On a nonhomogeneous and singular quasilinear equation involving critical growth in R2

This paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form −Δu+V(x)u−Δ(u2)u=g(x,u)|x|a+h(x)inR2,where a∈[0,2), V(x) is a continuous positive potential bounded away from zero and which can be “large” at infinity, the nonlinearity g(x,s) is allowed to enjoy the critical exponential growth with respect to the Trudinger-Moser inequality and the nonhomogeneous term h belongs to Lq(R2) for some q∈(1,2]. By combining variational arguments in a nonstandard Orlicz space context with a singular version of the Trudinger-Moser inequality, we obtain the existence of two distinct solutions when ‖h‖q is sufficiently small. Schrödinger equations of this type have been studied as models of several physical phenomena.