Abstract
We consider the equation \(-\Delta u = |u| ^{\frac{4}{n-2}}u + \varepsilon f(x) \) under zero Dirichlet boundary conditions in a bounded domain \(\Omega \) in \(\mathbb {R}^{n}\), \(n \ge 3\), with \(f\ge 0\), \(f\ne 0\). We find sign-changing solutions with large energy. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem
recently constructed in del Pino et al. (J Differ Equ 251(9):2568–2597, 2011).
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References
Ali, I., Castro, A.: Positive solutions for a semilinear elliptic problem with critical exponent. Nonlinear Anal. 27, 327–338 (1996)
Aubin, T.: Problèmes isopérimetriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)
Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41, 253–294 (1988)
Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437–477 (1983)
Brézis, H., Nirenberg, L.: A minimization problem with critical exponent and nonzero data. In: Symmetry in Nature. Scuola Norm. Sup. Pisa, pp. 437–477 (1989)
Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42, 271–297 (1989)
Clapp, M., del Pino, M., Musso, M.: Multiple solutions for a non-homogeneous elliptic equation at the critical exponent. Proc. R. Soc. Edinb. Sect. A 134(1), 69–87 (2004)
Coron, J.M.: Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Ser. I Math. 299, 209–212 (1984)
Dancer, E.N.: A note on an equation with critical exponent. Bull. Lond. Math. Soc. 20, 600–602 (1988)
del Pino, M., Felmer, P.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149(1), 245–265 (1997)
del Pino, M., Felmer, P., Musso, M.: Two-bubble solutions in the super-critical Bahri–Coron’s problem. Calc. Var. PDE 16(2), 113–145 (2003)
del Pino, M., Felmer, P., Musso, M.: Multi-peak solutions for super-critical elliptic problems in domains with small holes. J. Differ. Equ. 182(2), 511–540 (2002)
del Pino, M., Felmer, P., Musso, M.: Multi-bubble solutions for slightly super-critial ellitpic problems in domains with symmetries. Bull. Lond. Math. Soc. 35(4), 513–521 (2003)
del Pino, M., Musso, M., Pacard, F., Pistoia, A.: Large energy entire solutions for the Yamabe equation. J. Differ. Equ. 251(9), 2568–2597 (2011)
del Pino, M., Musso, M., Pacard, F., Pistoia, A.: Torus action on \(S^n\) and sign changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(1), 209–237 (2013)
Ding, W.: On a conformally invariant elliptic equation on \(R^n\). Commun. Math. Phys. 107, 331–335 (1986)
Ding, W.: Positive solutions of \(\Delta u + u^{N+2 \over N-2} =0\) on contractible domains. J. Partial Differ. Equ. 2(4), 83–88 (1989)
Duyckaerts, T., Kenig, C., Merle, F.: Solutions of the focusing nonradial critical wave equation with the compactness property. arxiv:1402.0365v1
Kazdan, J., Warner, F.W.: Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28, 567–597 (1975)
Musso, M., Wei, J.: Nondegeneracy of nonradial nodal solutions to Yamabe problem. Commun. Math. Phys. 340(3), 1049–1107 (2015)
Musso, M., Wei, J.: Sign-changing blowing-up solutions for supercritical Bahri–Coron’s problem. Calc. Var. Partial Differ. Equ. 55(1), 39 (2016)
Obata, M.: Conformal changes of Riemannian metrics on a Euclidean sphere. Differential geometry (in honor of Kentaro Yano), pp. 347–353. Kinokuniya, Tokyo (1972)
Passaseo, D.: Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscr. Math. 65(2), 147–165 (1989)
Passaseo, D.: New nonexistence results for elliptic equations with supercritical nonlinearity. Differ. Integral Equ. 8(3), 577–586 (1995)
Pohozaev, S.: Eigenfunctions of the equation \(\Delta u + \lambda f(u) = 0\). Sov. Math. Dokl. 6, 1408–1411 (1965)
Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, RI (1986)
Rey, O.: Concentration of solutions to elliptic equations with critical nonlinearity. Ann. Inst. Henri Poincaré Anal. Non Linéaire 9, 201–218 (1992)
Talenti, G.: Best constants in Sobolev inequality. Annali di Matematica 10, 353–372 (1976)
Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire. 9, 281–304 (1992)
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With admiration and gratitude to Professor Paul Rabinowitz.
This research has been partly supported by Fondecyt Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017.
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Musso, M. Sign-changing blowing-up solutions for a non-homogeneous elliptic equation at the critical exponent. J. Fixed Point Theory Appl. 19, 345–361 (2017). https://doi.org/10.1007/s11784-016-0356-2
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DOI: https://doi.org/10.1007/s11784-016-0356-2