On a Class of Semipositone Problems in Orlicz–Sobolev Spaces Involving Critical Exponential Growth

Abstract This paper investigates the existence of positive solution for a class of semipositone elliptic problems driven by a $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> -Laplacian operator in bounded smooth domains. The nonlinear term is assumed to exhibit critical exponential growth in the framework of Orlicz–Sobolev spaces. By introducing an auxiliary problem and employing variational methods, we overcome the difficulties caused by the negative parameter that characterizes the semipositone nature of the equation. Under suitable conditions on the $$\mathcal {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -function $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> and on the nonlinearity, we establish the existence of at least one weak positive solution for sufficiently small values of the parameter.