On some nonlinear elliptic systems with coercive perturbations in \(\mathbb R^ N\). (Q1432788)

The authors give some existence and regularity results concerning the following class of nonlinear-elliptic systems in \(\mathbb{R}^d\), \(d\geq 2\), \[ \begin{cases} -\Delta_p u+ a(x)|u|^{p-2}u= f(x)|u|^{\alpha-1} u|v|^{\beta+1}\quad &\text{in }\mathbb{R}^d,\\ -\Delta_q v+ b(x)|v|^{q-2} v= f(x)|u|^{\alpha+1}|v|^{\beta-1} v\quad &\text{in }\mathbb{R}^d,\\ \lim_{|x|\to\infty} u(x)= \lim_{|x|\to\infty} v(x)= 0,\end{cases}\tag{1} \] where \(1< p< d\), \(1< q<d\) and \(\alpha\), \(\beta\) are real constants. Under same suitable assumptions the authors prove existence of nontrivial solutions for (1) and study the regularity of obtained solutions. To this end they use critical point theory.