THE EXISTENCE OF A NONTRIVIAL SOLUTION TO A NONLINEAR ELLIPTIC PROBLEM OF LINKING TYPE WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we study the existence of a nontrivial solution to the following nonlinear elliptic problem: (0.1) {-Delta u - a(x)u = f (x, u), x is an element of Omega, u/partial derivative Omega = 0, where Omega is a bounded domain of R(N) and a is an element of L(N/2) (Omega), N >= 3, f is an element of C(0) (Omega) over bar x R(1), R(1)) is superlinear at t = 0 and subcritical at t = infinity. Under suitable conditions, (0.1) possesses the so-called linking geometric structure. We prove that the problem (0.1) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a recent result of Miyagaki and Souto given in [14] for (0.1) with a(x) = 0 and possessing the mountain-pass geometric structure.