Singular p-biharmonic problems involving the Hardy-Sobolev exponent
DOI:
https://doi.org/10.58997/ejde.2023.61Keywords:
p-Laplacian operator; p-Biharmonic equation; Variational method; Existence of solutions; Hardy potential; Critical Hardy-Sobolev exponent; Ekeland variational principle; Mountain pass geometryAbstract
This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods combined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and example.
For mote information see https://ejde.math.txstate.edu/Volumes/2023/61/abstr.html
References
R. Alsaedi, A. Dhifli, A. Ghanmi; Low perturbations of p-biharmonic equations with competing nonlinearities, Complex Var. Elliptic Equ. 66 (2021), no. 4, 642.657.
A. Ambrosett, P.H. Rabinowitz; Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349.381.
M. Bhakta, A. Biswas, D. Ganguly, L. Montoro; Integral representation of solutions using the Green function for fractional Hardy equations, J. Differ. Equ. 269(2020), no. 7, 5573.5594.
M. Bhakta, S. Chakraborty, P. Pucci; Fractional Hardy-Sobolev equations with nonhomogeneous terms, Adv. Nonlinear Anal. 10 (2021), no. 1, 1086.1116.
H. Brezis, E. Lieb; A relation between point convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486.490.
C. Bucur, E. Valdinoci; Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital., 20, Springer, Cham, 2016.
M. M. Chaharlang, A. Razani; A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese J. Math. 23 (2019), 589.599.
Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383.1406.
W. Chen, V. D. R.adulescu, B. Zhang; Fractional Choquard-Kirchhoff problems with critical nonlinearity and Hardy potential, Anal. Math. Phys. 132 (2021), no. 11, https: //doi.org/10.1007/s13324-021-00564-7.
N. T. Chung, A. Ghanmi, T. Kenzizi; Multiple solutions to p-biharmonic equations of Kirchhoff type with vanishing potential, Numer. Funct. Anal. Optim. 44(2023), no. 3, 202-220
E. Davies, A. Hinz; Explicit constants for Rellich inequalities in Lp(¶), Math. Z. 227 (1998), 511.523.
A. Dhifli, R. Alsaedi; Existence and multiplicity of solution for a singular problem involving the p-biharmonic operator in RN, J. Math. Anal. Appl. 499 (2021), 125049.
N. Ghoussoub, C. Yuan; Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703.5743.
W. Guan, V. D. R.adulescu, D. B. Wang; Bound states of fractional Choquard equations with Hardy-Littlewood-Sobolev critical exponent, J. Differ. Equa. 355 (2023), 219.247.
Y. Huang, X. Liu; Sign-changing solutions for p-biharmonic equations with Hardy potential, J. Math. Anal. Appl. 412 (2014), 142.154.
A. Lazer, P. McKenna; Large-amplitude periodic oscillations in suspension bridges. Some new connections with nonlinear analysis, SIAM Rev. 32 (1990), 537.578.
E. Mitidieri; A simple approach to Hardyfs inequalities, Math. Notes 67 (2000), 479.486.
N. S. Papageorgiou, V. D. R.adulescu, D. D. Repov.s; Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.
K. Perera, W. Zou; p-Laplacian problems involving critical Hardy.Sobolev exponents, NoDEA Nonlinear Differ. Equ. Appl. 25 (2018), no. 25, https://doi.org/10.1007/s00030-018-0517-7.
M. PLerez-Llanos, A. Primo; Semilinear biharmonic problems with a singular term, J. Differ. Equ. 257 (2014), 3200.3225.
F. Rellich; Perturbation Theory of Eigenvalue Problems, Courant Institute of Mathematical Sciences, New York University, New York, 1954.
M. Ru.zi.cka; Electrorheological Fluids: Modelling and Mathematical Theory, Springer, Berlin, 2000.
J. Sun, J. Chu, T.F. Wu; Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differ. Equ. 262 (2017), 945.977.
J. Sun, T.F. Wu; Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential, Appl. Math. Lett. 66 (2017), 61.67.
J. Sun, T.F. Wu; The Nehari manifold of biharmonic equations with p-Laplacian and singular potential, Appl. Math. Lett., 88 (2019), 156.163.
W. Wang; p-biharmonic equation with Hardy.Sobolev exponent and without the Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differ. Equ. Appl. 42 (2020), 1.16.
W. Wang, P. Zhao; Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl. 348 (2008), 730.738.
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Copyright (c) 2023 Amor Drissi, Abdeljabbar Ghanmi, Dusan D. Repovs
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