Abstract
We verify the existence of radial positive solutions for the semilinear equation
where N ≥ 3, p is close to p* ≔ (N+ 2)/(N − 2), and V is a radial smooth potential. If q is super-critical, namely q > p*, we prove that this problem has a radial solution behaving like a superposition of bubbles blowing-up at the origin with different rates of concentration, provided V(0) < 0. On the other hand, if N/(N − 2) < q < p*, we prove that this problem has a radial solution behaving like a super-position of flat bubbles with different rates of concentration, provided limr→∞V(r) < 0.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), 573–598.
R. Bamón, M. del Pino and I. Flores, Ground states of semilinear elliptic equations: A geometric approach, Ann. Inst. H. Poincaré 17 (2000), 551–581.
V. Benci and G. Cerami, Existence of positive solutions of the equation −Δu+a(x)u = u(N+2)/(N−2)in ℝn, J. Funct. Anal. 88 (1990), 91–117.
L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations involving critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271–297.
J. Campos, “Bubble-Tower” phenomena in a semilinear elliptic equation with mixed Sobolev growth, Nonlinear Anal. 68 (2008), 1382–1397.
C. C. Chen and C. -S. Lin, Blowing up with infinite energy of conformal metrics on Sn, Comm. Partial Differential Equations24 (1999), 785–799.
A. Contreras and M. del Pino, Nodal Bubble-Tower solutions to radial elliptic problems near criticality, Discrete Contin. Dyn. Syst. 16 (2006), 525–539.
J. Davila and I. Guerra, Slowly decaying radial solutions of an elliptic equations with subcritical and supercritical exponents, J. Analyse Math. 129 (2016), 367–391.
W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration Mech. Anal. 91 (1986), 283–308.
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equation with bounded potential, J. Funct. Anal. 69 (1986), 397–408.
R. H. Fowler, Further studies on Emden’s and similar differential equations, Q. J. Math. 2 (1931), 259–288.
A. M. Micheletti, M. Musso and A. Pistoia, Super-position of spikes for a slightly super-critical elliptic equation in ℝn, Discrete Contin. Dyn. Syst. 12 (2005), 747–760.
A. M. Micheletti and A. Pistoia, Existence of blowing-up solutions for a slightly sub-critical or a slightly supercritical non-linear elliptic equation on ℝn, Nonlinear Anal. 52 (2003), 173–195.
M. del Pino, J. Dolbeault and M. Musso, “Bubble-Tower” radial solutions in the slightly super critical Brezis—Nirenberg problem, J. Differential Equations 193 (2003), 280–306.
M. del Pino, M. Musso and A. Pistoia, Super critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincare 22 (2005), 45–82.
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353–372.
Acknowledgements
The first author is supported by FONDECYT Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017. The second author was supported by FAPESP (Brazil) Grant #2016/04925-7.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Musso, M., Pimentel, J. A semilinear elliptic equation with competing powers and a radial potential. JAMA 140, 283–298 (2020). https://doi.org/10.1007/s11854-020-0089-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-020-0089-4