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.

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doi.org/10.1007/s10231-018-0813-1
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