On a quasilinear nonhomogeneous elliptic equation with critical growth in RNRN

In this paper, Ekeland variational principle, mountain-pass theorem and a suitable Trudinger–Moser inequality are employed to establish sufficient conditions for the existence of solutions of quasilinear nonhomogeneous elliptic partial differential equations of the form−ΔNu+V(x)|u|N−2u=f(x,u)+εh(x)inRN,N⩾2, where V:RN→RV:RN→R is a continuous potential, f:RN×R→Rf:RN×R→R behaves like exp(α|u|N/(N−1))exp(α|u|N/(N−1)) when |u|→∞|u|→∞ and h∈(W1,N(RN))∗=W−1,N′h∈(W1,N(RN))∗=W−1,N′, h≢0h≢0. As an application of this result we have existence of two positive solutions for the following elliptic problem involving critical growth−Δu+V(x)u=λu(eu2−1)+εh(x)inR2, where λ>0λ>0 is large, ε>0ε>0 is a small parameter and h∈H−1(R2)h∈H−1(R2), h⩾0h⩾0.