On a Class of Elliptic Systems in R(N)

Abstract

We consider a class of variational systems in ℝ^N of the form { <pre>−∆u + a(x)u = F_u(x, u, v)</pre> <pre>−∆v + b(x)v = F_v(x, u, v),</pre>

<p>where a, b : ℝ^N → 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(.) and F_v(.), the (weak) solutions are precisely the critical points of a related functional defined on a Hilbert space of functions u, v in H¹(ℝ^N).</p>

<p>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.</p>

<p>Our approach allows situations in which a(.) and b(.) may assume negative values, and the potential F (x, s) may grow either faster of slower than |s|².