(N,q)$(N,q)$‐Laplacian equations with one‐sided critical exponential growth

We prove the existence of two non‐trivial weak solutions for a class of quasilinear, non‐homogeneous elliptic problems driven by the (N,q)$(N,q)$‐Laplacian with one‐sided critical exponential growth in a bounded domain Ω⊂RN$\Omega \subset \mathbb {R}^{N}$. 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 C1$C^{1}$ topology is also a local minimum in the natural W1,N$W^{1,N}$ 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.