Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions

References

• Kirchhoff G. Mechanik. Leipezig: Teubner; 1983.
   Google Scholar
• Lions JL. On some questions in boundary value problems of mathematical physics. North-Holland Mathematics Studies. 1978. p. 284–346. (Contemporary developments in continuum mechanics and partial differential equations; vol. 30).
   Google Scholar
• Chipot M, Lovat B. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 1997;30(7):4619–4627.
   Web of Science ®Google Scholar
• Arosio A, Pannizi S. On the well-posedness of the Kirchhoff string. Trans Am Math Soc. 1996;348:305–330.
   Web of Science ®Google Scholar
• Cavalcanti MM, Cavacanti VN, Soriano JA. Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv Differ Equ. 2001;6:701–730.
   Google Scholar
• Corrêa FJSA, Nascimento RG. On a nonlocal elliptic system of p-Kirchhoff type under Neumann boundary condition. Math Comput Model. 2008;DOI:10.1016/j.mcm.2008.03.013 .
   Google Scholar
• Heidarkhani S, Henderson J. Infinitely many solutions for nonlocal elliptic systems of (p1,…,pn)-Kirchhoff type. Electron J Differ Equ. 2012;2012(69):1–15.
   Google Scholar
• Miao Q. Infinitely many solutions for nonlocal elliptic systems of (p1(x),…,pn(x))-Kirchhoff type. Math Methods Appl Sci. 2016;39:2325–2333.
   Web of Science ®Google Scholar
• Perera K, Zhang Z. Nontrivial solutions of Kirchhoff type problems via the Yang index. J Differ Equ. 2006;221:246–255.
   Web of Science ®Google Scholar
• Abbas MI, Ragusa MA. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry. 2021;13(2):264.
   Google Scholar
• Akdemir AO, Butt SI, Nadeem M, et al. New general variants of Chebyshev type inequalities via generalized fractional integral operators. Mathematics. 2021;9(2):122.
   Web of Science ®Google Scholar
• Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. Heidelberg: Springer-Verlag; 2011. (Lecture notes in mathematics 2017).
   Google Scholar
• Gala S, Ragusa MA. Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices. Appl Anal. 2016;95(6):1271–1279.
   Web of Science ®Google Scholar
• Goodrich CS, Ragusa MA. Holder continuity of weak solutions of p-Laplacian PDEs with VMO coefficients. Nonlinear Anal. 2019;185:336–355.
   Web of Science ®Google Scholar
• Halsey TC. Electrorheological fluids. Science. 1992;258:761–766.
   PubMed Web of Science ®Google Scholar
• Papageorgiou NS, Scapellato A. Nonlinear Robin problems with general potential and crossing reaction. Rend Lincei-Mat Appl. 2019;30:1–29.DOI:10.4171/RLM/833 .
   Google Scholar
• Ragusa MA, Tachïkawa A. Regularity for minimizers for functionals of double phase with variable exponents. Adv Nonlinear Anal. 2020;9:710–728.
   Web of Science ®Google Scholar
• Ružika M. Flow of shear dependent electro-rheological fluids. C R Acad Sci Paris Ser. 1999;I 329:393–398.
   Web of Science ®Google Scholar
• Ružika M. Electro-rheological fluids: modeling and mathematical theory. Berlin: Springer; 2000. (Lecture notes in mathematics).
   Google Scholar
• He C, Li G. The regularity of weak solutions to nonlinear scalar field elliptic equations containing p−q-Laplacians. Ann Acad Sci Fenn. 2008;33:337–371.
   Google Scholar
• Mahshid M, Razani A. A weak solution for a (p(x),q(x))-Laplacian elliptic problem with a singular term. Bound Value Probl. 2021;2021:327.
   Web of Science ®Google Scholar
• Ni. W, Serrin J. Existence and nonexistence theorems for ground states of quasilinear partial differential equations. Att Convegni Lincei. 1985;77:231–257.
   Google Scholar
• Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pure Appl Math. 1983;36:437–477.
   Web of Science ®Google Scholar
• Lions PL. The concentration-compactness principle in calculus of variation, the limit case part 1 and 2. Rev Mat Ibroamericana. 1985;1(1):145–201.
   Google Scholar
• Fu YQ. The principle of concentration compactness in Lp(x) spaces and its application. Nonlinear Anal. 2009;71:1876–1892.
   Web of Science ®Google Scholar
• Li S, Zou W. Remarks on a class of elliptic problems with critical exponents. Nonlinear Anal. 1998;32:769–774.
   Web of Science ®Google Scholar
• Djellit A, Tas S. Quasilinear elliptic systems with critical Sobolev exponents in RN. Nonlinear Anal. 2007;66(7):1485–1497.
   Web of Science ®Google Scholar
• Lalilia H, Tasa S, Djellitb A. Existence of solutions for critical systems with variable exponents. Math Model Anal. 2018;23(4):596–610.
   Web of Science ®Google Scholar
• Chems Eddine N. Existence of solutions for a critical (p1(x),…,pn(x))-Kirchhoff-type potential systems. Appl Anal. 2020;
   Google Scholar
• Chems Eddine N. Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent. Appl Anal. 2021;
   Web of Science ®Google Scholar
• Alves CO, Barreiro JP. 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
• Hurtado EJ, Miyagaki OH, Rodrigues RS. Existence and asymptotic behaviour for a Kirchhoff type equation with variable critical growth exponent. Milan J Math. 2010;77:127–150.
   Google Scholar
• Zhang X, Fu Y. Solutions of p(x)-Laplacian equations with critical exponent and perturbations in RN. Electron J Differ Equ. 2012;21(120):1–14.
   Google Scholar
• Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. Electron J Differ Equ. 2010;141:1–18.
   Google Scholar
• Bonder JF, Saintier N, Silva A. On the Sobolev trace theorem for variable exponent spacesin the critical range. Ann Mat Pura Appl (4). 2014;193(6):1607–1628.
   Web of Science ®Google Scholar
• Kováčik O, Rákosník J. On spaces Lp(x)(Ω) and W1,p(Ω). Czechoslovak Math J. 1991;41:592–618.
   Web of Science ®Google Scholar
• Fan X, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(Ω). J Math Anal Appl. 2001;263:424–446.
   Web of Science ®Google Scholar
• Edmunds DE, Rakosnik J. Sobolev embeddings with variable exponent. Studia Math. 2000;143:267–293.
   Web of Science ®Google Scholar
• Harjulehto P, Hästö P, Koskenoja M, et al. The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal. 2006;25(3):205–222.
   Web of Science ®Google Scholar
• Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. CBMS regional conference series in mathematics, vol. 65, American Mathematical Society, Providence, RI; 1986.
   Google Scholar
• Bonder JF, Martínez S, Rossi JD. Existence results for gradient elliptic systems withnonlinear boundary conditions. NoDEA. 2017;14(1–2):153–179.
   Web of Science ®Google Scholar
• Chen YM, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 2006;66:1383–1406.
   Web of Science ®Google Scholar
• Chems Eddine N, Idrissi AA. Multiple solutions to a (p1(x),…,pn(x))-Laplacian-type systems in unbounded domain. Azerbaijan J Math. 2020;10(1):3–20.
   Web of Science ®Google Scholar
• DiBenedetto E. Degenerate parabolic equations. New York: Springer-Verlag; 1993.
   Google Scholar
• Ragusa MA, Razani A, Safari F. Existence of radial solutions for a -Laplacian Dirichlet problem. Adv Differ Equ. 2021;2021:215.
   Web of Science ®Google Scholar