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This paper deals with the existence and multiplicity of solutions for thegeneralized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x,u)|\nabla u|^{q-2}\nabla u) \\ &\quad\qquad+\frac1q B_t(x, u)|\nabla u|^q +V(x)|u|^{p-2} u+ W(x)|u|^{q-2} u= g(x, u)\quad\qquad\mbox{ in } \mathbb{R}^N,\end{align*} where $1<q\le p< N$, $A,B:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ are suitable $C^1$ Carath\'eodoryfunctions with $A_t(x, u)=\frac{\partial A}{\partial t}(x, u), B_t(x,u)=\frac{\partial B}{\partial t}(x, u)$, $V, W:\mathbb{R}^N\to\mathbb{R}$ areproper ``weight functions" and $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ isa Carath\'eodory map. Notwithstanding the occurrence of some coefficients whichrely upon the solution itself makes the use of variational techniques morechallenging, under suitable assumptions on the involved functions, we are ableto exploit the variational nature of our problem. In particular, the existenceof a nontrivial solution is derived via a generalized version of theAmbrosetti-Rabinowitz Mountain Pass Theorem, based on a weaker version of theclassical Cerami-Palais-Smale condition. Finally, the multiplicity result,which is thoroughly new also even in the simpler case $q=p$, is gained undersymmetry assumptions and a sharp decomposition of the ambient space.

On existence and multiplicity of solutions for generalized (p, q)-Laplacian equations on unbounded domains