Existence results for a supercritical Neumann problem with a convex–concave non-linearity

Abstract

We shall consider the following semi-linear problem with a Neumann boundary condition $$\begin{aligned} -\Delta u + u= a(|x|)|u|^{p-2}u- b(|x|)|u|^{q-2}u, \quad x\in B_1, \end{aligned}$$ where \(B_1\) is the unit ball in \({\mathbb {R}}^N\), \(N\ge 2\), 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.