A borderline analysis of the Nehari manifold method for concave-convex system
DOI:
https://doi.org/10.12775/TMNA.2025.018Keywords
Nehari manifold, variational methods, extremal curve, concave-convex systemAbstract
The aim of this paper is to obtain an existence and multiplicity result for a strongly coupled concave-convex system for an {\it optimal} choice of involved real parameters via the Nehari manifold method. In the paper, we have obtained the parametric region which is optimal in the sense that the constraint minimization idea based on the Nehari manifold is no longer applicable if the parameters lie in the exterior of the optimal region. By applying a finer analysis of fibering maps, we have shown the existence of atleast two positive solutions for the parameters lying below and even above the parametric optimal curve, characterized variationally via nonlinear generalized Rayleigh quotient. The main result of the paper is complemented by the study of the problem for negative parameter values.References
K. Adriouch and A.E. Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal. 64 (2006), 2149–2167.
C.O. Alves, D.C. de M. Filho and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000), 771–787.
A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.
A. Ambrosetti, J.G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.
K.J. Brown and T.F. Wu, A fibering map approach to a semilinear elliptic boundary value problems, Electron. J. Differential Equations 69 (2007), 1–9.
K.J. Brown and T.F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations 22 (2009), 1097–1114.
L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia 7 (2012), 37–52.
A. Caãda, P. Magal and J.A. Montero, Optimal control of harvesting in a nonlinear elliptic system arising from population dynamics, J. Math. Anal. Appl. 254 (2001), 571–586.
W. Chen and S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Anal. 27 (2016), 80–92.
C.Y. Chen, Y.C. Kuo and T.F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), 1876–1908.
J.M. do Ó, J. Giacomoni and P.K. Mishra, Nehari manifold for fractional Kirchhoff system with critical nonlinearity, Milan J. Math. 87 (2019), 201–231.
P. Drabek and S.I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 703–726.
X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Commun. Pure Appl. Anal. 15 (2016), 1285–1308.
T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concaveconvex nonlinearities, Nonlinear Anal. 71 (2009), 2688–2698.
Y. Il’yasov, On non-local existence results for elliptic operator with convex-concave nonlinearity, Nonlinear Anal. 61 (2005), 211–236.
Y. I’yasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017), 683–714.
N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000), 298–305.
G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
P.K. Mishra and K. Sreenadh, Fractional p-Kirchhoff system with sign changing nonlinearities, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 111 (2017), 281–296.
P.K. Mishra and V.M. Tripathi, A study of extremal parameter for fractional singular Choquard problem, Math. Nachr. 296 (2023), 5259–5287.
P.K. Mishra and V.M. Tripathi, Nehari manifold approach for fractional Kirchhoff problems with extremal value of the parameter, Fract. Calc. Appl. Anal. 27 (2024), 919–943.
T. Ouyang, On the positive solutions of semilinear equations ∆u + λu + hup = 0 on the compact manifolds, Part II, Indiana Univ. Math. J. 40 (1991), 1083–1141.
P. Pucci and V. Rădulescu, Prosress in nonlinear Kirchhoff problems, Nonlinear Anal. 186 (2019), 1-5.
K. Silva and A. Macedo, Local minimizer over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity, J. Differential Equations 265 (2018), 1894–1921.
Y. Sire and E. Valdinoci, Fractional Laplacian phase transition and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 1842–1864.
T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006), 253–270.
T.F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in RN involving sign-changing weight, J. Funct. Anal. 258 (2010), 99–131.
T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal. 68 (2008), 1733–1745.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
