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Paper   IPM / M / 15564
School of Mathematics
  Title:   Existence results for a super-critical Neumann problem with a convex-concave non-linearity
  Author(s):  Leila Salimi (Joint with A. Moameni)
  Status:   To Appear
  Journal: Annali di Matematica Pura ed Applicata
  Supported by:  IPM
We shall consider the following semi-linear problem with a Neumann boundary condition

−∆u + u = a(|x|)|u|p−2ub(|x|)|u|q−2u,     xB1,
where B1 is the unit ball in \mathbbRN, N ≥ 2, a, b are non-negative 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 established recently by one of the authors that allows one to deal with problem with super-critical Sobolev non-linearities. Indeed, we first find a critical point of the Euler-Lagrange functional associated with the equation, i.e.
E(u): = 1


(|∇u|2+ u2 ) dx+ 1


b(|x|) uq dx 1


a(|x|)|u|p dx,
over a suitable closed and convex subset of H1(B1). Then we shall use this new variational principle to deduce that the restricted critical point of E is an actual critical point.

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