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On a class of elliptic systems in \(\mathbb R^n\). (English) Zbl 0809.35020
Summary: We consider a class of variational systems in \(\mathbb{R}^ N\) of the form \[ -\Delta u+ a(x)u= f(x,u,v),\qquad -\Delta v+ b(x)v= g(x,u,v) \] where \(a,b: \mathbb{R}^ N\to \mathbb{R}\) are continuous functions which are coercive; i.e., \(a(x)\) and \(b(x)\) approach plus infinity as \(x\) approaches plus infinity. Under appropriate growth and regularity conditions on the nonlinearities \(F_ u(\cdot)\) and \(F_ v(\cdot)\), the (weak) solutions are precisely the critical points of a related functional defined on a Hilbert space of functions \(u\), \(v\) in \(H^ 1 (\mathbb{R}^ N)\).
By considering a class of potentials \(F(x,u,v)\) which are nonquadratic at infinity, we show that a weak version of the Palais-Smale condition holds true and that a nontrivial solution can be obtained by the generalized mountain pass theorem. Our approach allows situations in which \(a(\cdot)\) and \(b(\cdot)\) may have negative values, and the potential \(F(x,s)\) may grow either faster or slower than \(| s|^ 2\).

35J60 Nonlinear elliptic equations
35J50 Variational methods for elliptic systems
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