×

Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics. (English) Zbl 0938.35056

Let \(p=(n+2)/(n-2)\) be the critical Sobolev exponent in \(\mathbb{R}^n\), \(n\geq 3\). This paper deals with equations of the form \[ -\Delta u=u^p+ \varepsilon F(x,u)\text{ on }\mathbb{R}^n,\tag{1} \] where \(F(x,u)\) is alternately equal to \(K(x)u^p\) (resp. \(K(r)u^p,r=|x|\), i.e. \(K\) is radial), \(K(x)u^p+h(x)u\), with \(n>4\) and \(h(x)u^q\) with \(1<q<p\), \(n\geq 3\). In each case, the authors study conditions on the data which imply the existence of at least one positive solution of (1), in a suitable space, when \(\varepsilon\) is small enough. Some results are related with the scalar curvature problem in \(\mathbb{R}^n\). Their approach is based on the abstract perturbation method in critical point theory discussed in previous papers by the first author and M. Badiale [Ann. Inst. H. Poincaré, Anal. Non Linéaire 15, 233-252 (1998) and Proc. R. Soc. Edinb., Sect. A, Math. 128 No, 6, 1131-1161 (1998; Zbl 0928.34029)].
Reviewer: D.Huet (Nancy)

MSC:

35J60 Nonlinear elliptic equations
35B20 Perturbations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrosetti, A.; Badiale, M., Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 233-252 (1998) · Zbl 1004.37043
[2] Ambrosetti, A.; Badiale, M., Variational Perturbative methods and bifurcation of bound states from the essential spectrum, Proc. Royal Soc. Edinburgh, 128A, 1131-1161 (1998) · Zbl 0928.34029
[3] Ambrosetti, A.; Coti Zelati, V.; Ekeland, I., Symmetry breaking in Hamiltonian systems, J. Differential Equations, 67, 165-184 (1987) · Zbl 0606.58043
[4] Aubin, T., Nonlinear Analysis on Manifolds. Monge-Ampere Equations (1982), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0512.53044
[5] Bahri, A.; Coron, J. M., The Scalar-Curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95, 106-172 (1991) · Zbl 0722.53032
[6] Bianchi, G.; Egnell, H., A variational approach to the equation \(Δu + Ku^{(N+2)/(N−2)} =0\) in \(R^N \), Arch. Rational Mech. Anal., 122, 159-182 (1993) · Zbl 0803.35033
[7] Bianchi, G.; Egnell, H., Local existence and uniqueness of positive solutions of the equation \(Δu +(1+ εϕ (r))u^{(n+2)/(n−2)} =0\), in \(R^n\) and a related equation, (Lloyd, N. G.; Ni, W. M.; Peletier, L. A.; Serrin, J., Nonlinear Diffusion Equations and Their Equilibrium States. Nonlinear Diffusion Equations and Their Equilibrium States, Proceedings Gregynog, 1989 (1992), Birkhäuser: Birkhäuser Basel), 111-128 · Zbl 0812.35038
[8] Brezis, H.; Kato, T., Remarks on the Schödinger operator with singular complex potentials, J. Math. Pures Appl., 58, 137-151 (1979) · Zbl 0408.35025
[9] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[10] Chang, S. A.; Gursky, M. J.; Yang, P., The scalar curvature equation on 2- and 3-spheres, Calc. Var., 1, 205-229 (1993) · Zbl 0822.35043
[11] Chang, K. C.; Liu, J. Q., A Morse theoretical approach to the prescribing gaussian curvature problem, (Ambrosetti, A.; Chang, K. C., Variational Methods in Nonlinear Analysis (1993), Gordon and Breach: Gordon and Breach New York), 55-62 · Zbl 0861.53034
[12] Chang, S. A.; Yang, P., Prescribing scalar curvature on \(S^2\), Acta Math., 159, 215-259 (1987) · Zbl 0636.53053
[13] Chang, S. A.; Yang, P., A perturbation result in prescribing scalar curvature on \(S^n\), Duke Math. J., 64, 27-69 (1991) · Zbl 0739.53027
[14] Ding, W. Y.; Ni, W. M., On the elliptic equation \(Δu + ku^{(n+2)/(n−2)=0}\) and related topics, Duke Math. J., 52, 485-506 (1985) · Zbl 0592.35048
[15] Escobar, J.; Schoen, R., Conformal metrics with prescribed scalar curvature, Invent. Math., 86, 243-254 (1986) · Zbl 0628.53041
[16] Folland, G. B., Fourier Analysis and Its Applications (1992), Wadsworth and Brooks/Cole: Wadsworth and Brooks/Cole Belmont · Zbl 0371.35008
[17] J. Garcı́a Azorero, E. Montefusco, and, I. Peral, Bifurcation for the \(pR^N\); J. Garcı́a Azorero, E. Montefusco, and, I. Peral, Bifurcation for the \(pR^N\)
[18] Kazdan, J. L., Prescribing the Curbature of a Riemannian Manifolds. Prescribing the Curbature of a Riemannian Manifolds, Regional Conf. Series in Math., 57 (1984), Amer. Math. Soc: Amer. Math. Soc Providence
[19] Kazdan, J. L.; Warner, F., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. of Math., 101, 317-331 (1971) · Zbl 0297.53020
[20] Li, Y. Y., Prescribing scalar curvature on \(S^n\) and related topics, Part I, J. Differential Equations, 120, 319-410 (1995) · Zbl 0827.53039
[21] Li, Y. Y., Prescribing scalar curvature on \(S^n\) and related topics, Part II, Existence and compactness, Comm. Pure Appl. Math., 49, 437-477 (1996)
[22] Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1, 541-597 (1985)
[23] Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana, 1, 45-121 (1985) · Zbl 0704.49006
[24] Moser, J., On a nonlinear problem in differential geometry, (Peixoto, M., Dynamical Systems (1973), Academic Press: Academic Press San Diego), 273-280
[25] Ni, W. M., On the Elliptic Equation \(Δu +K(x)u^{(n+2)/(n−2)} =0\), its Generalizations, and Applications to Geometry, Indiana Univ. Math. J., 31, 493-529 (1982) · Zbl 0496.35036
[26] Nirenberg, L., Monge-Ampére equations and some associated problems in geometry, Proceedings of the International Congress of Mathematicians (1974), Vancouver, p. 275-279
[27] Rey, O., The role of the Green’s Function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89, 1-52 (1990) · Zbl 0786.35059
[28] Schoen, R. S., The existence of weak solutions with prescribed singular behavior for a conformaly invariant scalar equation, Comm. Pure Appl. Math., 41, 317-392 (1988) · Zbl 0674.35027
[29] Schoen, R. S., Variational theory for the total scalar curvature functional for riemannian metrics and related topics, (Gaquinta, M., Topics on Calculus of Variations. Topics on Calculus of Variations, Lecture Notes in Mathematics, 1365 (1989), Springer-Verlag: Springer-Verlag Berlin/New York), 120-154
[30] Trudinger, N., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22, 265-274 (1968) · Zbl 0159.23801
[31] Yau, S. T., Survey on partial differential equations in differential geometry, Ann. of Math. Stud., 102, 3-71 (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.