Elliptic bifurcation problems that are singular in the dependent and in the independent variables

Abstract

We consider singular problems of the form $-\mathrm{\Delta }u=k\left(.,u\right)+\lambda h\left(.,u\right)$ in Ω, u=0 on $\mathrm{\partial }\mathrm{\Omega },$ u>0 in Ω, where Ω is a bounded $C^{1,1}$ domain in ${\mathbb{R}}^n$, $n\ge 2$, $h:\mathrm{\Omega }\times \left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ and $k:\mathrm{\Omega }\times \left(0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ are Carathéodory functions such that $h\left(x,.\right)$ is nondecreasing, and $k\left(x,.\right)$ is nonincreasing and singular at the origin $a.e.$ $x\in \mathrm{\Omega }.$ Additionally, $k\left(.,s\right)$ and $h\left(.,s\right)$ are allowed to be singular at $\mathrm{\partial }\mathrm{\Omega }$ for $s>0.$ Under suitable additional hypotheses on h and k, we prove that there is a positive ${\lambda }^{\ast }$ such that, for any $\lambda \in \left(0,{\lambda }^{\ast }\right),$ a minimal positive weak solution $u_{\lambda }\in H_0^1\left(\mathrm{\Omega }\right)\cap C\left(\overline{\mathrm{\Omega }}\right)$ exists. The monotonicity of $\lambda \to u_{\lambda },$ and the behaviour of $u_{\lambda }$ at $\mathrm{\partial }\mathrm{\Omega },$ is addressed. In addition, we prove that no weak solution exists if $\lambda >{\lambda }^{\ast }.$

COMMUNICATED BY:

• V. Rădulescu

Keywords:

• Singular elliptic problems
• variational problems
• sub-supersolutions method
• positive solutions

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35J75
• Secondary: 35D30
• 35J20

Disclosure statement

No potential conflict of interest was reported by the authors.