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A Nonexistence of Solutions to a Supercritical Problem
Pure and Applied Mathematics Journal
Volume 2, Issue 6, December 2013, Pages: 184-190
Received: Dec. 4, 2013; Published: Jan. 10, 2014
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Kamal Ould Bouh, Department of Mathematics, Taibah University, Almadinah Almunawwarah, KSA
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In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ) in Ω ; u >0 in Ω and u=0 on ∂ Ω where is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for small, (Pε) has no positive solutions which blow up at one critical point of the function K.
Nonlinear Elliptic Equations, Critical Exponent, Variational Problem
To cite this article
Kamal Ould Bouh, A Nonexistence of Solutions to a Supercritical Problem, Pure and Applied Mathematics Journal. Vol. 2, No. 6, 2013, pp. 184-190. doi: 10.11648/j.pamj.20130206.13
A. Bahri, Critical point at infinity in some variational problems, Pitman Res. Notes Math. Ser. 182, Longman Sci. Tech. Harlow 1989.
A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the Sobolev exponent : the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294
A. Bahri, Y.Y. Li and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. and Part. Diff. Equ. 3 (1995), 67-94.
M. Ben Ayed, K. El Mehdi, M. Grossi and O. Rey, A Nonexistence result of single peaked solutions to a supercritical nonlinear problem, Comm. Contenporary Math., 2 (2003), 179-195.
M. Ben Ayed, K. Ould Bouh, Nonexistence results of sign-changing solutions to a supercritical nonlinear problem, Comm. Pure Applied Anal, 5 (2007), 1057-1075.
M. Del Pino, P. Felmer and M. Musso, Two bubles solutions in the supercritical Bahri-Coron’s problem, Calc. Var. Part. Diff. Equat., 16 (2003), 113–145.
Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincare (Analyse non-linear) 8(1991),159-174.
K. Ould Bouh, Nonexistence result of sign-changing solutions for a supercritical problem of the scalar curvature type , Advance in Nonlinear Studies (ANS), 12 (2012), 149-171.
O. Rey, The role of Green’s function in a nonlinear elliptic equation involving critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52.
O. Rey, The topological impact of critical points at infinity in a variational problem with lack of compactness : the dimension 3, Adv. Diff. Equ. 4 (1999), 581-616.
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