Abstract
This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent : in Ω, on ∂ Ω, where Ω is a smooth bounded domain in , , and ε is a positive real parameter. We show that, for ε small, has no sign-changing solutions with low energy which blow up at exactly three points. Moreover, we prove that has no bubble-tower sign-changing solutions. MSC:35J20, 35J60.
Highlights
Introduction and resultsWe consider the following semi-linear elliptic problem with supercritical nonlinearity:⎧ ⎨ u = |u|p– +εu in, (Pε) ⎩ u = u = on ∂, where is a smooth bounded domain in Rn, n ≥, ε is a positive real parameter and p+ = n n– is the criticalSobolev exponent for the embedding of H ( ) ∩ H ( ) into Lp+ ( ).When the biharmonic operator in (Pε) is replaced by the Laplacian operator, there are many works devoted to the study of the counterpart of (Pε); see for example [ – ], and the references therein.When ε
When we study problem ( . ) in a bounded smooth domain, we need to introduce the function Pδ(a,λ) which is the projection of δ(a,λ) on H ( )
We prove that d /d
Summary
J satisfies the Palais-Smale condition in the subcritical case, while this condition fails in the critical case Such a failure is due to the functions λ(n– )/ δ(a,λ)(x) = c ( + λ |x – a| )(n– )/ , c = n(n – ) n – (n– )/ , λ > , a ∈ Rn. c is chosen so that δ(a,λ) is the family of solutions of the following problem:. If is a regular value of ρ(x), there exists ε > , such that, for each ε ∈ ( , ε ), problem (Pε) has no signchanging solutions uε which satisfy uε = Pδ(aε, ,λε, ) – Pδ(aε, ,λε, ) + Pδ(aε, ,λε, ) + vε, with |uε|ε∞ is bounded and ⎧ ⎨aε,i ∈ , λε,id(aε,i, ∂ ) → ∞ for i = , , , ⎩ Pδ(aε,i,λε,i), Pδ(aε,j,λε,j) → for i = j and vε → as ε →.
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