Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms
Nikolaos S. Papageorgiou, Youpei Zhang
Abstract
Abstract We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation $f(z,x,y)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:math> which is only locally defined in $x \in {\mathbb {R}} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:math> . Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.