| Summary: | Abstract In this paper, a semipositone anisotropic p-Laplacian problem − Δ p → u = λ 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 ) ≤ f ( u ) ≤ 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 ≤ − 1 $u\leq -1$ . It is proved that there exists λ ∗ > 0 $\lambda ^{*}>0$ such that if λ ∈ ( 0 , λ ∗ ) $\lambda \in (0,\lambda ^{*})$ , then the problem has a positive weak solution u λ ∈ L ∞ ( Ω ‾ ) $u_{\lambda}\in L^{\infty}(\overline{\Omega})$ via combining Mountain-Pass arguments, comparison principles, and regularity principles.
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