Abstract
We discuss the asymptotic behavior of the least energy solution of a half-Laplacian operator with supercritical exponents on an interval. This extends the results obtained by Ren–Wei (Trans Am Math Soc 343(2):749–763, 1994) to the fractional Laplacian case.
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
Adimurthi, Grossi, M.: Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity. Proc. Am. Math. Soc. 132(4), 1013–1019 (2004)
Bass, R.F., Cranston, M.: Exit times for symmetric stable processes in \({\mathbb{R}}^N\). Ann. Probab. 11(3), 578–588 (1983)
Bogdan, K., Byczkowski, T., Nowak, A.: Gradient estimates for harmonic and \(q\)-harmonic functions of symmetric stable processes. Ill. J. Math. 46(2), 541–556 (2002)
Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Stat. 20(2), 293–335 (2000)
Ben Ayed, M., El Mehdi, M., Grossi, M.: Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four. Indiana Univ. Math. J. 55(5), 1723–1749 (2006)
Blumenthal, R.M., Getoor, R., Ray, R.: On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc. 99, 540–554 (1961)
Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications. In: Lecture Notes of the Unione Matematica Italiana. Springer (2016)
Bisci, G., Radulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2016)
Da Lio, F., Martinazzi, L., Rivière, T.: Blow-up analysis of a nonlocal Liouville-type equation. Anal. PDE 8(7), 1757–1805 (2015)
Demengel, F., Demengel, G.: Functional Spaces for the Theory of Elliptic Partial Differential Equations, vol. 110, pp. 179–228. Birkhauser, Basel (2012)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)
Fernández-Real, X., Ros-Oton, X.: Boundary regularity for the fractional heat equation. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. 110(1), 49–64 (2016)
Han, Z.C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. A.I.H.P. 8(2), 159–174 (1991)
Martinazzi, L.: Fractional Adams–Moser–Trudinger type inequalities. Nonlinear Anal. 127, 263–275 (2015)
Mosconi, S., Squassina, M.: Nonlocal problems at nearly critical growth. Nonlinear Anal. 136, 84–1015 (2016)
Parini, E., Ruf, B.: On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces. Rend. Linci Mat. Appl. 29, 315–319 (2018)
Paul, S., Santra, S.: On the ground state solution of a fractional Schrödinger equation. Preprint (2018)
Ren, X., Wei, J.: On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans. Am. Math. Soc. 343(2), 749–763 (1994)
Ren, X., Wei, J.: Single-point condensation and least-energy solutions. Proc. Am. Math. Soc. 124(1), 111–120 (1996)
Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2), 587–628 (2014)
Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101(3), 275–302 (2014)
Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)
Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
Takahashi, F.: Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem. Calc. Var. Partial Differ. Equ. 29(4), 509–520 (2007)
Acknowledgements
The author is very grateful to the unknown referee for the valuable comments and suggestion which helped in improving the presentation of the paper. Part of the paper was written when the author was visiting Universidade de Aveiro. He would like to the thank the Department of Mathematics for its support and warm hospitality.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Santra, S. Existence and shape of the least energy solution of a fractional Laplacian. Calc. Var. 58, 48 (2019). https://doi.org/10.1007/s00526-019-1494-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-019-1494-3