Fountain Theorem over Cones and Applications

In [2] and [3], fountain theorems and their dual forms in Banach space were established respectively. They are effective tools in studying the existence of infinitely many solutions. It should be noted that a decomposition of the Banach space plays an important role in proving these theorems. The decomposition allows one to apply Borsuk-Ulam theorem to establish a proper intersection lemma. Such a decomposition in many cases is done by using the eigenspaces of operators concerned. However, there are many operators, for instance, the p-Laplacian operator -Δp, whose spectrum are not very well understood. In recent works [5], [6], [7], a linking theorem over cones was obtained, and solutions for a quasilinear elliptic problem were found. In the use of the theorem, it does not require a complete decomposition of spaces. In this paper, we first establish a fountain theorem over cones in Banach spaces.