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- ΔN u + V (x) | u |N - 2 u = f (x, u) + ε h (x) in RN, N ≥ 2, where V : RN → R is a continuous potential, f : RN × R → R behaves like exp (α | u |N / (N - 1)) when | u | → ∞ and h ∈ (W1, N (RN))* = W- 1, N′, h ≢ 0. As an application of this result we have existence of two positive solutions for the following elliptic problem involving critical growth- Δ u + V (x) u = λ u (eu2 - 1) + ε h (x) in R2, where λ > 0 is large, ε > 0 is a small parameter and h ∈ H-1 (R2), h ≥ 0. © 2008 Elsevier Inc. All rights reserved.
Marcos do Ó, J., Medeiros, E., & Severo, U. (2009). On a quasilinear nonhomogeneous elliptic equation with critical growth in RN. Journal of Differential Equations, 246(4), 1363–1386. https://doi.org/10.1016/j.jde.2008.11.020