On the existence of multiple solutions for fractional Brezis-Nirenberg-type equations. (English) Zbl 1523.35288
Summary: This paper studies the nonlocal fractional analog of the famous paper of H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)]. Namely, we focus on the following model:
\[
(\mathcal{P})
\begin{cases}
(-\Delta )^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s -2}u \quad \text{in}\quad \Omega, \\
u=0\quad \text{in}\quad \mathbb{R}^N \setminus \Omega,
\end{cases}
\]
where \((-\Delta )^s\) is the fractional Laplace operator, \(s \in (0,1)\), with \(N > 2s\), \(2<p<2^*_s\), \(\beta >0\), \(\lambda, \alpha \in \mathbb{R}\), and establish the existence of nontrivial solutions and sign-changing solutions for the problem \((\mathcal{P})\).
{© 2022 Wiley-VCH GmbH.}
{© 2022 Wiley-VCH GmbH.}
MSC:
| 35R11 | Fractional partial differential equations |
| 35B33 | Critical exponents in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
Keywords:
critical exponent; elliptic equation; fractional Laplacian; multiple solutions; sign-changing solutionsCitations:
Zbl 0541.35029References:
| [1] | A.Ambrosetti and P. H.Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973), 349-381. https://doi.org/10.1016/0022‐1236(73)90051‐7. · Zbl 0273.49063 · doi:10.1016/0022‐1236(73)90051‐7 |
| [2] | C.Bandle and R.Benguria, The Brézis‐Nirenberg problem on \(\mathbb{S}^3\), J. Differential Equations178 (2002), no. 1, 264-279. https://doi.org/10.1006/jdeq.2001.4006. · Zbl 0995.35027 · doi:10.1006/jdeq.2001.4006 |
| [3] | V.Benci and G.Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations2 (1994), no. 1, 29-48. https://doi.org/10.1007/BF01234314. · Zbl 0822.35046 · doi:10.1007/BF01234314 |
| [4] | H.Brézis and L.Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983), no. 4, 437-477. https://doi.org/10.1002/cpa.3160360405. · Zbl 0541.35029 · doi:10.1002/cpa.3160360405 |
| [5] | A.Capozzi, D.Fortunato, and G.Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire2 (1985), no. 6, 463-470. · Zbl 0612.35053 |
| [6] | P. C.Carrião, et al., A Brezis‐Nirenberg problem on hyperbolic spaces, Electron. J. Differential Equations (2019), Paper No. 67, 15. · Zbl 1418.35167 |
| [7] | G.Cerami, D.Fortunato, and M.Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire1 (1984), no. 5, 341-350. http://www.numdam.org/item?id=AIHPC_1984__1_5_341_0. · Zbl 0568.35039 |
| [8] | G.Cerami, S.Solimini, and M.Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal.69 (1986), no. 3, 289-306. https://doi.org/10.1016/0022‐1236(86)90094‐7. · Zbl 0614.35035 · doi:10.1016/0022‐1236(86)90094‐7 |
| [9] | K.‐S.Chen, M.Montenegro, and X.Yan, The Brezis‐Nirenberg problem for fractional elliptic operators, Math. Nachr.290 (2017), no. 10, 1491-1511. https://doi.org/10.1002/mana.201600072. · Zbl 1375.35176 · doi:10.1002/mana.201600072 |
| [10] | E.Colorado and A.Ortega, The Brezis‐Nirenberg problem for the fractional Laplacian with mixed Dirichlet‐Neumann boundary conditions, J. Math. Anal. Appl.473 (2019), no. 2, 1002-1025. https://doi.org/10.1016/j.jmaa.2019.01.006. · Zbl 1516.35455 · doi:10.1016/j.jmaa.2019.01.006 |
| [11] | G.Cora and A.Iacopetti, On the structure of the nodal set and asymptotics of least energy sign‐changing radial solutions of the fractional Brezis‐Nirenberg problem, Nonlinear Anal.176 (2018), 226-271. https://doi.org/10.1016/j.na.2018.07.001. · Zbl 1401.35094 · doi:10.1016/j.na.2018.07.001 |
| [12] | A.Cotsiolis and N. K.Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl.295 (2004), no. 1, 225-236. https://doi.org/10.1016/j.jmaa.2004.03.034. · Zbl 1084.26009 · doi:10.1016/j.jmaa.2004.03.034 |
| [13] | G.Devillanova, Multiscale weak compactness in metric spaces, J. Elliptic Parabol. Equ.2 (2016), no. 1‐2, 131-144. https://doi.org/10.1007/BF03377397. · Zbl 1392.46033 · doi:10.1007/BF03377397 |
| [14] | G.Devillanova and G. C.Marano, A free fractional viscous oscillator as a forced standard damped vibration, Fract. Calc. Appl. Anal.19 (2016), no. 2, 319-356. https://doi.org/10.1515/fca‐2016‐0018. · Zbl 1499.34041 · doi:10.1515/fca‐2016‐0018 |
| [15] | G.Devillanova and S.Solimini, Some remarks on profile decomposition theorems, Adv. Nonlinear Stud.16 (2016), no. 4, 795-805. https://doi.org/10.1515/ans‐2015‐5049. · Zbl 1361.46025 · doi:10.1515/ans‐2015‐5049 |
| [16] | G.Devillanova, S.Solimini, and C.Tintarev, On weak convergence in metric spaces, Nonlinear analysis and optimization, Contemp. Math., vol. 659, Amer. Math. Soc., Providence, RI, 2016. 43-63. https://doi.org/10.1090/conm/659/13160. · Zbl 1356.46018 · doi:10.1090/conm/659/13160 |
| [17] | E. DiNezza, G.Palatucci, and E.Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.136 (2012), no. 5, 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004. · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 |
| [18] | L. F. O.Faria et al., The Brezis‐Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal.5 (2016), no. 1, 85-103. https://doi.org/10.1515/anona‐2015‐0114. · Zbl 1343.47054 · doi:10.1515/anona‐2015‐0114 |
| [19] | A.Fiscella and E.Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal.94 (2014), 156-170. https://doi.org/10.1016/j.na.2013.08.011. · Zbl 1283.35156 · doi:10.1016/j.na.2013.08.011 |
| [20] | P.Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var.3 (1998), 213-233. https://doi.org/10.1051/cocv:1998107. · Zbl 0907.46027 · doi:10.1051/cocv:1998107 |
| [21] | G.Molica Bisci, V. D.Radulescu, and R.Servadei, Variational methods for nonlocal fractional problems, Encyclopedia of Mathematics and its Applications, vol. 162, pp. xvi+383, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316282397, with a foreword by Jean Mawhin. · Zbl 1356.49003 · doi:10.1017/CBO9781316282397 |
| [22] | S.Mosconi, et al., The Brezis‐Nirenberg problem for the fractional p‐Laplacian, Calc. Var. Partial Differential Equations55 (2016), no. 4, Art. 105, 25. https://doi.org/10.1007/s00526‐016‐1035‐2. · Zbl 1361.35198 · doi:10.1007/s00526‐016‐1035‐2 |
| [23] | G.Palatucci and A.Pisante, Improved Sobolev embeddings, profile decomposition, and concentration‐compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations50 (2014), no. 3‐4, 799-829. https://doi.org/10.1007/s00526‐013‐0656‐y. · Zbl 1296.35064 · doi:10.1007/s00526‐013‐0656‐y |
| [24] | G.Palatucci and A.Pisante, A global compactness type result for Palais‐Smale sequences in fractional Sobolev spaces, Nonlinear Anal.117 (2015), 1-7. https://doi.org/10.1016/j.na.2014.12.027. · Zbl 1312.35092 · doi:10.1016/j.na.2014.12.027 |
| [25] | X.Ros‐Oton and J.Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations50 (2014), no. 3‐4, 723-750. https://doi.org/10.1007/s00526‐013‐0653‐1. · Zbl 1301.35204 · doi:10.1007/s00526‐013‐0653‐1 |
| [26] | R.Servadei and E.Valdinoci, Mountain pass solutions for non‐local elliptic operators, J. Math. Anal. Appl.389 (2012), no. 2, 887-898. https://doi.org/10.1016/j.jmaa.2011.12.032. · Zbl 1234.35291 · doi:10.1016/j.jmaa.2011.12.032 |
| [27] | R.Servadei and E.Valdinoci, A Brezis‐Nirenberg result for non‐local critical equations in low dimension, Commun. Pure Appl. Anal.12 (2013), no. 6, 2445-2464. https://doi.org/10.3934/cpaa.2013.12.2445. · Zbl 1302.35413 · doi:10.3934/cpaa.2013.12.2445 |
| [28] | R.Servadei and E.Valdinoci, Variational methods for non‐local operators of elliptic type, Discrete Contin. Dyn. Syst.33 (2013), no. 5, 2105-2137. https://doi.org/10.3934/dcds.2013.33.2105. · Zbl 1303.35121 · doi:10.3934/dcds.2013.33.2105 |
| [29] | R.Servadei and E.Valdinoci, The Brezis‐Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc.367 (2015), no. 1, 67-102. https://doi.org/10.1090/S0002‐9947‐2014‐05884‐4. · Zbl 1323.35202 · doi:10.1090/S0002‐9947‐2014‐05884‐4 |
| [30] | R.Servadei and E.Valdinoci, Fractional Laplacian equations with critical Sobolev exponent, Rev. Mat. Complut.28 (2015), no. 3, 655-676. https://doi.org/10.1007/s13163‐015‐0170‐1. · Zbl 1338.35481 · doi:10.1007/s13163‐015‐0170‐1 |
| [31] | S.Solimini and C.Tintarev, Concentration analysis in Banach spaces, Commun. Contemp. Math.18 (2016), no. 3, 1550038, 33. https://doi.org/10.1142/S0219199715500388. · Zbl 1351.46015 · doi:10.1142/S0219199715500388 |
| [32] | J.Tan, The Brezis‐Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations42 (2011), no. 1‐2, 21-41. https://doi.org/10.1007/s00526‐010‐0378‐3. · Zbl 1248.35078 · doi:10.1007/s00526‐010‐0378‐3 |
| [33] | C.Tintarev, Concentration compactness functional‐analytic theory of concentration phenomena, series, De Gruyter Ser. Nonlinear Anal. Appl.33 (2020). https://doi.org/10.1515/9783110532432. · Zbl 1452.35003 · doi:10.1515/9783110532432 |
| [34] | K.Tintarev and K.‐H.Fieseler, Concentration compactness, pp. xii+264, Imperial College Press, London, 2007. https://doi.org/10.1142/p456, functional‐analytic grounds and applications. · Zbl 1118.49001 · doi:10.1142/p456 |
| [35] | E.Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA (2009), no. 49, 33-44. · Zbl 1242.60047 |
| [36] | M.Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, pp. x+162, Birkhäuser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978‐1‐4612‐4146‐1. · Zbl 0856.49001 · doi:10.1007/978‐1‐4612‐4146‐1 |
| [37] | B.Xuan, S.Su, and Y.Yan, Existence results for Brezis‐Nirenberg problems with Hardy potential and singular coefficients, Nonlinear Anal.67 (2007), no. 7, 2091-2106. https://doi.org/10.1016/j.na.2006.09.018. · Zbl 1387.35261 · doi:10.1016/j.na.2006.09.018 |
| [38] | X.Yue and W.Zou, Remarks on a Brezis‐Nirenberg’s result, J. Math. Anal. Appl.425 (2015), no. 2, 900-910. https://doi.org/10.1016/j.jmaa.2015.01.011. · Zbl 1312.35104 · doi:10.1016/j.jmaa.2015.01.011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
