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Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group

  • Farzaneh Safari and Abdolrahman Razani EMAIL logo
Published/Copyright: June 11, 2022
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Abstract

The existence of at least one positive radial solution of the p-biharmonic problem

Δ H n ( w ( ξ ) | Δ H n u | p 2 Δ H n u ) + R ( ξ ) w ( ξ ) | u | p 2 u = i = 1 m a i ( | ξ | H n ) | u | q i 2 u j = 1 k b j ( | ξ | H n ) | u | r j 2 u ,

with Navier boundary condition on a Korányi ball is proved, where wAs is a Muckenhoupt weight function and ΔHn,p2 is the Heisenberg p-biharmonic operator.

  1. (Communicated by Alberto Lastra)

References

[1] Badiale, M.—Serra, M.: Semilinear Elliptic Equations for Beginners, Existence Results via the Variational Approach, Springer Science and Business Media, 2010.10.1007/978-0-85729-227-8Search in Google Scholar

[2] Behboudi, F.—Razani, A.—Oveisiha, M.: Existence of a mountain pass solution for a nonlocal fractional (p, q)-Laplacian problem, Bound. Value Probl. 2020 (2020), Art. 149.10.1186/s13661-020-01446-wSearch in Google Scholar

[3] Ben Haddouch, K.—Tsouli, N.—El Miloud, H.—El Allali, Z.: Resonance problems for the Weighted p-biharmonic operator, Appl. Math. Sci. 9(89) (2015), 4433–4441.10.12988/ams.2015.48658Search in Google Scholar

[4] Bisci, G. M.—Repovš, D.: Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 59(2) (2014), 271–284.10.1080/17476933.2012.734301Search in Google Scholar

[5] Cowan, C.—Razani, A.: Singular solutions of a p-Laplace equation involving the gradient, J. Differential Equations 269 (2020), 3914-3942.10.1016/j.jde.2020.03.017Search in Google Scholar

[6] D’Ambrosio, L.: Some Hardy inequalities on the Heisenberg group, J. Differential Equations 40 (2004), 552–564.10.1023/B:DIEQ.0000035792.47401.2aSearch in Google Scholar

[7] Dong, X.-Y.—Lu, G.-Z.—Sun, L.-J.: Global Poincaré inequalities on the Heisenberg group and applications, Acta Math. Sin. (Engl. Ser.) 23(4) (2007), 735–744.10.1007/s10114-005-0874-0Search in Google Scholar

[8] Dwivedi, G.—Tyagi, J.: Stability of positive solution to biharmonic equations with logistic-type nonlinearities on Heisenberg group, arXiv:1606.06413, (2019).10.1619/fesi.62.61Search in Google Scholar

[9] Evans, L.: Partial Differential Equations, Sobolev Spaces, Academic Press, New York, 1998.Search in Google Scholar

[10] Fabes, E. B.—Kenig, C. E.—Serapioni, R. P.: The local regularity of solutions of degenerate elliptic equations, Commun. Stat. Theory Methods 7(1) (1982), 77–116.10.1080/03605308208820218Search in Google Scholar

[11] Heidari, S.—Razani, A.: Infinitely many solutions for (p(x), q(x))-Laplacian-like systems, Commun. Korean Math. Soc. 36(1) (2021), 51-62.Search in Google Scholar

[12] Makvand Chaharlang, M.—Ragusa, M. A.—Razani, A.: A sequence of radially symmetric weak solutions for some nonlocal elliptic problem in ℝN, Mediterr. J. Math. 17(2) (2020), Art. No. 53.10.1007/s00009-020-1492-xSearch in Google Scholar

[13] Makvand Chaharlang, M.—Razani, A.: A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese J. Math. 23(3) (2019), 589–599.10.11650/tjm/180906Search in Google Scholar

[14] Makvand Chaharlang, M.—Razani, A.: Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28(3) (2021), 429–438.10.1515/gmj-2019-2077Search in Google Scholar

[15] Pólya, G.—Szegö, G.: Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions, Springer-Verlag Berlin Heidelberg New York, 1998.10.1007/978-3-642-61905-2Search in Google Scholar

[16] Rabinowitz, P. H.: Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 2(1) (1978), 215–223.Search in Google Scholar

[17] Ragusa, M. A.—Razani, A.—Safari, F.: Existence of radial solutions for a p(x)-Laplacian Dirichlet problem, Adv. Difference Equ. 2021:215 (2021), 14 pp.10.1186/s13662-021-03369-xSearch in Google Scholar

[18] Razani, A.: Subsonic detonation waves in porous media, Phys. Scr. 94 (2019), Art No. 085209, 6 pp.10.1088/1402-4896/ab029bSearch in Google Scholar

[19] Ricciotti, D.: p-Laplace Equation in the Heisenberg Group: Regularity of Solutions, Springer, 2015.10.1007/978-3-319-23790-9Search in Google Scholar

[20] Romanovskiĭ, N. N.: Integral representations and embedding theorems for functions defined on the Heisenberg groups ℍn, St. Petersburg Math. J. 16(2) (2005), 349–375.10.1090/S1061-0022-05-00854-XSearch in Google Scholar

[21] Safari, F.—Razani, A.: Existence of positive radial solutions for Neumann problem on the Heisenberg group, Bound. Value Probl. 2020 (2020), Art No. 88.10.1186/s13661-020-01386-5Search in Google Scholar

[22] Safari, F.—Razani, A.: Existence of radial solutions of the Kohn-Laplacian problem, Complex Var. Elliptic Equ. 2020 (2020).10.1080/17476933.2020.1818733Search in Google Scholar

[23] Safari, F.—Razani, A.: Positive weak solutions of a generalized supercritical Neumann problem, Iran J. Sci. Technol. Trans. Sci. 2020 (2020).10.1007/s40995-020-00996-zSearch in Google Scholar

[24] Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3(2) (1986), 77–109.10.1007/978-94-009-4632-3_32Search in Google Scholar

Received: 2020-11-05
Accepted: 2021-05-14
Published Online: 2022-06-11
Published in Print: 2022-06-27

© 2022 Mathematical Institute Slovak Academy of Sciences

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