Existence of nontrivial solutions for perturbed elliptic system in \(\mathbb{R}^N\) (Q1722449)

Summary: We consider the perturbed nonlinear elliptic system \(- \varepsilon^2 \Delta u + V(x) u = K(x) | u |^{2^* - 2} u + H_u(u, v)\), \(x \in \mathbb{R}^N\), \(- \varepsilon^2 \Delta v + V(x) v = K(x) | v |^{2^* - 2} v + H_v(u, v)\), \( x \in \mathbb{R}^N\), where \(N \geq 3\), \(2^* = 2 N /(N - 2)\) is the Sobolev critical exponent. Under proper conditions on \(V\), \(H\), and \(K\), the existence result and multiplicity of the system are obtained by using variational method provided \(\varepsilon\) is small enough.