March  2017, 37(3): 1691-1706. doi: 10.3934/dcds.2017070

A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition

Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Kingdom of Saudi Arabia

Received  June 2016 Revised  October 2016 Published  December 2016

In this paper we consider the following nonlinear critical problem: $-Δ u= (1+\varepsilon_0 K_0(x)) u^\frac{n+2}{n-2}$, $u>0$ in $Ω$, $u=0$, on $\partial Ω$, where $Ω$ is a bounded domain of $\mathbb{R}^n$, $K_0$ is a given function and $\varepsilon_0$ is a small parameter. Under the assumption that $K_0$ is flat near its critical points, we prove an existence result in terms of the Euler-Hopf index. We believe that it is the very first result in this direction that we do not need any restrictions on the flatness coefficient.

Citation: Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
References:
[1]

A. AmbrosettiJ. Garcia Azorero and I. Peral, Perturbation of $ - \Delta u + {u^{\frac{{\left( {N + 2} \right)}}{{\left( {N - 2} \right)}}}} = 0 $, the Scalar Curvature Problem in $ {\mathbb{R}^N} $ and related topics, Journal of Functional Analysis, 165 (1999), 117-149.  doi: 10.1006/jfan.1999.3390.

[2]

A. Ambrosetti and M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. Henri. Poincaré. ANL, 15 (1998), 233-252.  doi: 10.1016/S0294-1449(97)89300-6.

[3]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Royal. Soc. Edinburgh., 128 (1998), 1131-1161.  doi: 10.1017/S0308210500027268.

[4]

A. Ambrosetti and A. Malchiodi, A multiplicity result for the Yamabe problem on Sn, Journal of Functional Analysis, 168 (1999), 529-561.  doi: 10.1006/jfan.1999.3458.

[5]

A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math, Ser, 182 Longman Sci. Tech. Harlow, 1989.

[6]

A. Bahri, An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J., 81 (1996), 323-466.  doi: 10.1215/S0012-7094-96-08116-8.

[7]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172.  doi: 10.1016/0022-1236(91)90026-2.

[8]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain, Comm. Pure Appli. Math., 41 (1988), 255-294.  doi: 10.1002/cpa.3160410302.

[9]

A. Bahri and P. H. Rabinowitz, Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649. 

[10]

M. Ben Ayed and M. Hammami, On a variational problem involving critical Sobolev growth in dimension four, Advances in Differential Equations, 9 (2004), 415-446. 

[11]

R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $ {\mathbb{S}^3} $, Annales de l'Institut Fourier, (Grenoble), 61 (2011), 971-986.  doi: 10.5802/aif.2634.

[12]

R. Ben Mahmoud and H. Chtioui, Prescribing the scalar curvature problem on higherdimensional manifolds, Discrete Contin. Dyn. Syst., 32 (2012), 1857-1879.  doi: 10.3934/dcds.2012.32.1857.

[13]

A. Bensouf and H. Chtioui, Conformal metrics with prescribed Q-curvature on Sn, Calc. Var. Partial Differential Equations, 41 (2011), 455-481.  doi: 10.1007/s00526-010-0372-9.

[14]

Z. Bouchech and H. Chtioui, Multiplicity and existence results for a nonlinear elliptic equation with Sobolev exponent, Advanced Nonlinear Studies, 10 (2010), 537-571.  doi: 10.1515/ans-2010-0302.

[15]

H. Brezis and J. M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89 (1985), 21-56.  doi: 10.1007/BF00281744.

[16]

D. CaoE. Noussair and S. Yang, On the scalar curvature equation $ - \Delta u = \left( {1 + \varepsilon K} \right){u^{\frac{{n + 2}}{{n - 2}}}}\;{\rm{in}}\;{\mathbb{R}^n} $, Calc. Var., 15 (2002), 403-419.  doi: 10.1007/s00526-002-0137-1.

[17]

S. Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on Sn, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.

[18]

S. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2 and 3 spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.  doi: 10.1007/BF01191617.

[19]

X. Chen and X. Xu, The scalar curvature flow on Sn-Perturbation theorem revisited, Inventiones Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.

[20]

H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.  doi: 10.1515/ans-2003-0404.

[21]

H. ChtiouiR. Ben Mahmoud and D. A. Abuzaid, Conformal transformation of metrics on the n-sphere, Nonlinear Analysis: TMA, 82 (2013), 66-81.  doi: 10.1016/j.na.2013.01.003.

[22]

J. M. Coron, Topologie et cas limite del injections de Sobolev, C.R. Acad. Sc. Paris, 299 (1984), 209-212. 

[23]

E. N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc., 20 (1988), 600-602.  doi: 10.1112/blms/20.6.600.

[24]

E. Hebey, La methode d'isometrie concentration dans le cas d'un probléme non linéaire sur les varietés compactes á bord avec exposant critique de Sobolev, Bulletin des Sciences Mathématiques, 116 (1992), 35-51. 

[25]

M. Ji, Scalar curvature equation on Sn, Part Ⅰ: Topological conditions, J. Diff. Equa., 246 (2009), 749-787.  doi: 10.1016/j.jde.2008.04.011.

[26]

Y. Y. Li, Prescribing Scalar Curvature on S3, S4 and Related Problems, Journal of Functional Analysis, 118 (1993), 43-118.  doi: 10.1006/jfan.1993.1138.

[27]

Y. Y. Li, Prescribing scalar curvature on $S^{n}$ and related topics, Part Ⅰ, Journal of Differential Equations, 120 (1995), 319-410.  doi: 10.1006/jdeq.1995.1115.

[28]

A. Malchiodi, The scalar curvature problem on Sn: An approach via Morse theory, Calc. Var., 14 (2002), 429-445.  doi: 10.1007/s005260100110.

[29]

S. Pohozaev, Eigenfunctions of the equation $Δ u + λ f(u) = 0$, Soviet Math. Dokl., 6 (1965), 1408-1411. 

[30]

M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

show all references

References:
[1]

A. AmbrosettiJ. Garcia Azorero and I. Peral, Perturbation of $ - \Delta u + {u^{\frac{{\left( {N + 2} \right)}}{{\left( {N - 2} \right)}}}} = 0 $, the Scalar Curvature Problem in $ {\mathbb{R}^N} $ and related topics, Journal of Functional Analysis, 165 (1999), 117-149.  doi: 10.1006/jfan.1999.3390.

[2]

A. Ambrosetti and M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. Henri. Poincaré. ANL, 15 (1998), 233-252.  doi: 10.1016/S0294-1449(97)89300-6.

[3]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Royal. Soc. Edinburgh., 128 (1998), 1131-1161.  doi: 10.1017/S0308210500027268.

[4]

A. Ambrosetti and A. Malchiodi, A multiplicity result for the Yamabe problem on Sn, Journal of Functional Analysis, 168 (1999), 529-561.  doi: 10.1006/jfan.1999.3458.

[5]

A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math, Ser, 182 Longman Sci. Tech. Harlow, 1989.

[6]

A. Bahri, An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J., 81 (1996), 323-466.  doi: 10.1215/S0012-7094-96-08116-8.

[7]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172.  doi: 10.1016/0022-1236(91)90026-2.

[8]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain, Comm. Pure Appli. Math., 41 (1988), 255-294.  doi: 10.1002/cpa.3160410302.

[9]

A. Bahri and P. H. Rabinowitz, Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 561-649. 

[10]

M. Ben Ayed and M. Hammami, On a variational problem involving critical Sobolev growth in dimension four, Advances in Differential Equations, 9 (2004), 415-446. 

[11]

R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $ {\mathbb{S}^3} $, Annales de l'Institut Fourier, (Grenoble), 61 (2011), 971-986.  doi: 10.5802/aif.2634.

[12]

R. Ben Mahmoud and H. Chtioui, Prescribing the scalar curvature problem on higherdimensional manifolds, Discrete Contin. Dyn. Syst., 32 (2012), 1857-1879.  doi: 10.3934/dcds.2012.32.1857.

[13]

A. Bensouf and H. Chtioui, Conformal metrics with prescribed Q-curvature on Sn, Calc. Var. Partial Differential Equations, 41 (2011), 455-481.  doi: 10.1007/s00526-010-0372-9.

[14]

Z. Bouchech and H. Chtioui, Multiplicity and existence results for a nonlinear elliptic equation with Sobolev exponent, Advanced Nonlinear Studies, 10 (2010), 537-571.  doi: 10.1515/ans-2010-0302.

[15]

H. Brezis and J. M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89 (1985), 21-56.  doi: 10.1007/BF00281744.

[16]

D. CaoE. Noussair and S. Yang, On the scalar curvature equation $ - \Delta u = \left( {1 + \varepsilon K} \right){u^{\frac{{n + 2}}{{n - 2}}}}\;{\rm{in}}\;{\mathbb{R}^n} $, Calc. Var., 15 (2002), 403-419.  doi: 10.1007/s00526-002-0137-1.

[17]

S. Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on Sn, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.

[18]

S. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2 and 3 spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.  doi: 10.1007/BF01191617.

[19]

X. Chen and X. Xu, The scalar curvature flow on Sn-Perturbation theorem revisited, Inventiones Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.

[20]

H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.  doi: 10.1515/ans-2003-0404.

[21]

H. ChtiouiR. Ben Mahmoud and D. A. Abuzaid, Conformal transformation of metrics on the n-sphere, Nonlinear Analysis: TMA, 82 (2013), 66-81.  doi: 10.1016/j.na.2013.01.003.

[22]

J. M. Coron, Topologie et cas limite del injections de Sobolev, C.R. Acad. Sc. Paris, 299 (1984), 209-212. 

[23]

E. N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc., 20 (1988), 600-602.  doi: 10.1112/blms/20.6.600.

[24]

E. Hebey, La methode d'isometrie concentration dans le cas d'un probléme non linéaire sur les varietés compactes á bord avec exposant critique de Sobolev, Bulletin des Sciences Mathématiques, 116 (1992), 35-51. 

[25]

M. Ji, Scalar curvature equation on Sn, Part Ⅰ: Topological conditions, J. Diff. Equa., 246 (2009), 749-787.  doi: 10.1016/j.jde.2008.04.011.

[26]

Y. Y. Li, Prescribing Scalar Curvature on S3, S4 and Related Problems, Journal of Functional Analysis, 118 (1993), 43-118.  doi: 10.1006/jfan.1993.1138.

[27]

Y. Y. Li, Prescribing scalar curvature on $S^{n}$ and related topics, Part Ⅰ, Journal of Differential Equations, 120 (1995), 319-410.  doi: 10.1006/jdeq.1995.1115.

[28]

A. Malchiodi, The scalar curvature problem on Sn: An approach via Morse theory, Calc. Var., 14 (2002), 429-445.  doi: 10.1007/s005260100110.

[29]

S. Pohozaev, Eigenfunctions of the equation $Δ u + λ f(u) = 0$, Soviet Math. Dokl., 6 (1965), 1408-1411. 

[30]

M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

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