Existence and Bifurcation Results for Fourth-Order Elliptic Equations Involving Two Critical Sobolev Exponents

Abstract

Abstract. Let be a smooth bounded domain in R N , with N ≥ 5. We provide existence and bifurcation results for the elliptic fourth-order equation 2 u − p u = f (λ, x, u) in , under the Dirichlet boundary conditions u = 0 and ∇u = 0. Here λ is a positive real number, 1 < p ≤ 2 # and f (., ., u) has a subcritical or a critical growth s, 1 < s ≤ 2 * , where 2 * := 2N N−4 and 2 # := 2N N−2 . Our approach is variational, and it is based on the mountain-pass theorem, the Ekeland variational principle and the concentration-compactness principle.AMS Subject Classification. 35J35, 35B33, 35G20, 35B32. 1.Introduction. An approach for confronting second-order critical semilinear elliptic equations in a bounded domain in R N was introduced in [2], where it was shown that the Palais-Smale compactness condition holds for certain levels of the associated functional. Therefore, under the appropriate assumptions, the mountainpass theorem could be applied to yield a solution to the critical problem.The existence of solutions of fourth-order critical elliptic problems can also be proved by using this approach, see [4,5,8,11,15] and the references therein.In this paper, we study problems of the form

Remarks on the Qualitative Questions for Biharmonic Operators

“…we refer to [14]. Equations of type (1.2) are also discussed on Riemannian manifold (M n , g), n ≥ 5, see [17], where the author obtain the existence of classical solutions to…”