Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity

• Jian Wang ^[1] ; Zhuoran Du ^[1]
  1. [1] Hunan University

     Hunan University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity ⎧ ⎪⎨ ⎪⎩ (−)s u(x) = K(x)u−α(x) + μu p−1(x) in , u > 0 in , u = 0 in c := RN \ , where s ∈ (0, 1), α > 0 and ⊂ RN is a bounded domain with smooth boundary ∂ and N > 2s. Under some appropriate assumptions of α, p, μ and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of C1,1 loc ∩ L∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C1,1 ∩ L∞ solutions are obtained for star-shaped domain under a condition of K.

• Referencias bibliográficas
  • 1. Barrios, B., Coloradoc, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst....
  • 2. Barrios, B., De Bonis, I., Madina, M., Peral, I.: Semilinear problems for the fractional Laplacian with a singular nonlinearity. Open Math....
  • 3. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
  • 4. Cai, M., Li, F.: Properties of positive solutions to nonlocal problems with negative exponents in unbounded domains. Nonlinear Anal. 200,...
  • 5. Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017)
  • 6. Coclite, M., Palmieri, G.: On a singular nonlinear Dirichlet problem. Commun. Partial Differ. Equ. 14, 1315–1327 (1989)
  • 7. Crandall, M., Rabinowitz, P., Tartar, L.: On a Dirichlet problem with a singalar nonlinearity. Commun. Partial Differ. Equ. 2, 193–222...
  • 8. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
  • 9. Ghanmi, A., Saoudi, K.: The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator. Fract. Differ....
  • 10. Ghanmi, A., Saoudi, K.: A multiplicity results for a singular problem involving the fractional pLaplacian operator. Complex Var. Elliptic...
  • 11. Ghergu, M., R˘adulescu, V.: Singular elliptic problems with lack of compactness. Ann. Mat. Pura Appl. 185, 63–79 (2006)
  • 12. Haitao, Y.: Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differ. Equ. 189,...
  • 13. Hernández, J., Mancebo, F., Vega, J.: Positive solutions for singular nonlinear elliptic equations. Proc. R. Soc. Edinburgh Sect A. 137,...
  • 14. Hernández, J., Karátson, J., Simon, P.: Multiplicity for semilinear elliptic equations involving singular nonlinearity. Nonlinear Anal....
  • 15. Hirano, H., Saccon, C., Shioji, N.: Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem....
  • 16. Hirano, H., Saccon, C., Shioji, N.: Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities....
  • 17. Li, Q., Zou, W.: The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical...
  • 18. Li, Q., Zou, W.: Normalized ground states for Sobolev critical nonlinear Schrödinger equation in the L2-supercritical case. Discrete Contin....
  • 19. Molica Bisci, G., Ra˘dulescu, V., Servadei. R.: Variational method for nonlocal fractional problems, with a foreword by Jean Mawhin. In:...
  • 20. Mukherjee, T., Sreenadh, K.: Fractional elliptic equations with critical growth and singular nonlinearities. Electron. J. Differ. Equ....
  • 21. Papageorgiou, N., R˘adulescu, V.: Combined effects of singular and sublinear nonlinearities in some elliptic problems. Nonlinear Anal....
  • 22. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101,...
  • 23. Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213, 587–628 (2014)
  • 24. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1966)
  • 25. Saoudi, K.: A critical fractional elliptic equation with singular nonlinearities. Fract. Calc. Appl. Anal. 20(6), 1507–1530 (2017)
  • 26. Saoudi, K., Ghosh, S., Choudhuri, D.: Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity....
  • 27. Servadei, R., Valdinoci, E.: Mountain passs solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
  • 28. Servadei, R., Valdinoci, E.: The Brezis–Nirenberg results for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)
  • 29. Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear in some singular boundary value problems. J. Differ. Equ. 176,...