Abstract
In this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.
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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.
B. Barrios, E. Colorado, A. de Pablo, and U. Sánchez, On some critical problems for the fractional Laplacian operator. J. Differential Equations 252, No 11 (2012), 6133–6162.
B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, No 4 (2015), 875–900.
G. Bonanno, A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75, No 5 (2012), 2992–3007.
G. Bonanno, G. D’Aguì and D. O’Regan, A local minimum theorem and critical nonlinearities. An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 24, No 2 (2016), 67–86.
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88, No 3 (1983), 486–490.
E. Colorado, A. de Pablo, and U. Sánchez, Perturbations of a critical fractional equation. Pacific J. Math. 271, No 1 (2014), 65–85.
A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, No 1 (2004), 225–236.
E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, No 5 (2012), 521–573.
M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems. J. Funct. Anal. 263, No 8 (2012), 2205–2227.
J. García-Azorero and A. I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Amer. Math. Soc. 323, No 2 (1991), 877–895.
J. García Azorero and A. I. Peral, Some results about the existence of a second positive solution in a quasilinear critical problem. Indiana Univ. Math. J. 43, No 3 (1994), 941–957.
L. Li, Existence of solutions for some non-local fractional p-Laplacian elliptic problems with parameter. J. Nonlinear Convex Anal. 17, No 12 (2016), 2501–2509.
L. Li, R. P. Agarwal, and C. Li, Nonlinear fractional equations with supercritical growth. Nonlinear Anal. Model. Control 22, No 4 (2017), 521–530.
L. Li, J. Sun, and S. Tersian, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian. Fract. Calc. Appl. Anal. 20, No 5 (2017), 1146–1164; DOI: 10.1515/fca-2017-0061; https://www.degruyter.com/view/j/fca.2017.20.issue-5/issue-files/fca.2017.20.issue-5.xml.
G. Molica Bisci, V. D. Radulescu, and R. Servadei, Variational Methods for Nonlocal Fractional Problems. Cambridge Univ. Press (2016).
G. Molica Bisci and V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differential Equations 54, No 3 (2015), 2985–3008.
P. Pucci and J. Serrin, The structure of the critical set in the mountain pass theorem. Trans. Amer. Math. Soc. 299, No 1 (1987), 115–132.
R. Servadei, A critical fractional Laplace equation in the resonant case. Topol. Methods Nonlinear Anal. 43, No 1 (2014), 251–267.
R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, No 2 (2012), 887–898.
R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12, No 6 (2013), 2445–2464.
J. Sun, L. Li, and M. Cencelj, B. Gabrovšek, Infinitely many sign-changing solutions for Kirchhoff type problems in R3. Nonlinear Anal. 186 (2019), 33–54.
M. Willem, Minimax Theorems. Birkhäuser Boston, Inc., Boston, MA (1996).d
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Li, L., Tersian, S. Fractional problems with critical nonlinearities by a sublinear perturbation. Fract Calc Appl Anal 23, 484–503 (2020). https://doi.org/10.1515/fca-2020-0023
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DOI: https://doi.org/10.1515/fca-2020-0023