Bifurcation analysis of a class of polyharmonic semilinear equations with perturbed potential

Many problems of mechanics were described by polyharmonic equations [9]. Moreover, the areas of physics and geometry where such operators occur, include the study of the Kirchhoff plate equation in the theory of elasticity, and the study of the Paneitz-Branson operator in conformal geometry, see [9]. Inverse spectral problems for a potential perturbation of the polyharmonic operator were studied in [12]. The polyharmonic operator (−Δ)m is the prototype of an elliptic operator of order 2m. A general theory for boundary value problems for linear elliptic operators of order 2m was developed by Agmon-Douglis-Nirenberg in [2, 3] and Berchio-Gazzola in [5]. Although the material is quite technical, it turns out that the Lp-theory can be developed to a large extent analogously to second order equations. As long as existence, regularity and stability results are concerned, the theory of semi-linear higher order problems is already quite well developed. This is no longer true as soon as qualitative properties of the solution related to the bifurcation problems are investigated.