We shall consider the following semi-linear problem with a Neumann boundary condition -Δu+u=a(|x|)|u|p-2u-b(|x|)|u|q-2u,x∈B1,where B1 is the unit ball in RN, N≥ 2 , a, b are nonnegative radial functions, and p, q are distinct numbers greater than or equal to 2. We shall assume no growth condition on p and q. Our plan is to use a new variational principle that allows one to deal with problems with supercritical Sobolev non-linearities. Indeed, we first find a critical point of the Euler–Lagrange functional associated with this equation over a suitable closed and convex set. Then we shall use this new variational principle to deduce that the restricted critical point of the Euler–Lagrange functional is an actual critical point.

Calculus of variations, Semi-linear elliptic problems, Variational principles
Annali di Matematica Pura ed Applicata
School of Mathematics and Statistics

Momeni, A, & Salimi, L. (Leila). (2018). Existence results for a supercritical Neumann problem with a convex–concave non-linearity. Annali di Matematica Pura ed Applicata. doi:10.1007/s10231-018-0813-1