Existence and multiplicity of solutions to fourth order elliptic equations with critical exponent on compact manifolds (Q617754)

Let \((M,g)\) be a Riemannian compact smooth \(n\)-manifold, \(n\geq5\), with metric \(g\). The authors study on perturbation of the so called prescribed scalar Q-curvature type equations on compact Riemannian manifolds; these equations are fourth order elliptic and of critical Sobolev growth. They investigate multiple solutions of the equation \(\Delta^2 u+\nabla^i(a(x)\nabla_iu)+h(x)u=f(x)|u|^{N-2}u+\lambda |u|^{q-2}u+\epsilon g(x)\). Sufficient conditions are given to have at least two distinct solutions first without using the concentration-compactness technic but with a suitable range of the parameters and secondly by using the concentration-compactness methods.