## Revista Matemática Iberoamericana

### Critical nonlinear elliptic equations with singularities and cylindrical symmetry

#### Abstract

Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in $\mathbb{R}^{N}$ with cylindrical symmetry and with singularities on a whole subspace of $\mathbb{R}^{N}$. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 20, Number 1 (2004), 33-66.

Dates
First available in Project Euclid: 2 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1080928419

Mathematical Reviews number (MathSciNet)
MR2076771

Zentralblatt MATH identifier
1330.35137

#### Citation

Badiale, Marino; Serra, Enrico. Critical nonlinear elliptic equations with singularities and cylindrical symmetry. Rev. Mat. Iberoamericana 20 (2004), no. 1, 33--66. https://projecteuclid.org/euclid.rmi/1080928419

#### References

• Badiale, M. and Tarantello, G.: A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163 (2002), no. 4, 259-293.
• Brézis, H. and Kato, T.: Remarks on the Schrödinger operators with singular complex potentials. J. Math. Pures Appl. (9) 58 (1979), 137-151.
• Brézis, H. and Lieb, E.: A relation between pointwise convergence of functionals and convergence of functionals. Proc. Amer. Math. Soc. 28 (1983), 486-490.
• Caffarelli, L., Kohn, R. and Nirenberg, L.: First order interpolation inequalities with weights. Compositio Math. 53 (1984), 259-275.
• Caldiroli, P. and Musina, R.: On the existence of extremal functions for a weighted Sobolev embedding with critical exponent. Calc. Var. Partial Differential Equations 8 (1999), no. 4, 365-387.
• Caldiroli, P. and Musina, R.: Stationary states for a two-dimensional singular Schrödinger equation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), no. 3, 609-633.
• Caldiroli, P. and Musina, R.: Existence and non existence results for a class of nonlinear singular Sturm-Liouville equations. Adv. Differential Equations 6 (2001), 303-326.
• Caldiroli, P. and Musina, R.: On a class of 2-dimensional singular elliptic problems. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 3, 479-497.
• Ciotti, L.: Dynamical models in astrophysics. Scuola Normale Superiore, Pisa, 2001.
• Egnell, H.: Elliptic boundary value problems with singular coefficients and critical nonlinearity. Indiana Univ. Math. J. 38 (1989), 235-251.
• Egnell, H.: Asymptotic results for finite energy solutions of semilinear elliptic equations. J. Differential Equations 98 (1992), 34-56.
• Ghoussoub, N. and Yuan, C.: Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 352 (2000), 5703-5743.
• Kuzin, I. and Pohozaev, S.: Entire solutions of semilinear elliptic equations. Progress in Nonlinear Differential Equations and their Applications 33. Birkhäuser Verlag, Basel, 1997.
• Ni, W. M.: On the equation $\Delta u + K(|x|)u^(n+2)/(n-2)=0$, its generalizations and applications in geometry. Indiana Univ. Math. J. 31 (1982), 439-529.
• Sandeep, K.: On a noncompact minimization problem of Hardy-Sobolev type. Adv. Nonlinear Stud. 2 (2002), no. 1, 81-91.
• Solimini, S.: A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 3, 319-337.
• Struwe, M.: Variational Methods. Springer-Verlag, Berlin Heidelberg New York, 1990.
• Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differential Equations 1 (1996), 241-264.