Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight

Abstract

Let \(p>0\) and \((-\Delta )^{s}\) is the fractional Laplacian with \(0< s<1\). The purpose of this paper is to establish a classification result for positive stable solutions to a fractional singular elliptic equation with weight

$$ (-\Delta )^{s} u=-h(x)u^{-p}\text{ in }\mathbb{R}^{N}. $$

Here \(N>2s\) and \(h\) is a nonnegative, continuous function satisfying \(h(x)\geq C|x|^{a}\), \(a\geq 0\), when \(|x|\) large. We prove the nonexistence of positive stable solutions of this equation under the condition

$$ N< 2s+\frac{2(a+2s)}{p+1}\left (p+\sqrt{p^{2}+p}\right ) $$

or equivalently

$$ p>p_{c}(N,s,a), $$

where

$$ p_{c}(N,s,a)= \textstyle\begin{cases} \frac{(N-2s)^{2}-2(N+a)(a+2s)+2\sqrt{(a+2s)^{3}(2N-2s+a)}}{(N-2s)(10s+4a-N)}&\text{ if }N< 10s+4a \\ +\infty &\text{ if }N\geq 10s+4a \end{cases}\displaystyle . $$

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References

  1. 1.

    Applebaum, D.: Lévy processes—from probability to finance and quantum groups. Not. Am. Math. Soc. 51(11), 1336–1347 (2004)

    MATH  Google Scholar 

  2. 2.

    Binlin, Z., Rădulescu, V.D., Wang, L.: Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian. Proc. R. Soc. Edinb., Sect. A 149(4), 1061–1081 (2019)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Chen, C., Song, H., Yang, H.: Liouville type theorems for stable solutions of \(p\)-Laplace equation in \(\mathbb{R}^{N}\). Nonlinear Anal. 160, 44–52 (2017)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cowan, C.: Liouville theorems for stable Lane-Emden systems with biharmonic problems. Nonlinearity 26(8), 2357–2371 (2013)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cowan, C.: Stability of entire solutions to supercritical elliptic problems involving advection. Nonlinear Anal. 104, 1–11 (2014)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Cowan, C., Fazly, M.: On stable entire solutions of semi-linear elliptic equations with weights. Proc. Am. Math. Soc. 140(6), 2003–2012 (2012)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Dancer, E.N., Farina, A.: On the classification of solutions of \(-\Delta u=e^{u}\) on \(\mathbb{R}^{N}\): stability outside a compact set and applications. Proc. Am. Math. Soc. 137(4), 1333–1338 (2009)

    MATH  Google Scholar 

  9. 9.

    Dávila, J., Dupaigne, L., Farina, A.: Partial regularity of finite Morse index solutions to the Lane-Emden equation. J. Funct. Anal. 261(1), 218–232 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Dávila, J., Dupaigne, L., Wei, J.: On the fractional Lane-Emden equation. Trans. Am. Math. Soc. 369(9), 6087–6104 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Devillanova, G.: Multiscale weak compactness in metric spaces. J. Elliptic Parabolic Equ. 2(1–2), 131–144 (2016)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Devillanova, G., Solimini, S.: Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case. Calc. Var. Partial Differ. Equ. 52(3–4), 857–898 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Du, Y., Guo, Z.: Positive solutions of an elliptic equation with negative exponent: stability and critical power. J. Differ. Equ. 246(6), 2387–2414 (2009)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Duong, A.T., Nguyen, V.H.: A Liouville type theorem for fractional elliptic equation with exponential nonlinearity (2019). arXiv:1911.05966

  15. 15.

    Duong, A.T., Nguyen, N.T., Nguyen, T.Q.: Liouville type theorems for two elliptic equations with advections. Ann. Pol. Math. 122(1), 11–20 (2019)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Dupaigne, L., Farina, A.: Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities. Nonlinear Anal. 70(8), 2882–2888 (2009)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Dupaigne, L., Farina, A.: Stable solutions of \(-\Delta u=f(u)\) in \(\mathbb{R}^{N}\). J. Eur. Math. Soc. 12(4), 855–882 (2010)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Farina, A.: On the classification of solutions of the Lane-Emden equation on unbounded domains of \(\mathbb{R}\mathbb{R}^{n}N\). J. Math. Pures Appl. (9) 87(5), 537–561 (2007)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Farina, A.: Stable solutions of \(-\Delta u=e^{u}\) on \(\mathbb{R}^{N}\). C. R. Math. Acad. Sci. Paris 345(2), 63–66 (2007)

    MathSciNet  Google Scholar 

  20. 20.

    Fazly, M., Ghoussoub, N.: On the Hénon-Lane-Emden conjecture. Discrete Contin. Dyn. Syst. 34(6), 2513–2533 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Fazly, M., Sire, Y.: Symmetry properties for solutions of nonlocal equations involving nonlinear operators. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 36(2), 523–543 (2019)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Fazly, M., Wei, J.: On stable solutions of the fractional Hénon-Lane-Emden equation. Commun. Contemp. Math. 18(5), 1650005 (2016)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Fazly, M., Wei, J.: On finite Morse index solutions of higher order fractional Lane-Emden equations. Am. J. Math. 139(2), 433–460 (2017)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Felmer, P., Quaas, A.: Fundamental solutions and Liouville type theorems for nonlinear integral operators. Adv. Math. 226(3), 2712–2738 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Guo, Y., Wei, J.: Nonexistence of positive finite Morse index solutions to an elliptic problem with singular nonlinearity. Methods Appl. Anal. 15(3), 391–403 (2008)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Hajlaoui, H., Harrabi, A., Mtiri, F.: Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete Contin. Dyn. Syst. 37(1), 265–279 (2017)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Hu, L.-G.: Liouville type results for semi-stable solutions of the weighted Lane-Emden system. J. Math. Anal. Appl. 432(1), 429–440 (2015)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Hu, L.-G.: Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: \(p\geq \vartheta >1\). NoDEA Nonlinear Differ. Equ. Appl. 25(1), 7 (2018)

    MathSciNet  Google Scholar 

  29. 29.

    Hu, L.-G., Zeng, J.: Liouville type theorems for stable solutions of the weighted elliptic system. J. Math. Anal. Appl. 437(2), 882–901 (2016)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Le, P., Nguyen, H.T., Nguyen, T.Y.: On positive stable solutions to weighted quasilinear problems with negative exponent. Complex Var. Elliptic Equ. 63(12), 1739–1751 (2018)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Leite, E.J.F., Montenegro, M.: A priori bounds and positive solutions for non-variational fractional elliptic systems. Differ. Integral Equ. 30(11–12), 947–974 (2017)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Ma, L., Wei, J.C.: Properties of positive solutions to an elliptic equation with negative exponent. J. Funct. Anal. 254(4), 1058–1087 (2008)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Mingqi, X., Rădulescu, V.D., Zhang, B.: A critical fractional Choquard-Kirchhoff problem with magnetic field. Commun. Contemp. Math. 21(4), 1850004 (2019)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Molica Bisci, G., Rădulescu, V.D.: Multiplicity results for elliptic fractional equations with subcritical term. NoDEA Nonlinear Differ. Equ. Appl. 22(4), 721–739 (2015)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Molica Bisci, G., Rădulescu, V.D.: A sharp eigenvalue theorem for fractional elliptic equations. Isr. J. Math. 219(1), 331–351 (2017)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Molica Bisci, G., Vilasi, L.: On a fractional degenerate Kirchhoff-type problem. Commun. Contemp. Math. 19(1), 1550088 (2017)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Molica Bisci, G., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016). With a foreword by Jean Mawhin

    MATH  Google Scholar 

  38. 38.

    Pan, N., Pucci, P., Zhang, B.: Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian. J. Evol. Equ. 18(2), 385–409 (2018)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Pucci, P., Xiang, M., Zhang, B.: Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional \(p\)-Laplacian. Adv. Calc. Var. 12(3), 253–275 (2019)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Rahal, B., Zaidi, C.: On the classification of stable solutions of the fractional equation. Potential Anal. 50(4), 565–579 (2019)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Xiang, M., Zhang, B., Rădulescu, V.D.: Existence of solutions for perturbed fractional \(p\)-Laplacian equations. J. Differ. Equ. 260(2), 1392–1413 (2016)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Xiang, M., Zhang, B., Rădulescu, V.D.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional \(p\)-Laplacian. Nonlinearity 29(10), 3186–3205 (2016)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Yang, H., Zou, W.: Symmetry of components and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian. Nonlinear Anal. 180, 208–224 (2019)

    MathSciNet  MATH  Google Scholar 

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Acknowledgement

We would like to express our gratitude to the referee for the careful reading and helpful suggestions. This research is funded by VNU - University of Education under Research Project number QS.19.07.

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Correspondence to Anh Tuan Duong.

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Duong, A.T., Luong, V.T. & Nguyen, T.Q. Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight. Acta Appl Math 170, 579–591 (2020). https://doi.org/10.1007/s10440-020-00347-5

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Keywords

  • Liouville type theorems
  • Stable solutions
  • Fractional singular elliptic equations
  • Negative exponent nonlinearity

Mathematics Subject Classification

  • 35B53
  • 35J60
  • 35B35