On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand–Liouville equation

Abstract

We consider the nonlocal Hénon-Gelfand–Liouville problem

$$\begin{aligned} (-\Delta )^s u = |x|^a e^u\quad \mathrm {in}\quad \mathbb {R}^n, \end{aligned}$$

for every \(s\in (0,1)\), \(a>0\) and \(n>2s\). We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when

$$\begin{aligned} \dfrac{\Gamma (\frac{n}{2})\Gamma (s)}{\Gamma (\frac{n-2s}{2})}\left( s+\frac{a}{2}\right) > \dfrac{\Gamma ^2(\frac{n+2s}{4})}{\Gamma ^2(\frac{n-2s}{4})}. \end{aligned}$$

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. 1.

    Ao, W.W., Yang, W.: On the classification of solutions of cosmic strings equation. Ann. Mat. Pura Appl. 198(6), 2183–2193 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chen, W.X., Li, C.M.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

    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  Article  Google Scholar 

  6. 6.

    Crandall, M.G., Rabinowitz, P.H.: Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rat. Mech. Anal. 58, 207–218 (1975)

    Article  Google Scholar 

  7. 7.

    Da Lio, F., Martinazzi, L., Rivière, T.: Blow-up analysis of a nonlocal Liouville-type equation. Anal. PDE 8(7), 1757–1805 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dancer, E.N., Du, Y., Guo, Z.: Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differ. Equ. 250, 3281–3310 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    Article  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  Article  Google Scholar 

  11. 11.

    Dávila, J., Dupaigne, L., Wang, K.L., Wei, J.: A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem. Adv. Math. 258, 240–285 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Du, Y., Guo, Z., Wang, K.: Monotonicity formula and \(\epsilon \)-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions. Calc. Var. Partial Differ. Equ. 50(3–4), 615–638 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Dupaigne, L., Ghergu, M., Goubet, O., Warnault, G.: The Gel’fand problem for the biharmonic operator. Arch. Ration. Mech. Anal. 208(3), 725–752 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regualarity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)

    Article  Google Scholar 

  15. 15.

    Fall, M.M.: Semilinear elliptic equations for the fractional Laplacian with Hardy potential. Nonlinear Anal. 193, 111311 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    Article  Google Scholar 

  18. 18.

    Fazly, M., Shahgholian, H.: Monotonicity formulas for coupled elliptic gradient systems with applications. Adv. Nonlinear Anal. 9, 479–495 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    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  Article  Google Scholar 

  20. 20.

    Fazly, M., Wei, J.: On stable solutions of the fractional Hénon-Lane-Emden equation. Commun. Contemp. Math. 18(5), 24 (2016)

    Article  Google Scholar 

  21. 21.

    Fazly, M., Wei, J., Yang, W.: Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation. Preprint, ArXiv:2006.06089 (2020)

  22. 22.

    Fazly, M., Yang, W.: On stable and finite Morse index solutions of the fractional Toda system. J. Funct. Anal. 280(4), 35 (2021)

  23. 23.

    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525–598 (1981)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hajlasz, P., Koskola, P.: Sobolev met poincare. Mem. Am. Math. Soc. 145(688), 106 (2000)

    MathSciNet  Google Scholar 

  25. 25.

    Herbst, I.W.: Spectral theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\). Commun. Math. Phys. 53(3), 285–294 (1977)

    Article  Google Scholar 

  26. 26.

    Huang, X.: Stable weak solutions of weighted nonlinear elliptic equations. Commun. Pure Appl. Anal. 13(1), 293–305 (2014)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Hyder, A.: Structure of conformal metrics on \(R^n\) with constant \(Q\)-curvature. Differ. Integral Equ. 32(7–8), 423–454 (2019)

    MathSciNet  Google Scholar 

  28. 28.

    Hyder, A., Yang, W.: Classification of stable solutions to a non-local Gelfand-Liouville equation. Int. Math. Res. Notices. https://doi.org/10.1093/imrn/rnaa236

  29. 29.

    Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49, 241–269 (1973)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Perez, C., Rela, E.: Degenerate Poincare-Sobolev inequalities. Trans. Am. Math. Soc. 372, 6087–6133 (2019)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Phan, Q.H., Souplet, Ph: Liouville-type theorems and bounds of solutions of Hardy-Hénon equations. J. Differ. Equ. 252, 2544–2562 (2012)

    Article  Google Scholar 

  32. 32.

    Ros-Oton, X., Serra, J.: The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50(3–4), 723–750 (2014)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Wang, C., Ye, D.: Some Liouville theorems for Hénon type elliptic equations. J. Funct. Anal. 262(4), 1705–1727 (2012)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Wang, K.: Partial regularity of stable solutions to the supercritical equations and its applications. Nonlinear Anal. 75(13), 5328–5260 (2012)

    MathSciNet  Google Scholar 

  35. 35.

    Wang, K.: Partial regularity of stable solutions to the Emden equation. Calc. Var. Partial Differ. Equ. 44(3–4), 601–610 (2012)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Wang, K.: Stable and finite Morse index solutions of Toda system. J. Differ. Equ. 268(1), 60–79 (2019)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Yafaev, D.: Sharp constants in the Hardy–Rellich inequalities. J. Funct. Anal. 168(1), 121–144 (1999)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The research of the third author is partially supported by NSFC No.11801550 and NSFC No.11871470. The third author thanks Ali Hyder for many stimulating discussions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mostafa Fazly.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by M. del Pino.

Appendix A

Appendix A

The purpose of this appendix is to provide necessary details for the proof of Proposition 2.5. First, we notice that the stability condition (1.3) can be extended to \(\overline{u}\). More precisely, if u is stable in \(\Omega \) then

$$\begin{aligned} \int _{\mathbb {R}_+^{n+1}}t^{1-2s}|\nabla \Phi |^2dxdt\ge \kappa _s\int _{\mathbb {R}^n}|x|^a e^u\varphi ^2dx, \end{aligned}$$
(A.1)

for every \(\Phi \in C_c^\infty ({\overline{\mathbb {R}_+^{n+1}}})\) satisfying \(\varphi (\cdot ):=\Phi (\cdot ,0)\in C_c^\infty (\Omega )\). Indeed, if \(\overline{\varphi }\) is the s-harmonic extension of \(\varphi \), we have

$$\begin{aligned} \int _{\mathbb {R}_+^{n+1}}t^{1-2s}|\nabla \Phi |^2dxdt&\ge \int _{\mathbb {R}_+^{n+1}}t^{1-2s}|\nabla {\overline{\varphi }} |^2dxdt\\ {}&=\kappa _s\int _{\mathbb {R}^n}\varphi (-\Delta )^s\varphi dx\ge \kappa _s\int _{\mathbb {R}^n}|x|^a e^u\varphi ^2dx. \end{aligned}$$

Before giving the conclusion of Moser’s iteration, we present the following lemma which will be used in the later context.

Lemma A.1

Let \(e^{\alpha u}\in L^1( \Omega ) \) for some \(\Omega \subset \mathbb {R}^n\). Then \(t^{1-2s}e^{\alpha \overline{u}}\in L^1_{\mathrm {loc}}(\Omega \times [0, \infty ))\).

Proof

Let \(\Omega _0\Subset \Omega \) be fixed. Since \( u\in L_s(\mathbb {R}^n)\), we have for \(x\in \Omega _0\) and \(t\in (0,R)\)

$$\begin{aligned} \overline{u}(x,t)\le C+\int _{\Omega }u(y)P(X,y)dy=C+\int _{\Omega }g(x,t)u(y)\frac{P(X,y)dy}{g(x,t)}, \end{aligned}$$

where \(C \le g(x,t):=\int _{\Omega }P(X,y)dy\le 1\) for some positive constant C depending solely on R, \(\Omega _0\) and \(\Omega \). Therefore, by the Jensen’s inequality

$$\begin{aligned}&\int _{\Omega _0}e^{\alpha \overline{u}(x,t)}dx \le C \int _{\Omega _0}\int _{\Omega }e^{\alpha g(x,t)u(y)}P(X,y)dydx\\&\quad \le C\int _{\Omega }\max \{e^{\alpha u(y)},1\}\int _{\Omega _0}P(X,y)dxdy \le C+C\int _{\Omega }e^{\alpha u(y)}dy, \end{aligned}$$

where the constant C depends on \(R,~\Omega _0\) and \(\Omega \), but not on t. Hence,

$$\begin{aligned} \int _{\Omega _0\times (0,R)} t^{1-2s}e^{\alpha \overline{u}(x,t)}dxdt \le \int _0^R t^{1-2s}\int _{\Omega _0}e^{\alpha \overline{u}(x,t)}dxdt <\infty . \end{aligned}$$

This finishes the proof. \(\square \)

The following lemma is the main conclusion of this appendix and it is essential in the proof of Proposition 2.5.

Lemma A.2

Let \(u\in L_s(\mathbb {R}^n)\cap \dot{H}^s_{\mathrm {loc}}(\mathbb {R}^n)\) be a solution to (1.1). Assume that u is stable in \(\Omega \subseteq \mathbb {R}^n\). Let \(\Phi \in C_c^\infty ({{\overline{\mathbb {R}_+^{n+1}}}})\) be of the form \(\Phi (x,t) =\varphi (x)\eta (t)\) for some \(\varphi \in C_c^\infty (\Omega )\) and \(\eta \equiv 1\) on [0, 1]. Then for every \(0<\alpha <2\) we have

$$\begin{aligned} \begin{aligned} (2-\alpha )\kappa _s\int _{\mathbb {R}^n}|x|^a e^{(1+2\alpha ) u}\varphi ^2dx&\le 2\int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha {\bar{u}}}|\nabla \Phi |^2dxdt\\&\quad -\frac{1}{2}\int _{\mathbb {R}_+^{n+1}}e^{2\alpha \bar{u}}\nabla \cdot [t^{1-2s}\nabla \Phi ^2]dxdt . \end{aligned} \end{aligned}$$

Proof

For \(k\in \mathbb {N}\) we set \(\overline{u}_k:=\min \{\overline{u},k\}\). Let \(u_k\) be the restriction of \(\overline{u}_k\) on \(\mathbb {R}^n\). It is straightforward to see that \(e^{2\alpha \overline{u}_k}\Phi ^2\) can be considered as a test function in (1.8). Therefore,

$$\begin{aligned} \begin{aligned}&\kappa _s\int _{\mathbb {R}^n}|x|^a e^u e^{2\alpha u_k}\varphi ^2dx\\&\quad =2\alpha \int _{\mathbb {R}_+^{n+1}}t^{1-2s} e^{2\alpha \overline{u}_k}\Phi ^2\nabla \overline{u}\cdot \nabla \overline{u}_kdxdt +\int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}\nabla \overline{u}\cdot \nabla \Phi ^2dxdt\\&\quad =2\alpha \int _{\mathbb {R}_+^{n+1}}t^{1-2s} e^{2\alpha \overline{u}_k}\Phi ^2\ | \nabla \overline{u}_k|^2dxdt +\int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}\nabla \overline{u}\cdot \nabla \Phi ^2dxdt. \end{aligned} \end{aligned}$$
(A.2)

Now, we assume that \(t^{1-2s}e^{2(\alpha +\varepsilon )\overline{u}}\in L^1_{\mathrm {loc}}(\Omega \times [0,\infty ))\) for some \(\varepsilon >0\). Then by [28, Lemma 3.5], up to a subsequence, we have

$$\begin{aligned} \kappa _s\int _{\mathbb {R}^n}|x|^a e^u e^{2\alpha u_k}\varphi ^2dx=(2\alpha +o(1))\int _{\mathbb {R}_+^{n+1}}t^{1-2s} e^{2\alpha \overline{u}_k}\Phi ^2\ | \nabla \overline{u}_k|^2dxdt+O(1). \end{aligned}$$

Taking \(e^{\alpha \overline{u}_k} \Phi \) as a test function in the stability inequality (A.1) yields

$$\begin{aligned} \begin{aligned}&\kappa _s \int _{\mathbb {R}^n}|x|^a e^ue^{2\alpha u_k}\varphi ^2dx\\&\quad \le \alpha ^2 \int _{\mathbb {R}_+^{n+1}}t^{1-2s}\Phi ^2e^{2\alpha \overline{u}_k}|\nabla \overline{u}_k|^2dxdt + \int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}|\nabla \Phi |^2dxdt\\&\qquad +\frac{1}{2} \int _{\mathbb {R}_+^{n+1}}t^{1-2s} \nabla e^{2\alpha \overline{u}_k}\nabla \Phi ^2dxdt\\&\quad = \alpha ^2 \int _{\mathbb {R}_+^{n+1}}t^{1-2s}\Phi ^2e^{2\alpha \overline{u}_k}|\nabla \overline{u}_k|^2dxdt + \int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}|\nabla \Phi |^2dxdt \\&\qquad -\frac{1}{2} \int _{\mathbb {R}_+^{n+1}}e^{2\alpha \overline{u}_k} \nabla \cdot [t^{1-2s}\nabla \Phi ^2]dxdt, \end{aligned} \end{aligned}$$
(A.3)

where the last equality follows from integration by parts. Notice that the boundary term vanishes as \(\eta (t)=1\) on [0, 1]. Combining the prior estimates, we obtain

$$\begin{aligned} \begin{aligned}&(2-\alpha -\varepsilon )\kappa _s\int _{\mathbb {R}^n}|x|^a e^ue^{2\alpha u_k}\varphi ^2dx \\&\quad \le (-2\alpha \varepsilon +o(1))\int _{\mathbb {R}_+^{n+1}}t^{1-2s} e^{2\alpha \overline{u}_k}\Phi ^2|\nabla \overline{u}_k|^2dxdt+O(1) \\&\quad \quad +2\int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}|\nabla \Phi |^2dx -\int _{\mathbb {R}_+^{n+1}}e^{2\alpha \overline{u}_k}\nabla \cdot [t^{1-2s}\nabla \Phi ^2]dxdt. \end{aligned} \end{aligned}$$
(A.4)

Concerning the last term in (A.4), one can notice that

$$\begin{aligned} \nabla \cdot [t^{1-2s}\nabla \Phi ^2]=t^{1-2s}\eta ^2\Delta _x\varphi ^2+\varphi ^2\partial _t(t^{1-2s}\partial _t\eta ^2). \end{aligned}$$

Again, as \(\eta =1\) on [0, 1], the second term in the right-hand side of the above expression is identically zero for \(0\le t\le 1\). Therefore, Lemma A.1 yields

$$\begin{aligned} \left| \int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}_k}|\nabla \Phi |^2dxdt\right| +\left| \int _{\mathbb {R}_+^{n+1}}e^{2\alpha \overline{u}_k}\nabla \cdot [t^{1-2s}\nabla \Phi ^2]dxdt\right| \le C. \end{aligned}$$

Thus,

$$\begin{aligned} (2-\alpha -\varepsilon )\int _{\mathbb {R}^n}e^ue^{2\alpha u_k}\varphi ^2dx\le C, \end{aligned}$$

provided \(\int _{\Omega }e^{2(\alpha +\varepsilon )u}dx<\infty .\) Now, choosing \(\alpha \in (0,\frac{1}{2})\) and \(0<\varepsilon <\frac{1}{2}-\alpha \) in the above estimate and then taking \(k\rightarrow \infty \) we conclude \(e^{(1+2\alpha )u}\in L^1_{\mathrm {loc}}(\Omega )\). By an iteration argument we conclude that \(e^{(1+2\alpha )u}\in L^1_{\mathrm {loc}}(\Omega )\) for every \(\alpha \in (0,2)\).

Now, send \(k\rightarrow \infty \) in (A.4) to get

$$\begin{aligned} \int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}}\Phi ^2|\nabla \overline{u}|^2dxdt<\infty . \end{aligned}$$

And, take limit in (A.2) and (A.3) as \(k\rightarrow \infty \). Then, the proof of lemma follows immediately since the second term on the right-hand side of (A.2), when \(k\rightarrow \infty \), can be re-written as

$$\begin{aligned} \int _{\mathbb {R}_+^{n+1}}t^{1-2s}e^{2\alpha \overline{u}}\nabla \overline{u}\cdot \nabla \Phi ^2dxdt&=\frac{1}{2\alpha } \int _{\mathbb {R}_+^{n+1}}t^{1-2s}\nabla e^{2\alpha \overline{u}}\cdot \nabla \Phi ^2dxdt\\&=-\frac{1}{2\alpha } \int _{\mathbb {R}_+^{n+1}} e^{2\alpha \overline{u}}\nabla \cdot [t^{1-2s}\nabla \Phi ^2]dxdt. \end{aligned}$$

Note that the boundary integral is zero as \(\eta =1\) on [0, 1]. This completes the proof. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fazly, M., Hu, Y. & Yang, W. On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand–Liouville equation. Calc. Var. 60, 11 (2021). https://doi.org/10.1007/s00526-020-01874-7

Download citation

Keywords

  • Hénon-Gelfand–Liouville equation
  • Fractional Laplacian
  • Stable and finite Morse index solutions
  • Monotonicity formula

Mathematics Subject Classification

  • 35B65
  • 35J60
  • 35B08
  • 35A15