This paper concentrates on establishing the existence of multiple weak solutions for a specific type of elliptic equations that involve a Hardy potential and have mixed boundary conditions. The main goal of the study is to establish an existence result of at least three different weak solutions thanks to variational techniques, Hardy inequality, and a particular theorem called the Bonanno–Marano type three critical points theorem.
Citation: Khaled Kefi, Mohammed M. Al-Shomrani. On multiple solutions for an elliptic problem involving Leray–Lions operator, Hardy potential and indefinite weight with mixed boundary conditions[J]. AIMS Mathematics, 2025, 10(3): 5444-5455. doi: 10.3934/math.2025251
This paper concentrates on establishing the existence of multiple weak solutions for a specific type of elliptic equations that involve a Hardy potential and have mixed boundary conditions. The main goal of the study is to establish an existence result of at least three different weak solutions thanks to variational techniques, Hardy inequality, and a particular theorem called the Bonanno–Marano type three critical points theorem.
[1] |
G. Bonanno, A. Chinnì, V. D. Rădulescu, Existence of two non-zero weak solutions for a $p(\cdot)$-biharmonic problem with Navier boundary conditions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 34 (2023), 727–743. https://doi.org/10.4171/RLM/1025 doi: 10.4171/RLM/1025
![]() |
[2] |
G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1–10. https://doi.org/10.1080/00036810903397438 doi: 10.1080/00036810903397438
![]() |
[3] |
G. Bonanno, G. D'Aguì, A. Sciammetta, Nonlinear elliptic equations involving the $ p $-Laplacian with mixed Dirichlet-Neumann boundary conditions, Opuscula Math., 39 (2019), 159–174. https://doi.org/10.7494/OpMath.2019.39.2.159 doi: 10.7494/OpMath.2019.39.2.159
![]() |
[4] |
E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions, J. Funct. Anal., 199 (2003), 468–507. https://doi.org/10.1016/S0022-1236(02)00101-5 doi: 10.1016/S0022-1236(02)00101-5
![]() |
[5] |
E. B. Davies, A. M. Hinz, Explicit constants for Rellich inequalities in $ L^{p}(\Omega) $, Math. Z., 227 (1998), 511–523. https://doi.org/10.1007/PL00004389 doi: 10.1007/PL00004389
![]() |
[6] |
D. E. Edmunds, J. Rákosník, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267–293. https://doi.org/10.4064/sm-143-3-267-293 doi: 10.4064/sm-143-3-267-293
![]() |
[7] |
X. L. Fan, Q. H. Zhang, Existence of solutions for $ p(x) $-Laplacian Dirichlet problem, Nonlinear Anal.-Theor., 52 (2003), 1843–1852. https://doi.org/10.1016/S0362-546X(02)00150-5 doi: 10.1016/S0362-546X(02)00150-5
![]() |
[8] | X. Fan, D. Zhao, On the generalized Orlicz-Sobolev space $ W^{k, p(x)}(\Omega) $, Journal of Gansu Education College, 12 (1998), 1–6. |
[9] |
X. L. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395–1412. https://doi.org/10.1016/j.jmaa.2007.08.003 doi: 10.1016/j.jmaa.2007.08.003
![]() |
[10] |
K. Kefi, Existence and multiplicity of triple weak solutions for a nonlinear elliptic problem with fourth-order operator and Hardy potential, AIMS Mathematics, 9 (2024), 17758–17773. https://doi.org/10.3934/math.2024863 doi: 10.3934/math.2024863
![]() |
[11] |
K. Kefi, N. Irzi, M. M. Al-Shomrani, Existence of three weak solutions for fourth-order Leray–Lions problem with indefinite weights, Complex Var. Elliptic, 68 (2023), 1473–1484. https://doi.org/10.1080/17476933.2022.2056887 doi: 10.1080/17476933.2022.2056887
![]() |
[12] |
O. Kováčik, J. Rákosník, On spaces $ L^{p(x)} $ and $ W^{k, p(x)} $, Czech. Math. J., 41 (1991), 592–618. https://doi.org/10.21136/CMJ.1991.102493 doi: 10.21136/CMJ.1991.102493
![]() |
[13] |
J. Liu, Z. Q. Zhao, Generalized solutions for singular double-phase elliptic equations under mixed boundary conditions, Nonlinear Anal.-Model., 29 (2024), 1051–1061. https://doi.org/10.15388/namc.2024.29.37845 doi: 10.15388/namc.2024.29.37845
![]() |
[14] | J. Simon, Régularité de la solution d'une équation non linéaire dans $ \mathbb{R}^N $, In: Journées d'analyse non linéaire, Berlin: Springer, 1978,205–227. https://doi.org/10.1007/BFb0061807 |
[15] | E. Zeidler, Nonlinear functional analysis and its applications II/B: nonlinear monotone operators, New York: Springer, 1990. https://doi.org/10.1007/978-1-4612-0981-2 |