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

Aranda, Carlos ; Godoy, Tomas

Electronic journal of differential equations, Tome 2004 (2004)

Résumé

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 $C(overlineOmega)capC^1(Omega)$ and that $lambda=0$ is a bifurcation point from infinity. The existence of positive solutions for problems of the form $-Delta_pu=K(x) g(u)+lambdah(u)+f(x)$ in $Omega, u=0$ on $partialOmega$ is also studied.$

Détail

Classification : 35J60, 35J65
Keywords: singular problems, p-Laplacian operator, nonlinear eigenvalue problems