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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 $$| s|^ 2$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35J50 Variational methods for elliptic systems
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