Infinitely many solutions for differential inclusion problems in \({\mathbb{R}^N}\) <

详细信息    查看全文

• 作者：Bin Ge ; Li-Li Liu
• 关键词：\({p(x)}\) ; Laplacian ; Differential inclusion problem ; Locally Lipschitz function ; Genus
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：67
• 期：1
• 全文大小：559 KB
• 参考文献：1.Chang K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)CrossRef MATH MathSciNet
  2.Ricceri B.: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401–410 (2000)CrossRef MATH MathSciNet
  3.Marano S., Motreanu D.: Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the \({p}\) -Laplacian. J. Differ. Equ. 182, 108–120 (2002)CrossRef MATH MathSciNet
  4.Kristály A.: Infinitely many solutions for a differential inclusion problem in \({\mathbb{R}^N}\) . J. Differ. Equ. 220, 511–530 (2006)CrossRef MATH MathSciNet
  5.Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1993)MATH
  6.Harjulehto P., Hästö P., Latvala V.: Minimizers of the variable exponent, non-uniformly convex Dirichlet energy.. J. Math. Pures Appl. 89, 174–197 (2008)CrossRef MATH MathSciNet
  7.Chen Y.M., Levine S., Murali R.: Variable exponent linear growth functionals in image restoration. SIAM J. Appl. Math. (4) 66, 1383–1406 (2006)CrossRef MATH MathSciNet
  8.Antontsev S.N., Rodrigues J.F.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, 19–36 (2006)CrossRef MATH MathSciNet
  9.Antontsev S.N., Shmarev S.I.: A model porous medium equation with variable exponent of nonlinearity: Existence uniqueness and localization properties of solutions. Nonlinear Anal. Theory Methods Appl. 60, 515–545 (2005)CrossRef MATH MathSciNet
  10.Ge B.: On the superlinear problems involving the \({p(x)}\) -Laplacian and a non-local term without AR-condition. Nonlinear Anal. Theory Methods Appl. 102, 133–143 (2014)CrossRef MATH MathSciNet
  11.Ge B., Zhou Q.M.: Multiple solutions to a class of inclusion problem with the \({p(x)}\) -Laplacian. Appl. Anal. 91, 895–909 (2012)CrossRef MATH MathSciNet
  12.Ge B., Zhou Q.M.: Infinitely many positive solutions for a differential inclusion problem involving the \({p(x)}\) -Laplacian. Math. Nachr. 285, 1303–1315 (2012)MATH MathSciNet
  13.Ge B., Xue X.P., Zhou Q.M.: Existence of at least five solutions for a differential inclusion problem involving the \({p(x)}\) -Laplacian. Nonlinear Anal. Real World Appl. 12, 2304–2318 (2011)CrossRef MATH MathSciNet
  14.Zhou Q.M.: On the superlinear problems involving \({p(x)}\) -Laplacian-like operators without AR-condition. Nonlinear Anal. Real World Appl. 21, 161–169 (2015)CrossRef MATH MathSciNet
  15.Perera K., Squassina M.: Asymptotic behavior of the eigenvalues of the \({p(x)}\) -Laplacian. Manuscr. Math. 144, 535–544 (2014)CrossRef MATH MathSciNet
  16.Liang S.H., Zhang J.H.: Infinitely many small solutions for the \({p(x)}\) -Laplacian operator with nonlinear boundary conditions. Ann. Mat. Pura Appl. 192, 1–16 (2013)CrossRef MATH MathSciNet
  17.Benouhiba N.: On the eigenvalues of weighted \({p(x)}\) -Laplacian on \({\mathbb{R}^N}\) . Nonlinear Anal. 74, 235–243 (2011)CrossRef MATH MathSciNet
  18.Dai G.W.: Infinitely many solutions for a differential inclusion problem in \({\mathbb{R}^N}\) involving the \({p(x)}\) -Laplacian. Nonlinear Anal. Theory Methods Appl. 71, 1116–1123 (2009)CrossRef MATH MathSciNet
  19.Dai G.W.: Three symmetric solutions for a differential inclusion system involving the \({p(x)}\) -Laplacian in \({\mathbb{R}^N}\) . Nonlinear Anal. Theory Methods Appl. 71, 1763–1771 (2009)CrossRef MATH MathSciNet
  20.Ge B., Xue X.P., Zhou Q.M.: The existence of radial solutions for differential inclusion problems in \({\mathbb{R}^N}\) involving the \({p(x)}\) -Laplacian. Nonlinear Anal. Theory Methods Appl. 73, 622–633 (2010)CrossRef MATH MathSciNet
  21.Ge B., Zhou Q.M., Xue X.P.: Infinitely many solutions for a differential inclusion problem in \({\mathbb{R}^N}\) involving \({p(x)}\) -Laplacian and oscillatory terms. Z. Angew. Math. Phys. 64, 691–711 (2012)CrossRef MATH MathSciNet
  22.Ge B., Zhou Q.M., Xue X.P.: Multiplicity of solutions for differential inclusion problems in \({\mathbb{R}^N}\) involving the \({p(x)}\) -Laplacian. Monatshefte Math. 168, 363–380 (2012)CrossRef MATH MathSciNet
  23.Allaoui M., El Amrouss A., Ourraoui A.: Existence results for a class of \({p(x)}\) -Laplacian problems in \({\mathbb{R}^N}\) . Comput. Math. Appl. 69, 582–591 (2015)CrossRef MathSciNet
  24.Zhang Q.Y., Wang Q.: Multiple solutions for a class of sublinear Schröinger equations. J. Math. Anal. Appl. 389, 511–518 (2012)CrossRef MATH MathSciNet
  25.Fan X.L., Han X.Y.: Existence and multiplicity of solutions for \({p(x)}\) -Laplacian equations in \({\mathbb{R}^N}\) . Nonlinear Anal. Theory Methods Appl. 59, 173–188 (2004)MATH MathSciNet
  26.Edmunds D., Rákosník J.: Sobolev embedding with variable exponent. Studia Math. 143, 267–293 (2000)MATH MathSciNet
  27.Fan X.L, Jishen S., Zhao D.: Sobolev embedding theorems for spaces \({W^{k,p(x)}(\Omega )}\) . J. Math. Anal. Appl. 262, 749–760 (2001)CrossRef MATH MathSciNet
  28.Kovacik O., Rakosuik J.: On spaces \({L^{p(x)}(\Omega)}\) and \({W^{k,p(x)}(\Omega)}\) . Czechoslov. Math. J. 41, 592–618 (1991)
  29.Fan X.L, Zhao D.: On the spaces \({L^{p(x)}(\Omega)}\) and \({W^{m,p(x)}(\Omega)}\) . J. Math. Anal. Appl. 263, 424–446 (2001)CrossRef MATH MathSciNet
  30.Fan X.L, Zhang Q.H.: Existence of solutions for \({p(x)}\) -laplacian Dirichlet problem. Nonlinear Anal. Theory Methods Appl. 52, 1843–1853 (2003)CrossRef MATH MathSciNet
  31.Chang K.C.: Critical Point Theory and Applications. Shanghai Scientific and technology Press, Shanghai (1996)
  32.Edmunds D., Rakosnik J.: Sobolev embeddings with variable exponent. Stud. Math. 143, 267–293 (2000)MATH MathSciNet
  33.Kandiakis D., Kourogenis N., papageorgiou N.S.: Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications. J. Glob. Optim. 34, 219–244 (2006)CrossRef MATH MathSciNet
  34.Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence, RI (1986)
  35.Papageorgiou N.S., Kyritsi-Yisllourou S. Th.: Handbook of Applied Analysis. Advance in Mechanics and Mathematics. Springer, New York (2009)
  36.Gasinski L., Papageorgiou N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman Hall and CRC Press, Boca Raton (2005)MATH
  
• 作者单位：Bin Ge (1)
  Li-Li Liu (1)
  
  1. Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, People’s Republic of China
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

In this paper we consider the differential inclusion problem in \({\mathbb{R}^N}\) involving the \({p(x)}\)-Laplacian of the type $$-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u)\,\,\,{\rm {in}} \, \mathbb{R}^N.$$Some new criteria to guarantee that the existence of infinitely many solutions for the considered problem is established by using the genus properties in nonsmooth critical point theory, which extend and complement previously known results in the literature. Keywords \({p(x)}\)-Laplacian Differential inclusion problem Locally Lipschitz function Genus Mathematics Subject Classification 35J20 35J70 35R70 This work is supported by the National Natural Science Foundation of China (No. 11201095), the Youth Scholar Backbone Supporting Plan Project of Harbin Engineering University (No. 307201411008), the Fundamental Research Funds for the Central Universities (No. 2016), the Postdoctoral Research Startup Foundation of Heilongjiang (No. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).