Positive solutions for a semipositone anisotropic p-Laplacian problem

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