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Some existence results for a paneitz type problem via the theory of critical points at infinity. (English) Zbl 1210.35102

The authors prove some existence results for the following nonlinear problem under the Navier boundary condition \[ \begin{gathered} \Delta^2 u= Hu^p\quad\text{in }\Omega,\\ u> 0\quad\text{in }\Omega,\\ u=\Delta u= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded domain (smooth) of \(\mathbb{R}^N\), \(N\geq 5\), \(p+1= {2n\over n-4}\), is the critical exponent of the embedding \(H^2\cap H^1_0(\Omega)\) into \(L^{p+1}(\Omega)\) and \(H\) is a \(C^3\)-positive function in \(\overline\Omega\). The aim of this paper is to give sufficient conditions on \(H\) such that (1) possesses a solution.

MSC:

35J61 Semilinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J35 Variational methods for higher-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Aubin, T.; Bahri, A., Méthodes de topologie algébrique pour le problème de la coubure scalaire prescrite, J. Math. Pures Appl., 76, 525-549 (1997) · Zbl 0886.58109
[2] Aubin, T.; Bahri, A., Une hypothèse topologique pour le problème de la courbure scalaire prescrite, J. Math. Pures Appl., 76, 843-850 (1997) · Zbl 0916.58041
[3] Bahri, A., Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182 (1989), Longman: Longman Harlow · Zbl 0676.58021
[4] Bahri, A., An invariant for Yamabe-type flows with application to scalar curvature problems in high dimension, Duke Math. J., 81, 323-466 (1996), A celebration of J.F. Nach Jr. · Zbl 0856.53028
[5] Bahri, A.; Coron, J. M., On a nonlinear elliptic equation involving the critical Dobolev exponent: the effect of the topology on the domain, Comm. Pure Appl. Math., 41, 253-294 (1988) · Zbl 0649.35033
[6] Bahri, A.; Rabinowitz, P. H., Periodic solution of Hamiltonian systems of three-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8, 561-649 (1991) · Zbl 0745.34034
[7] Ben Ayed, M.; Chen, Y.; Chtioui, H.; Hammami, M., On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84, 633-667 (1996) · Zbl 0862.53034
[8] Ben Ayed, M.; Chtioui, H.; Hammami, M., The scalar curvature problem on higher dimensional spheres, Duke Math. J., 93, 379-424 (1998) · Zbl 0977.53035
[9] M. Ben Ayed, K. El Mehdi, The Paneitz curvature problem on lower dimensional spheres, The Abdus Salam ICTP Preprint IC/2003/48, Trieste, Italy; M. Ben Ayed, K. El Mehdi, The Paneitz curvature problem on lower dimensional spheres, The Abdus Salam ICTP Preprint IC/2003/48, Trieste, Italy
[10] Ben Ayed, M.; El Mehdi, K., Existence of conformal metrics on spheres with prescribed Paneitz curvature, Manuscripta Math., 114, 211-228 (2004) · Zbl 1067.53021
[11] M. Ben Ayed, M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Preprint (2003); M. Ben Ayed, M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Preprint (2003) · Zbl 1104.35010
[12] Bernis, F.; Garcia-Azorero, J.; Peral, I., Existence and multiplicity of nontrivial solutions in semilinear critical problems, Adv. Differential Equations, 1, 219-240 (1996) · Zbl 0841.35036
[13] Brezis, H.; Coron, J. M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89, 21-56 (1985) · Zbl 0584.49024
[14] Chang, S. A., On a Paneitz operator—a fourth order differential operator—in conformal geometry, (Christ, M.; Kenig, C.; Sadorsky, C., Harmonic Analysis and Partial Differential Equations; Essays in honor of Alberto P. Calderon. Harmonic Analysis and Partial Differential Equations; Essays in honor of Alberto P. Calderon, Chicago Lectures in Math. (1999), Univ. of Chicago Press: Univ. of Chicago Press Chicago, IL), 127-150 · Zbl 0982.53036
[15] Edmunds, D. E.; Fortunato, D.; Janelli, E., Critical exponents, critical dimension and the biharmonic operator, Arch. Rational Mech. Anal., 112, 269-289 (1990) · Zbl 0724.35044
[16] Ebobisse, F.; Ould Ahmedou, M., On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52, 1535-1552 (2003) · Zbl 1022.35012
[17] Djadli, Z.; Hebey, E.; Ledoux, M., Paneitz type operators and applications, Duke Math. J., 104, 129-169 (2000) · Zbl 0998.58009
[18] Djadli, Z.; Malchiodi, A.; Ould Ahmedou, M., Prescribing a fourth order conformal invariant on the standard sphere, Part I: a perturbation result, Commun. Contemp. Math., 4, 375-408 (2002) · Zbl 1023.58020
[19] Djadli, Z.; Malchiodi, A.; Ould Ahmedou, M., Prescribing a fourth order conformal invariant on the standard sphere, Part II: blow-up analysis and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5, 387-434 (2002) · Zbl 1150.53012
[20] Felli, V., Existence of conformal metrics on \(S^n\) with prescribed fourth-order invariant, Adv. Differential Equations, 7, 47-76 (2002) · Zbl 1054.53061
[21] Gazzola, F.; Grunau, H. C.; Squassina, M., Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18, 117-143 (2003) · Zbl 1290.35063
[22] Hulshof, J.; Van Der Vorst, R. C.A. M., Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114, 32-58 (1993) · Zbl 0793.35038
[23] Lin, C. S., A classification of solutions of a conformally invariant fourth order equation in \(R^n\), Comment. Math. Helv., 73, 206-231 (1998) · Zbl 0933.35057
[24] Lions, P. L., The concentration compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1 (1985), I: 165-201; II: 45-121 · Zbl 0704.49005
[25] Milnor, J., Lectures on \(h\)-Cobordism Theorem (1965), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[26] Noussair, E. S.; Swanson, C. A.; Jianfu, Y., Critical semilinear biharmonic equations in \(R^n\), Proc. Royal Soc. Edinburgh Sect. A, 121, 139-148 (1992) · Zbl 0779.35044
[27] Peletier, L.; Van Der Vorst, R. C.A. M., Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5, 747-767 (1992) · Zbl 0758.35029
[28] Pucci, P.; Serrin, J., Critical exponents and critical dimensions for polyharmonic operator, J. Math. Pures Appl., 69, 55-83 (1990) · Zbl 0717.35032
[29] Rey, O., The role of Green’s function in a nonlinear elliptic equation involving critical Sobolev exponent, J. Funct. Anal., 89, 1-52 (1990) · Zbl 0786.35059
[30] Struwe, M., A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187, 511-517 (1984) · Zbl 0535.35025
[31] Van Der Vorst, R. C.A. M., Fourth order elliptic equations with critical growth, C. R. Acad. Sci. Paris Sér. I, 320, 295-299 (1995) · Zbl 0834.35053
[32] Van Der Vorst, R. C.A. M., Best constant for the embedding of the space \(H^2 \cap H_0^1(\Omega)\) into \(L^{2 n /(n - 4)}(\Omega)\), Differential Integral Equations, 6, 259-276 (1993) · Zbl 0801.46033
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