Two types of solutions to a class of \((p, q)\)-Laplacian systems with critical Sobolev exponents in \(\mathbb{R}^{\mathbb{N}}\) (Q1629263)

Summary: We focus on the following elliptic system with critical Sobolev exponents: \[ -\mathrm{div}\big(|\nabla u|^{p-2}\nabla u\big) +m(x)| u|^{p-2}u=\lambda| u|^{p^*-2}u+ (1/\eta)G_u(u,\nu),\quad x\in \mathbb R^N; \] \[ -\mathrm{div}\big(|\nabla v|^{q-2}\nabla v\big) +n(x)| v|^{q-2}v=\mu| v|^{q^*-2}v+ (1/\eta)G_v(u,v),\quad x\in \mathbb R^N, \] where \(\mu,\lambda>0\), \(1<p\leq q<N\), either \(\eta \in(1, p)\) or \(\eta \in(q, p^\ast)\), and critical Sobolev exponents \(p^\ast = p N /(N - p)\) and \(q^\ast = q N /(N - q)\). Conditions on potential functions \(m(x), n(x)\) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with \(\eta \in(q, p^\ast)\) or \(\eta \in(1, p)\) will be established.