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Paper
IPM / M / 15564 |
School of Mathematics
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Title: |
Existence results for a super-critical Neumann problem with a convex-concave non-linearity
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Author(s): |
Leila Salimi (Joint with A. Moameni)
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Status: |
To Appear
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Journal: |
Annali di Matematica Pura ed Applicata
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Supported by: |
IPM
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Abstract: |
We shall consider the following semi-linear problem with a Neumann boundary condition
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−∆u + u = a(|x|)|u|p−2u− b(|x|)|u|q−2u, x ∈ B1, |
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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.
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E(u): = |
1
2
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⌡
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B1
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(|∇u|2+ u2 ) dx+ |
1
q
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⌡
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B1
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b(|x|) uq dx− |
1
p
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⌡
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B1
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a(|x|)|u|p dx, |
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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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