Abstract
In the present paper, by using variational methods, we study the existence of multiple nontrivial weak solutions for parametric nonlocal equations, driven by the fractional Laplace operator \({(-\Delta)^{s}}\) , in which the nonlinear term has a sublinear growth at infinity. More precisely, a critical point result for differentiable functionals is exploited, in order to prove the existence of an open interval of positive eigenvalues for which the treated problem admits at least two nontrivial weak solutions in a suitable fractional Sobolev space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \({\mathbb{R}^N}\). J. Differ. Equ. 255, 2340–2362 (2013)
Barrios, B., Colorado, E., De Pablo, A., Sanchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)
Brezis, H.: Analyse Fonctionnelle Théorie et Applications. Masson, Paris (1983)
Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009)
Caffarelli, L., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)
Capella, A.: Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains. Commun. Pure Appl. Anal. 10(6), 1645–1662 (2011)
Dipierro, S., Pinamonti, A.: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ. 255, 85–119 (2013)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Fiscella, A., Servadei, R.: Valdinoci E A resonance problem for non-local elliptic operators. Z. Anal. Anwendungen (to appear)
Kristály, A., Radulescu, V., Varga, Cs.: Variational principles in mathematical physics, geometry, and economics. Qualitative analysis of nonlinear equations and unilateral problems. With a foreword by Jean Mawhin. In: Encyclopedia of Mathematics and its Applications, vol. 136. Cambridge University Press, Cambridge (2010)
Molica Bisci, G.: Fractional equations with bounded primitive. Appl. Math. Lett. 27, 53–58 (2014)
Molica Bisci, G.: Sequences of weak solutions for fractional equations. Math. Res. Lett. 21, 1–13 (2014)
Molica Bisci, G., Pansera, B.A.: Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14, 591–601 (2014)
Molica Bisci, G., Servadei R.: A bifurcation result for nonlocal fractional equations. Anal. Appl. (to appear). doi:10.1142/S0219530514500067
Ricceri, B.: On a three critical points theorem. Archiv der Mathematik (Basel) 75, 220–226 (2000)
Ricceri, B.: Existence of three solutions for a class of elliptic eigenvalue problems. Math. Comput. Model. 32, 1485–1494 (2000)
Servadei, R.: The Yamabe equation in a non-local setting. Adv. Nonlinear Anal. 2, 235–270 (2013)
Servadei, R.: Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity. Contemp. Math. 595, 317–340 (2013)
Servadei, R., Valdinoci, E.: Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29(3), 1091–1126 (2013)
Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
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.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Amer. Math. Soc. (to appear)
Tan, J.: The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var. Partial Differ. Equ. 36(1–2), 21–41 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bisci, G.M., Rădulescu, V.D. Multiplicity results for elliptic fractional equations with subcritical term. Nonlinear Differ. Equ. Appl. 22, 721–739 (2015). https://doi.org/10.1007/s00030-014-0302-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-014-0302-1