Document Type : Research Paper

**Authors**

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.

**Abstract**

Here, we consider a fourth-order elliptic problem involving singularity and p(x)- biharmonic operator. Using Hardy’s inequality, S+-condition, and Palais-Smale condition, the existence of weak solutions in a bounded domain in RN is proved. Finally, we percent some examples.

**Keywords**

[1] F. Abdolrazaghi and A. Razani, On the weak solutions of an overdetermined system of nonlinear fractional partial integro-differential equations, Miskolc Math. Notes, 20(1) (2019), 3–16, DOI:10.18514/MMN.2019.2755.

[2] R. Ayazoghlu, G. Alisay, and I. Ekincioglu, Existence of one weak solution for p(x)−biharmonic equations involving a concave-convex nonlinearity, Matematicki Vesnik, 69 (2017), 296–307.

[3] F. Behboudi and A. Razani, Two weak solutions for a singular (p, q)-Laplacian problem, Filomat, 33(11) (2019), 3399-3407, https://doi.org/10.2298/FIL1911399B.

[4] G. Bonanno, Relation between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205–220.

[5] G. Bonanno and A. Chinni, Discontinuous elliptic problems involving the p(x)-Laplacian, Math. Nacher., 284 (2011), 639–652.

[6] G. Bonanno and A. Chinni, Existence and multiplicity of weak solution for elliptic dirichlet problems with variable exponent, J. Math. Anal. Appl., 418 (2014), 812–827.

[7] G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Apple. Anal., 89 (2010), 1–10.

[8] C. C. Chen and C. S. Lin, Local bihavior of singular positive solutions of semilinear elliptic equations with sobolev exponent, Duke Math. J., 78 (1995), 315–334.

[9] P. Dr´abek, Strongly nonlinear degenerated and singular elliptic problems, Pitman Research Notes in Mathematics Series, 343 (1996), 112–146.

[10] X. L. Fan and D. Zhao, On the space Lp(x)(Ω) and W m,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424–446.

[11] X. L. Fan and Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2011), 1843-1852.

[12] N. Ghoussoub and C. Yuan, Multiple solutions for Quashi-linear PDEs involving the critical Sobolev and Hardy exponents, Amer. Math. Soc., 352 (2000), 5703–5743.

[13] M. Khodabakhshi, A. M. Aminpour, G. A. Afrouzi, and A. Hadjian, Existence of two weak solutions for some singular elliptic problems, Rev. R. Acad. Cienc. Ex. Fis. Nat. Ser. A. Mat. 110 (2016), 385–393.

[14] L. Kong, Multiple solutions fourth order elliptic problems with p(x)-biharmonic operators, Opuscula Math., 36 (2016), 253–264.

[15] L. Li and C. Tang, Existence and multiplicity of solutions for a class of p(x)-biharmonic equations, Acta Math. Sci. Ser. A Chin Ed, 33 (2013), 155-170.

[16] R. Mahdavi Khanghahi and A. Razani, Existence of at least three weak solutions for a singular fourth-order elliptic problems, J. Math. Anal., 8 (2017), 45–51.

[17] R. Mahdavi Khanghahi, and A. Razani, Solutions for a singular elliptic problem involving the p(x)-Laplacian, Filomat, 32(14) (2018), 4841-4850.

[18] M. Makvand Chaharlang and A. Razani, Infinitely many solutions for a fourth order singular elliptic problem, Filomat, 32(14) (2018), 5003-5010.

[19] M. Makvand Chaharlang and A. Razani, A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese J. Math., 23(3) (2019), 589-599, doi:10.11650/tjm/180906, https://projecteuclid.org/euclid.twjm/1537927424.

[2] R. Ayazoghlu, G. Alisay, and I. Ekincioglu, Existence of one weak solution for p(x)−biharmonic equations involving a concave-convex nonlinearity, Matematicki Vesnik, 69 (2017), 296–307.

[3] F. Behboudi and A. Razani, Two weak solutions for a singular (p, q)-Laplacian problem, Filomat, 33(11) (2019), 3399-3407, https://doi.org/10.2298/FIL1911399B.

[4] G. Bonanno, Relation between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205–220.

[5] G. Bonanno and A. Chinni, Discontinuous elliptic problems involving the p(x)-Laplacian, Math. Nacher., 284 (2011), 639–652.

[6] G. Bonanno and A. Chinni, Existence and multiplicity of weak solution for elliptic dirichlet problems with variable exponent, J. Math. Anal. Appl., 418 (2014), 812–827.

[7] G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Apple. Anal., 89 (2010), 1–10.

[8] C. C. Chen and C. S. Lin, Local bihavior of singular positive solutions of semilinear elliptic equations with sobolev exponent, Duke Math. J., 78 (1995), 315–334.

[9] P. Dr´abek, Strongly nonlinear degenerated and singular elliptic problems, Pitman Research Notes in Mathematics Series, 343 (1996), 112–146.

[10] X. L. Fan and D. Zhao, On the space Lp(x)(Ω) and W m,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424–446.

[11] X. L. Fan and Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2011), 1843-1852.

[12] N. Ghoussoub and C. Yuan, Multiple solutions for Quashi-linear PDEs involving the critical Sobolev and Hardy exponents, Amer. Math. Soc., 352 (2000), 5703–5743.

[13] M. Khodabakhshi, A. M. Aminpour, G. A. Afrouzi, and A. Hadjian, Existence of two weak solutions for some singular elliptic problems, Rev. R. Acad. Cienc. Ex. Fis. Nat. Ser. A. Mat. 110 (2016), 385–393.

[14] L. Kong, Multiple solutions fourth order elliptic problems with p(x)-biharmonic operators, Opuscula Math., 36 (2016), 253–264.

[15] L. Li and C. Tang, Existence and multiplicity of solutions for a class of p(x)-biharmonic equations, Acta Math. Sci. Ser. A Chin Ed, 33 (2013), 155-170.

[16] R. Mahdavi Khanghahi and A. Razani, Existence of at least three weak solutions for a singular fourth-order elliptic problems, J. Math. Anal., 8 (2017), 45–51.

[17] R. Mahdavi Khanghahi, and A. Razani, Solutions for a singular elliptic problem involving the p(x)-Laplacian, Filomat, 32(14) (2018), 4841-4850.

[18] M. Makvand Chaharlang and A. Razani, Infinitely many solutions for a fourth order singular elliptic problem, Filomat, 32(14) (2018), 5003-5010.

[19] M. Makvand Chaharlang and A. Razani, A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese J. Math., 23(3) (2019), 589-599, doi:10.11650/tjm/180906, https://projecteuclid.org/euclid.twjm/1537927424.

[20] M. Makvand Chaharlang and A. Razani, Existence of infinitely many solutions for a class of nonlocal problems with Dirichlet boundary condition, Commun. Korean Math. Soc., 34(1) (2019), 155–167, https://doi.org/10.4134/CKMS.c170456.

[21] E. Mitidier, A simple approach to Hardy inequalities, Math. Notes, 67 (2000), 479–486.

[22] E. Montefusco, Lower semicontinuity of functional via concentration-compactness principle, J. Math. Anal. Appl., 263 (2001), 264–276.

[23] A. Razani, Weak and strong detonation profiles for a qualitative model, J. Math. Anal. Appl., 276 (2002), 868–881, https://doi.org/10.1016/S0022-247X(02)00459-6.

[24] A. Razani, Chapman-Jouguet detonation profile for a qualitative model, Bull. Austral. Math. Soc., 66 (2002), 393–403, https://doi.org/10.1017/S0004972700040259.

[25] A. Razani, Existence of Chapman-Jouguet detonation for a viscous combustion model, J. Math. Anal. Appl., 293 (2004), 551–563, https://doi.org/10.1016/j.jmaa.2004.01.018.

[26] A. Razani, On the existence of premixed laminar flames, Bull. Austral. Math. Soc., 69 (2004), 415–427, https://doi.org/10.1017/S0004972700036194.

[27] A. Razani, Shock waves in gas dynamics, Surv. Math. Appl., 2 (2007), 59-89.

[28] A. Razani, An existence theorem for ordinary differential equation in Menger probabilistic metric space, Miskolc Mathematical Notes, 15 (2014), No. 2, 711–716, DOI: 10.18514/MMN.2014.640.

[29] A. Razani, Chapman-Jouguet travelling wave for a two-steps reaction scheme, Ital. J. Pure Appl. Math., 39 (2018), 544–553.

[30] A. Razani, Subsonic detonation waves in porous media, Phys. Scr., 94(8) (2019), 6 pages, https://doi.org/10.1088/1402-4896/ab029b.

[31] S. Shokooh and G. Afrouzi, Existene results of infinity many solutions for a class of p(x)- biharmonic problems, Computational Methods for Differential Equations, 5 (2017), 310–323.

[32] J. Simon, Regularite de la solution d’une equation non lineaire dans RN, Jornees d’Analyse NonLineaire, 665 (1978), 205–227.

[33] S. Taarabti, Z. El Allali, and K. Hadddouch, Eigenvalues of p(x)-biharmonic operator with indefinite weight under Neumann boundary condition, Bol. Soc. Paran. Mat., 36 (2018), 195–213.

[34] Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109–3125.

[35] H. Xie and J. Wang, Infinitely many solutions for p-Harmonic equation with singular term, J. Inequal. Appl., 2013(9) (2013), https://doi.org/10.1186/1029-242X-2013-9.

[36] H. Yin and Y. Liu, Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic, Bull. Korean Math. soc., 50 (2013), 1817-1826.

[37] E. Zeidler, Nonlinear functional analysis and its application, Springer Verlage, BerlinHeidelberg-Newyork, 1986.

[21] E. Mitidier, A simple approach to Hardy inequalities, Math. Notes, 67 (2000), 479–486.

[22] E. Montefusco, Lower semicontinuity of functional via concentration-compactness principle, J. Math. Anal. Appl., 263 (2001), 264–276.

[23] A. Razani, Weak and strong detonation profiles for a qualitative model, J. Math. Anal. Appl., 276 (2002), 868–881, https://doi.org/10.1016/S0022-247X(02)00459-6.

[24] A. Razani, Chapman-Jouguet detonation profile for a qualitative model, Bull. Austral. Math. Soc., 66 (2002), 393–403, https://doi.org/10.1017/S0004972700040259.

[25] A. Razani, Existence of Chapman-Jouguet detonation for a viscous combustion model, J. Math. Anal. Appl., 293 (2004), 551–563, https://doi.org/10.1016/j.jmaa.2004.01.018.

[26] A. Razani, On the existence of premixed laminar flames, Bull. Austral. Math. Soc., 69 (2004), 415–427, https://doi.org/10.1017/S0004972700036194.

[27] A. Razani, Shock waves in gas dynamics, Surv. Math. Appl., 2 (2007), 59-89.

[28] A. Razani, An existence theorem for ordinary differential equation in Menger probabilistic metric space, Miskolc Mathematical Notes, 15 (2014), No. 2, 711–716, DOI: 10.18514/MMN.2014.640.

[29] A. Razani, Chapman-Jouguet travelling wave for a two-steps reaction scheme, Ital. J. Pure Appl. Math., 39 (2018), 544–553.

[30] A. Razani, Subsonic detonation waves in porous media, Phys. Scr., 94(8) (2019), 6 pages, https://doi.org/10.1088/1402-4896/ab029b.

[31] S. Shokooh and G. Afrouzi, Existene results of infinity many solutions for a class of p(x)- biharmonic problems, Computational Methods for Differential Equations, 5 (2017), 310–323.

[32] J. Simon, Regularite de la solution d’une equation non lineaire dans RN, Jornees d’Analyse NonLineaire, 665 (1978), 205–227.

[33] S. Taarabti, Z. El Allali, and K. Hadddouch, Eigenvalues of p(x)-biharmonic operator with indefinite weight under Neumann boundary condition, Bol. Soc. Paran. Mat., 36 (2018), 195–213.

[34] Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109–3125.

[35] H. Xie and J. Wang, Infinitely many solutions for p-Harmonic equation with singular term, J. Inequal. Appl., 2013(9) (2013), https://doi.org/10.1186/1029-242X-2013-9.

[36] H. Yin and Y. Liu, Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic, Bull. Korean Math. soc., 50 (2013), 1817-1826.

[37] E. Zeidler, Nonlinear functional analysis and its application, Springer Verlage, BerlinHeidelberg-Newyork, 1986.

July 2021

Pages 818-829

**Receive Date:**23 July 2019**Revise Date:**03 May 2020**Accept Date:**10 May 2020