Abstract
In this paper, we study an approximate subcritical problem of the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We construct some sign-changing solutions which blow up at two different points on the boundary. We construct also some sign-changing solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.
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Ambrosetti A., Garcia Azorero J., Peral A.: Perturbation of \({\Delta u + u^{\frac{n+2}{n-2}} = 0}\), the scalar curvature problem in \({\mathbb{R}^n}\) and related topics. J. Funct. Anal. 165, 117–149 (1999)
Bahri A.: An invarient for Yamabe-type flows with applications to scalar curvature problems in high dimension, a celebration of J. F. Nash Jr.. Duke Math. J. 81, 323–466 (1996)
Bahri A., Coron J.M.: The scalar curvature problem on the standard three dimensional spheres. J. Funct. Anal. 95, 106–172 (1991)
Bahri A., Li Y.Y., Rey O.: On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc. Var. Partial Differ. Equ. 3, 67–94 (1995)
Ayed M.B., Chen Y., Chtioui H., Hammami M.: On the prescribed scalar curvature problem on 4-manifolds. Duke Math. J. 84, 633–677 (1996)
Ayed M.B., El Mehdi K., Ahmedou M.O.: The scalar curvature problem on the four dimensional half sphere. Calc. Var. 22, 465–482 (2005)
Ayed M.B., Ghoudi R., Bouh K.O.: Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere. Nonlinear Differ. Equ. Appl. NoDEA 19, 629–662 (2012)
Bianchi G., Pan X.B.: Yamabe equations on half spheres. Nonlinear Anal. 37, 161–186 (1999)
Chang S.A., Yang P.: A perturbation result in prescribing scalar curvature on S n. Duke Math. J. 64, 27–69 (1991)
Cherrier P.: Problèmes de Neumann non linéaires sur les variétés Riemaniennes. J. Funct. Anal. 57, 154–207 (1984)
Djadli Z., Malchiodi A., Ould Ahmedou M.: Prescribing the scalar and the boundary mean curvature on the three dimensional half sphere. J. Geom. Anal. 13, 233–267 (2003)
Escobar J., Schoen R.: Conformal metrics with prescribed scalar curvature. Invent. Math. 86, 243–254 (1986)
Li Y.Y.: Prescribing scalar curvature on S n and related topics, Part I. J. Differ. Equ. 120, 319–410 (1995)
Li Y.Y.: Part II, existence and compactness. Commun. Pure Appl. Math. 49, 437–477 (1996)
Rey O.: The topological impact of critical points at infinity in a variational problem with lack of compactness: the dimension 3. Adv. Differ. Equ. 4, 581–616 (1999)
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Bouh, K.O. Blowing up of sign-changing solutions to a subcritical problem. manuscripta math. 146, 265–279 (2015). https://doi.org/10.1007/s00229-014-0700-z
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DOI: https://doi.org/10.1007/s00229-014-0700-z