Abstract and Applied Analysis

Critical singular problems on unbounded domains

D. C. de Morais Filho and O. H. Miyagaki

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We present some results of existence for the following problem: Δu=a(x)g(u)+u|u|22, xN(N3), uD1,2(N), where the function a is a sign-changing function with a singularity at the origin and g has growth up to the Sobolev critical exponent 2=2N/(N2).

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Abstr. Appl. Anal., Volume 2005, Number 6 (2005), 639-653.

First available in Project Euclid: 3 October 2005

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Filho, D. C. de Morais; Miyagaki, O. H. Critical singular problems on unbounded domains. Abstr. Appl. Anal. 2005 (2005), no. 6, 639--653. doi:10.1155/AAA.2005.639. https://projecteuclid.org/euclid.aaa/1128345943

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  • S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 95--115.
  • S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439--475.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349--381.
  • M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), no. 4, 259--293.
  • A. K. Ben-Naoum, C. Troestler, and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal. 26 (1996), no. 4, 823--833.
  • V. Benci and G. Cerami, Existence of positive solutions of the equation $ - \Delta u + a(x)u = u^(N + 2)/(N - 2) $ in $\mathbf R^N$, J. Funct. Anal. 88 (1990), no. 1, 90--117.
  • H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553--572.
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313--345.
  • H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486--490.
  • H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437--477.
  • S. Cingolani and J. L. Gámez, Positive solutions of a semilinear elliptic equation on $\mathbf R^N$ with indefinite nonlinearity, Adv. Differential Equations 1 (1996), no. 5, 773--791.
  • D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbbR^N $, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 159--189.
  • G. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • J. L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conference Series in Mathematics, vol. 57, American Mathematical Society, Rhode Island, 1985.
  • Y. Li and W.-M. Ni, On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations, Arch. Ration. Mech. Anal. 108 (1989), no. 2, 175--194.
  • F. Munyamarere and M. Willem, Multiple solutions of semi-linear elliptic equations on $\mathbf R^N $, J. Math. Anal. Appl. 187 (1994), no. 2, 526--537.
  • W.-M. Ni and S. Yotsutani, On Matukuma's equation and related topics, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 7, 260--263.
  • E. S. Noussair and C. A. Swanson, Solutions of Matukuma's equation with finite total mass, Indiana Univ. Math. J. 38 (1989), no. 3, 557--561.
  • X. B. Pan, Positive solutions of the elliptic equation $\Delta u + u^(N + 2)/(N - 2) + K(x)u^q = 0$ in $\mathbf R^N $ and in balls, J. Math. Anal. Appl. 172 (1993), no. 2, 323--338.
  • P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Rhode Island, 1986.
  • M. Ramos, S. Terracini, and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal. 159 (1998), no. 2, 596--628.
  • H. T. Tehrani, Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Comm. Partial Differential Equations 21 (1996), no. 3-4, 541--557.
  • --------, Solutions for indefinite semilinear elliptic equations in exterior domains, J. Math. Anal. Appl. 255 (2001), no. 1, 308--318.