BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

doi:10.1016/S0252-9602(15)30072-2

Abstract

Abstract:

In this paper, we consider the following nonlinear elliptic problem : △²u = |u|^8/(n-4)u + μ|u|^q-1u, in Ω, △u = u = 0 on ∂Ω, where Ω is a bounded and smooth domain in Rⁿ, n ∈ {5, 6, 7}, μ is a parameter and q ∈[4/(n 4), (12 n)/(n 4)]. We study the solutions which concentrate around two points of Ω. We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded. For Ω = (Ω_a)_a a smooth ringshaped open set, we establish the existence of positive solutions which concentrate at two points of Ω. Finally, we show that for μ > 0, large enough, the problem has at least many positive solutions as the LjusternikSchnirelman category of Ω.

Key words: fourth order elliptic equations, critical Sobolev exponent, blow up solution

CLC Number:

35J20

Siwar AMMAR, Mokhles HAMMAMI. BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY[J].Acta mathematica scientia,Series B, 2015, 35(6): 1511-1546.

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