positive solutions for problems of the form $-\Delta_{p}u=K(x) g(u)+\lambda h(u)+f(x)$ in $\Omega$, $u=0$ on $\partial\Omega$ is also studied.

Submitted June 18, 2004. Published November 16, 2004.
Math Subject Classifications: 35J60, 35J65.
Key Words: Singular problems; p-laplacian operator; nonlinear eigenvalue problems.

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Carlos Aranda
Departamento de Matematica de la Facultad de Ciencias
Universidad de Tarapaca
Av. General Velasquez 1775
Casilla 7-D, Arica, Chile
email: caranda@uta.cl
  Tomas Godoy
FaMAF, Universidad Nacional de Cordoba
Medina Allende y Haya de la Torre
Ciudad Universitaria, 5000 Cordoba, Argentina
email: godoy@mate.uncor.edu

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