Existence and multiplicity of solutions for critical nonlocal equations with variable exponents

References

• Applebaum D. Lévy processes from probability to finance and quantum groups. Notices Amer Math Soc. 2004;51:1336–1347.
   Google Scholar
• Caffarelli L. Nonlocal diffusions, drifts and games. Nonlinear Partial Differ Equ, Abel Symposia. 2012;7:37–52.
   Google Scholar
• Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differ Equ. 2007;32:1245–1260.
   Web of Science ®Google Scholar
• Molica Bisci G, Rădulescu V, Servadei R. Variational methods for nonlocal fractional equations. Encyclopedia of Mathematics and its Applications, 162, Cambridge: Cambridge University Press; 2016.
   Google Scholar
• Dipierro S, Medina M, Valdinoci E. Fractional elliptic problems with critical growth in the whole of Rn, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie) 15, 2017, viii+152.
   Google Scholar
• Chen Y, Levine S, Rao R. Functionals with p(x) growth in image processing. SIAM J Appl Math. 2006;66:1383–1406.
   Web of Science ®Google Scholar
• Ruzicka M. Electrorheological fluids modeling and mathematical theory. Berlin: Springer-Verlag; 2002.
   Google Scholar
• Fan X. Sobolev embeddings for unbounded domain with variable exponent having values across N. Math Inequal Appl. 2010;13:123–134.
   Web of Science ®Google Scholar
• Kováčik O, Rákosnik J. On spaces Lp(x) and W1,p(x). Czech Math J. 1991;41:592–618.
   Web of Science ®Google Scholar
• Yao J, Wang X. Compact imbeddings between variable exponent spaces with unbounded underlying domain. Nonlinear Anal. 2009;70:3472–3482.
   Web of Science ®Google Scholar
• Zhikov VV. On some variational problems. Russ J Math Phys. 1997;5:105–116.
   Web of Science ®Google Scholar
• Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. In: Lecture Notes in Mathematics; Vol. 2017, Heidelberg: Springer-Verlag; 2011.
   Google Scholar
• Kaufmann U, Rossi JD, Vidal R. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electron J Qual Theory Differ Equ. 2017;76:1–10.
   Web of Science ®Google Scholar
• Bahrouni A. Comparison and sub-supersolution principles for the fractional p(x)-Laplacian. J Math Anal Appl. 2018;458:1363–1372.
   Web of Science ®Google Scholar
• Bahrouni A, Rădulescu V. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin Dyn Syst Ser S. 2018;11:379–389.
   Web of Science ®Google Scholar
• Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal. 1973;14:349–381.
   Google Scholar
• Ho K, Kim YH. A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p(⋅)-Laplacian. Nonlinear Anal. 2019;188:179–201.
   Web of Science ®Google Scholar
• Azroul E, Benkirane A, Shimi M. Existence and multiplicity of solutions for fractional p(x,⋅)-Kirchhoff-type problems in RN. Appl Anal. 2021;100:2029–2048.
   Web of Science ®Google Scholar
• Willem M. Minimax theorems. Boston (MA): Birkhäuser; 1996.
   Google Scholar
• Xiang M, Zhang B, Yang D. Multiplicity results for variable order fractional Laplacian with variable growth. Nonlinear Anal. 2019;178:190–204.
   Web of Science ®Google Scholar
• Xiang M, Yang D, Zhang B. Homoclinic solutions for Hamiltonian systems with variable-order fractional derivatives. Complex Var Elliptic Equ. 2020;65:1412–1432.
   Web of Science ®Google Scholar
• Wang L, Zhang B. Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. Appl Anal. 2021;100:2418–2435.
   Web of Science ®Google Scholar
• Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc. 1983;88:437–477.
   Web of Science ®Google Scholar
• Lions PL. The concentration compactness principle in the calculus of variations. The locally compact case, Part I. Ann Inst H Poincaré Anal Non Lineaire. 1984;1:109–145.
   Web of Science ®Google Scholar
• Lions PL. The concentration compactness principle in the calculus of variations. The limit case, Part II. Ann Inst H Poincaré Anal Non Lineaire. 1984;1:223–283.
   Web of Science ®Google Scholar
• Ben-Naouma AK, Troestler C, Willem M. Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 1996;26:823–833.
   Web of Science ®Google Scholar
• Bianchi G, Chabrowski J, Szulkin A. Symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 1995;25:41–59.
   Web of Science ®Google Scholar
• Chabrowski J. Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc Var Partial Differ Equ. 1995;3:493–512.
   Web of Science ®Google Scholar
• Alves CO, Barreiro JLP. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. J Math Anal Appl. 2013;403:143–154.
   Web of Science ®Google Scholar
• Bhakta M, Mukherjee D. Multiplicity results for (p,q) fractional elliptic equations involving critical nonlinearities. Adv Differ Equ. 2019;24:185–228.
   Web of Science ®Google Scholar
• Bonder JF, Silva A. Concentration-compactness principle for variable exponent spaces and applications. Electron J Differ Equ. 2010;141:1–18.
   Google Scholar
• Bonder JF, Saintier N, Silva A. The concentration-compactness principle for fractional order Sobolev spaces in unbounded-domains and applications to the generalized fractional Brezis-Nirenberg problem. Nonlinear Differ Equ Appl. 2018;25.
   Web of Science ®Google Scholar
• Chabrowski J, Fu YQ. Existence of solutions for p(x)-Laplacian problems on a bounded domain. J Math Anal Appl. 2005;306:604–618.
   Web of Science ®Google Scholar
• Fu YQ, Zhang X. Multiple solutions for a class of p(x)-Laplacian equations in Rn involving the critical exponent. Proc A. 2010;466:1667–1686.
   Google Scholar
• Fu YQ. Existence of solutions for p(x)-Laplacian problem on an unbounded domain. Topol Methods Nonlinear Anal. 2007;30:235–249.
   Web of Science ®Google Scholar
• Liang S, Zhang J. Multiple solutions for a noncooperative p(x)-Laplacian equations in RN involving the critical exponent. J Math Anal Appl. 2013;403:344–356.
   Web of Science ®Google Scholar
• Liang S, Zhang J. Infinitely many small solutions for the p(x)-Laplacian operator with nonlinear boundary conditions. Ann Mat Pura Appl. 2013;192:1–16.
   Web of Science ®Google Scholar
• Mawhin J, Molica Bisci G. A Brezis-Nirenberg type result for a nonlocal fractional operator. J Lond Math Soc. 2017;95:73–93.
   Web of Science ®Google Scholar
• Ho K, Kim YH. The concentration-compactness principles for Ws,p(⋅,⋅)(RN) and application. Adv Nonlinear Anal. 2021;10:816–848.
   Web of Science ®Google Scholar
• Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014;94:156–170.
   Web of Science ®Google Scholar
• D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math. 1992;108:247–262.
   Web of Science ®Google Scholar
• Pucci P, Xiang M, Zhang B. Existence and multiplicity of entire solutions for fractional p–Kirchhoff equations. Adv Nonlinear Anal. 2016;5:27–55.
   Web of Science ®Google Scholar
• Autuori G, Fiscella A, Pucci P. Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity. Nonlinear Anal. 2015;125:699–714.
   Web of Science ®Google Scholar
• Caponi M, Pucci P. Existence theorems for entire solutions of stationary Kirchhoff fractional p–Laplacian equations. Ann Mat Pura Appl. 2016;195:2099–2129.
   Web of Science ®Google Scholar
• Mingqi X, Molica Bisci G, Tian G, et al. Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p–Laplacian. Nonlinearity. 2016;29:357–374.
   Web of Science ®Google Scholar
• Fiscella A, Pucci P. Degenerate Kirchhoff (p,q)-fractional systems with critical nonlinearities. Fract Calc Appl Anal. 2020;23:723–752.
   Web of Science ®Google Scholar
• Palatucci G, Pisante A. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc Var Partial Differ Equ. 2014;50:799–829.
   Web of Science ®Google Scholar
• Pucci P, Xiang M, Zhang B. Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p–Laplacian in RN. Calc Var Partial Differ Equ. 2015;54:2785–2806.
   Web of Science ®Google Scholar
• Pucci P, Saldi S. Critical stationary Kirchhoff equations in RN involving nonlocal operators. Rev Mat Iberoam. 2016;32:1–22.
   Web of Science ®Google Scholar
• Xiang M, Zhang B, Ferrara M. Existence of solutions for Kirchhoff type problem involving the non-local fractional p–Laplacian. J Math Anal Appl. 2015;424:1021–1041.
   Web of Science ®Google Scholar
• Fan XL, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J Math Anal Appl. 2001;263:424–446.
   Web of Science ®Google Scholar
• Fan X, Shen J, Zhao D. Sobolev embedding theorems for spaces Wk,p(x)(Ω). J Math Anal Appl. 2001;262:749–760.
   Web of Science ®Google Scholar
• Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573.
   Web of Science ®Google Scholar
• Fonseca I, Leoni G. Modern methods in the calculus of variations: $L^p$ spaces. New York: Springer; 2007.
   Google Scholar
• Ho K, Kim YH, Sim I. Existence results for Schrödinger p(⋅)-Laplace equations involving critical growth in RN. Nonlinear Anal. 2019;182:20–44.
   Web of Science ®Google Scholar
• Aubin JP, Ekeland I. Applied nonlinear analysis. New York (NY): Wiley; 1984.
   Google Scholar
• Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. In: CBME Regional Conference Series in Mathematics; Vol. 65, Providence (RI): American Mathematical Society; 1986.
   Google Scholar
• Wei W, Wu X. A multiplicity result for quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 1992;18:559–567.
   Web of Science ®Google Scholar