Multiple solutions to Bahri-Coron problem involving fractional $p$-Laplacian in some domain with nontrivial topology
DOI:
https://doi.org/10.12775/TMNA.2022.033Keywords
Non-local problem, fractional p-Laplacian, critical exponent, global compactnessAbstract
In this article, we establish the existence of positive and multiple sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity \begin{equation} \label{Ppomegas-a}\tag{P$_{p,\Omega}^{s}$} \begin{cases} (-\Delta)_{p}^s u =|u|^{p_s^*-2} u& \text{in }\Omega, \\ u =0 & \text{on }\Omega^{c}, \end{cases} \end{equation} in a bounded domain $\Omega\subset \mathbb{R}^{N}$ for $s\in (0,1)$, $p\in (1,\infty)$, and the fractional critical Sobolev exponent $p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions. We study Struwe's type global compactness results for the Palais-Smale sequence in the presence of symmetries.References
W. Abdelhedi, H. Chtioui and H. Hajaiej, The Bahri–Coron theorem for fractional Yamabe-type problems, Adv. Nonlinear Stud. 18 (2018), no. 2, 393–407.
D. Applebaum, Lévy processes – from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), no. 11, 1336–1347.
A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent:the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294.
T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc. (3) 91 (2005), no1̇, 129–152.
L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355.
L. Brasco, M. Squassina and Y. Yang, Global compactness results for nonlocal problems, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 391–424.
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, Cham, Unione Matematica Italiana, Bologna, 2016.
L.A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), no. 3, 1903–1930.
X. Chang, Z. Nie and Z.-Q. Wang, Sign-changing solutions of fractional p-Laplacian problems, Adv. Nonlinear Stud. 19 (2019), no. 1, 29–53.
X. Chang and Z.Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Eqnuations 256 (2014), no. 8, 2965–2992.
W. Chen, S. Mosconi and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018), no. 11, 3065–3114.
M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear Equations: Methods, Models and Applications (Bergamo, 2001), Progr. Nonlinear Differential Equations Appl., vol. 54, Birkhäuser, Basel, 2003, pp. 117–126.
M. Clapp and J. Faya, Multiple solutions to the Bahri–Coron problem in some domains with nontrivial topology, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4339–4344.
M. Clapp and J. Faya, Multiple solutions to anisotropic critical and supercritical problems in symmetric domains, Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl., vol. 86, Birkhäuser–Springer, Cham, 2015, pp. 99–120.
M. Clapp and F. Pacella, Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size, Math. Z. 259 (2008), no. 3, 575–589.
M. Clapp and S. Tiwari, Multiple solutions to a pure supercritical problem for the pLaplacian, Calc. Var. Partial Differential Equations 55 (2016), no. 1, art. 7, 23 pp.
M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var. Partial Differential Equations 21 (2004), no. 1, 1–14.
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Anal. 137 (2016), 357–380.
S. Dipierro, O. Savin and E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Art. 86, 25 pp. [20] G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.) 5 (2014), no. 2, 373–386.
C. Mercuri and F. Pacella, On the pure critical exponent problem for the p-Laplacian, Calc. Var. Partial Differential Equations 49 (2014), no. 3–4, 1075–1090.
S. Mosconi, N. Shioji and M. Squassina, Nonlocal problems at critical growth in contractible domains, Asymptot. Anal. 95 (2015), no. 1–2, 79–100.
S. Mosconi and M. Squassina, Nonlocal problems at nearly critical growth, Nonlinear Anal. 136 (2016), 84–101.
S Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Mathematical Analysis Seminar 2016, Bruno Pini Math. Anal. Semin., vol. 7, Univ. Bologna, Alma Mater Stud., Bologna, 2016, pp. 147–164.
R.S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30.
S. Patrizi and E. Valdinoci, Relaxation times for atom dislocations in crystals, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 71, 44 pp.
P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conference Board of the Mathematical Sciences, CBMS Reg. Conf. Ser. Math., vol. 5, Washington, DC; Amer.Math. Soc., Providence, RI, 1986.
X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 587–628.
S. Secchi, N. Shioji and M. Squassina, Coron problem for fractional equations, Differential Integral Equations 28 (2015), no. 1–2, 103–118.
J.L. Vázquez, The Dirichlet problem for the fractional p-Laplacian evolution equation, J. Differential Equations 260 (2016), no. 7, 6038–6056.
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