Abstract We prove the existence of two non‐trivial weak solutions for a class of quasilinear, non‐homogeneous elliptic problems driven by the ‐Laplacian with one‐sided critical exponential growth in a bounded domain . The first solution is obtained as a local minimizer of the associated energy functional; to justify this, we establish that any local minimum in the topology is also a local minimum in the natural topology. This minimization result is proved in a more general setting and may be useful in related problems. The second solution is given by minimax methods.
Gloss et al. (Mon,) studied this question.