Multiplicity results of an elliptic equation with non-homogeneous boundary conditions

Abstract

where Ω is an open smooth bounded subset of R , N ≥ 2, g : ∂Ω→ R is a given continuous function and p > 2 is fixed. If g ≡ 0, it is well known that (1.1) has infinitely many distinct solutions for 2 2 if N = 2. Such results have been proved by using variational methods also for more general odd nonlinearities at the beginning of 70’s (see e.g. [2], [3], [6], [9], [11] and references therein). In all these papers a fundamental role is played by the fact that the energy functional is even in a Banach space, hence it is possible to use a modified version of the classical Ljusternik–Schnirelman theory and the properties of the genus for symmetric sets. On the contrary, if g 6≡ 0 the more general boundary value problem (1.1) loses its symmetry and the previous recalled arguments do not hold. In fact, it is well