MULTIPLICITY AND CONCENTRATION OF NONTRIVIAL SOLUTIONS FOR GENERALIZED EXTENSIBLE BEAM EQUATIONS IN RN

In this article, we study a class of generalized extensible beam equations with a superlinear nonlinearity Delta(2)u - M (parallel to del u parallel to(2)(L2))Delta u + lambda V(x)u = f(x,u) in R-N, u is an element of H-2(R-N), where N >= 3, M(t) = at(delta) + b with a, delta > 0 and b is an element of R, lambda > 0 is a parameter, V is an element of C(R-N, R) and f is an element of C(R-N x R, R). Unlike most other papers on this problem, we allow the constant b to be non-positive, which has the physical significance. Under some suitable assumptions on V(x) and f (x, u), when a is small and lambda is large enough, we prove the existence of two nontrivial solutions u(a,lambda)((1) )and u(a,lambda)((2)), one of which will blow up as the nonlocal term vanishes. Moreover, u(a,lambda)((1)) -> u(infinity)((1)) and u(a,lambda)((2)) -> u(infinity)((2) )strongly in H-2(R-N) as lambda -> infinity, where u(infinity)((1)) not equal u(infinity)((2)) is an element of H-0(2)(Omega) are two nontrivial solutions of Dirichlet BVPs on the bounded domain Omega. Also, the nonexistence of nontrivial solutions is also obtained for a large enough.