Solutions for perturbed biharmonic equations with critical nonlinearity?

摘要

In this paper, we study the perturbed biharmonic equations {∈~~4?~2u+V(x)u = P(x)|u|~(p-2)u+Q(x)|u|~(2**-2)u, x∈R~N, u∈H~2(R~N), u(x)→0, as |x| →∞, where ?~2 is the biharmonic operator, N ≥ 5, 2~**) = (2N)/(N-4) is the Sobolev critical exponent, p ∈ (2, 2~(**)), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that ∈ ≤ ε and that for any n~* ∈ N, it has at least n~* pairs solutions if ∈ ≤ ε_(n*), where ε and ε_(n*) are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in H~2(R~N) as ∈ → 0.