# On a biharmonic equation involving nearly critical exponent

@article{Ayed2004OnAB,
title={On a biharmonic equation involving nearly critical exponent},
author={Mohamed Ben Ayed and Khalil O. El Mehdi},
journal={Nonlinear Differential Equations and Applications NoDEA},
year={2004},
volume={13},
pages={485-509}
}
• Published 7 January 2004
• Mathematics
• Nonlinear Differential Equations and Applications NoDEA
Abstract.This paper is concerned with a biharmonic equation under the Navier boundary condition $${(P_{\mp\varepsilon}): \Delta^{2}u = u^{\frac{n+4}{n-4}\mp\varepsilon}}$$ , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in $${\mathbb{R}^n}$$ , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x0 ∈Ω as ε → 0, moreover x0 is a…
An Asymptotic Nondegeneracy Result for a Biharmonic Equation with the Nearly Critical Growth
Abstract.We consider the problem $$\Delta^{2}u = c_{0}u^{{p}\varepsilon}$$, u > 0 in Ω, $$u = \Delta u =0$$ on ∂Ω, where Ω is a smooth bounded domain in $$\mathbb {R}^{N}(N \geq 5)$$, c0 = (N − 4)(N
Single Blow-up Solutions for a Slightly Subcritical Biharmonic Equation
We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent ( P e ): ∆ 2 u = u 9 − e , 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> u > 0 in Ω Asymptotic behavior of least energy solutions for a biharmonic problem with nearly critical growth B boundary condition of (Pε,K) is called as the Navier boundary condition for the second-order Laplacian-case problem ⎧⎨ ⎩ −Δu = N (N − 2)K(x)u(N+2)/(N −2)−ε in Ω, u = 0 on ∂Ω. Blowing up solutions for a biharmonic equation with critical nonlinearity • Mathematics • 2004 In this paper we consider the following biharmonic equation with critical exponent (P") : � 2 u = Ku n+4 n 4 −" , u > 0 in and u = �u = 0 on @, where is a smooth bounded domain in R n , n � 5, " is a Concentration of solutions for a fourth order elliptic equation in$\mathbb{R}^N$In this paper, we study the following fourth order elliptic problem $$\Delta^2 u=(1+\epsilon K(x)) u^{2^*-1}, \quad x\in \mathbb{R}^N$$ where$2^*=\frac{2N}{N-4}$,$N\geq5$,$ \epsilon>0$. We prove Infinitely many peak solutions for a biharmonic equation involving critical exponent In this paper, we study the following biharmonic equation Δ2u=K(|y|)up,u>0,inB1(0),u=Δu=0,on∂B1(0), where p=N+4N−4 , K(1) > 0,K′(1) > 0, B1(0) is the unit ball in RN (N≥6). We show that the On a biharmonic equation involving slightly supercritical exponent We consider the biharmonic equation with supercritical nonlinearity (Pε) : ∆ u = K|u|8/(n−4)+εu in Ω, ∆u = u = 0 on ∂Ω, where Ω is a smooth bounded domain in R , n ≥ 5, K is a C positive function, PROFILE AND EXISTENCE OF SIGN-CHANGING SOLUTIONS TO AN ELLIPTIC SUBCRITICAL EQUATION • Mathematics • 2008 In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pe) : Δ2 u = |u|(8/(n-4))-eu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We On a fourth order elliptic equation with supercritical exponent This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent ( P e ) : Δ 2 u = | u | 8 / ( n − 4 ) + e u in Ω, Δ u = u = 0 on ∂ Ω, where Ω is a smooth bounded ## References SHOWING 1-10 OF 33 REFERENCES A classification of solutions of a conformally invariant fourth order equation in Rn Abstract. In this paper, we consider the following conformally invariant equations of fourth order¶$ \cases {\Delta^2 u = 6 e^{4u} &in $\bf {R}^4,$ \cr e^{4u} \in L^1(\bf {R}^4),\cr}$(1)¶and¶$
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In this paper a fourth order equation involving critical growth is considered under Navier boundary condition :∆u = Ku, u > 0 in Ω, u = ∆u = 0 on ∂Ω, where K is a positive function, Ω is a bounded
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RésuméNous étudions les problèmes sous-critiquesPɛ:−Δu=up−ɛ,u > 0 surΩ;u=0 sur ∂Ω−oùΩ est un domaine borné et régulier de ℝN,N−3,p + 1=2N/N −2 est l'exposant critique de Sobolev, et ɛ>0 tend vers
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This paper is concerned with the nonlinear elliptic problem (Pe): -Δu = up+e, u > 0 in Ω; u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, p + 1 = 2n/(n - 2) is the critical Sobolev
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Soit Ω un ensemble ouvert borne regulier et connexe de R N , N≥3. On considere u:Ω→R telle que −Δu=u (N+2)/(N−2) dans Ω, u>0 dans Ω, u=0 sur ∂Ω. On note par Hd(Ω; Z 2 ) l'homologie de diemnsion d de