Existence of solutions for nonlocal elliptic systems involving p(x)-Laplace operator

This article concerns the existence and multiplicity of solutions for a p(x)-Kirchhoff-type systems with a nonstandard growth condition. By a direct variational approach exponent Sobolev spaces, under growth conditions on the reaction terms, we establish the existence and multiplicity of solutions.

[1]  F. Cammaroto,et al.  Multiple solutions for a Neumann problem involving the p(x)-Laplacian , 2009 .

[2]  G. Bonanno,et al.  Discontinuous elliptic problems involving the p(x)‐Laplacian , 2011 .

[3]  M. N. Nkashama,et al.  STEKLOV SPECTRUM AND NONRESONANCE FOR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS , 2010 .

[4]  S. Heidarkhani,et al.  Multiple solutions for Kirchhoff‐type problems with variable exponent and nonhomogeneous Neumann conditions , 2018 .

[5]  Dušan D. Repovš,et al.  Existence and multiplicity of solutions for fractional Schrödinger–Kirchhoff equations with Trudinger–Moser nonlinearity , 2019, Nonlinear Analysis.

[6]  To Fu Ma,et al.  Positive solutions for a quasilinear elliptic equation of Kirchhoff type , 2005 .

[7]  Vicenţiu D. Rădulescu,et al.  Nonlinear Analysis - Theory and Methods , 2019, Springer Monographs in Mathematics.

[8]  Xianling Fan,et al.  Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , 2004 .

[9]  Stefano Panizzi,et al.  On the Well-Posedness of the Kirchhoff String , 1996 .

[10]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[11]  Djairo G. de Figueiredo.  Semilinear Elliptic Systems: A survey of superlinear problems , 1996 .

[12]  Jaime E. Muñoz Rivera,et al.  Positive solutions for a nonlinear nonlocal elliptic transmission problem , 2003, Appl. Math. Lett..

[13]  Shao-Gao Deng,et al.  Eigenvalues of the p(x)-Laplacian Steklov problem☆ , 2008 .

[14]  Guowei Dai,et al.  Infinitely many positive solutions for a p(x)-Kirchhoff-type equation , 2009 .

[15]  Xiaoguang Qi,et al.  Some results on difference polynomials sharing values , 2012, Advances in Difference Equations.

[16]  Dušan D. Repovš.  Stationary waves of Schr\"odinger-type equations with variable exponent , 2015 .

[17]  Qihu Zhang,et al.  Existence of solutions for p(x) -Laplacian dirichlet problem , 2003 .

[18]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[19]  Chao Ji.  Remarks on the existence of three solutions for the p ( x ) -Laplacian equations , 2011 .

[20]  Xianling Fan,et al.  On the Spaces Lp(x)(Ω) and Wm, p(x)(Ω) , 2001 .

[21]  J. Rodrigues,et al.  On stationary thermo-rheological viscous flows , 2006 .

[22]  Peter Hästö,et al.  Minimizers of the variable exponent, non-uniformly convex Dirichlet energy , 2008 .

[23]  A. El Hamidi,et al.  Existence results to elliptic systems with nonstandard growth conditions , 2004 .

[24]  G. Caristi,et al.  Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions , 2016 .

[25]  V. Zhikov,et al.  AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY , 1987 .

[26]  Xianling Fan,et al.  Sobolev Embedding Theorems for Spaces Wk, p(x)(Ω) , 2001 .

[27]  Xianling Fan,et al.  On the sub-supersolution method for p(x)-Laplacian equations , 2007 .

[28]  Dušan D. Repovš,et al.  Existence and multiplicity results for a newp(x)-Kirchhoff problem , 2019, Nonlinear Analysis.

[29]  Biagio Ricceri,et al.  On a three critical points theorem , 2000 .

[30]  G. Bonanno,et al.  Existence results of infinitely many solutions for p(x)-Laplacian elliptic Dirichlet problems , 2012 .

[31]  Dusan Repovs,et al.  On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity , 2017, Comput. Math. Appl..

[32]  Giuseppina D'Aguì,et al.  Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions , 2012 .

[33]  Guowei Dai,et al.  Existence of solutions for a p(x)-Kirchhoff-type equation , 2009 .

[34]  Dušan D. Repovš,et al.  Nonlocal Kirchhoff Superlinear Equations with Indefinite Nonlinearity and Lack of Compactness , 2016, 1610.02184.

[35]  S. Samko.  On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators , 2005 .

[36]  Xianling Fan,et al.  Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations , 2007 .

[37]  Giuseppe Mingione,et al.  Regularity Results for Stationary Electro-Rheological Fluids , 2002 .

[38]  Dušan D. Repovš,et al.  Fractional magnetic Schrödinger‐Kirchhoff problems with convolution and critical nonlinearities , 2020, Mathematical Methods in the Applied Sciences.

[39]  Qihu Zhang,et al.  Eigenvalues of p(x)-Laplacian Dirichlet problem , 2005 .

[40]  Francisco Júlio S. A. Corrêa,et al.  On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition , 2009, Math. Comput. Model..

[41]  Dušan D. Repovš,et al.  On a PDE involving the Ap(⋅)-Laplace operator , 2012 .

[42]  M. Willem.  Minimax Theorems , 1997 .

[43]  G. Bonanno,et al.  Multiple solutions for elliptic problems involving the p(x)-Laplacian , 2011 .

[44]  G. Bonanno,et al.  Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian , 2003 .

[45]  S. Heidarkhani,et al.  Critical Points Approaches to Elliptic Problems Driven by a p(x)-Laplacian , 2015 .

[46]  Vicentiu D. Radulescu,et al.  Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis , 2015 .

[47]  Marcelo M. Cavalcanti,et al.  Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation , 2001, Advances in Differential Equations.

[48]  Guowei Dai.  Infinitely many non-negative solutions for a Dirichlet problem involving p(x)-Laplacian , 2009 .

[49]  M. Ferrara,et al.  Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions , 2016 .

[50]  Giuseppe Mingione,et al.  Gradient estimates for the p (x)-Laplacean system , 2005 .