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}
}
  • M. Ayed, K. Mehdi
  • 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
TLDR
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
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
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¶ $
Proof of two conjectures of H. Brezis and L.A. Peletier
AbstractIn this paper, we consider the problem: −Δu=N(N−2)up−ɛ, u>0 on Ω; u=0 on ∂Ω, where Ω is a smooth and bounded domain inRN, N≥3, p= $$\frac{{N + 2}}{{N - 2}}$$ , and ε>0. We prove a conjecture
Some Existence Results for a Fourth Order Equation Involving Critical Exponent
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
Existence and nonexistence results for critical growth biharmonic elliptic equations
Abstract.We prove existence of nontrivial solutions to semilinear fourth order problems at critical growth in some contractible domains which are perturbations of small capacity of domains having
Asymptotics for Elliptic Equations Involving Critical Growth
Consider the problem $$ \left( I \right){\rm{ }}\left\{ {\matrix{ { - u - \lambda u = 3{u^{5 - \in }}} & {in{\rm{ }}\Omega } \cr {u >0} & {in{\rm{ }}\Omega } \cr {u = 0} & {on{\rm{ }}\partial
On a variational problem with lack of compactness: the topological effect of the critical points at infinity
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
A NONEXISTENCE RESULT OF SINGLE PEAKED SOLUTIONS TO A SUPERCRITICAL NONLINEAR PROBLEM
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
On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain
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
...
...