Positive solutions for a semipositone anisotropic p-Laplacian problem

Abstract

In this paper, a semipositone anisotropic p-Laplacian problem $$ -\Delta _{\overrightarrow{p}}u=\lambda f(u), $$ − Δ p → u = λ f ( u ) , on a bounded domain with the Dirchlet boundary condition is considered, where $A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)$ A ( u q − 1 ) ≤ f ( u ) ≤ B ( u q − 1 ) for $u>0$ u > 0 , $f(0)<0$ f ( 0 ) < 0 and $f(u)=0$ f ( u ) = 0 for $u\leq -1$ u ≤ − 1 . It is proved that there exists $\lambda ^{*}>0$ λ ∗ > 0 such that if $\lambda \in (0,\lambda ^{*})$ λ ∈ ( 0 , λ ∗ ) , then the problem has a positive weak solution $u_{\lambda}\in L^{\infty}(\overline{\Omega})$ u λ ∈ L ∞ ( Ω ‾ ) via combining Mountain-Pass arguments, comparison principles, and regularity principles.