Abstract
In this work we study the following nonlocal problem
where \(\varOmega \subset \mathbb R^N\) is open and bounded with smooth boundary, \(N>2s, s\in (0, 1), M(t)=a+bt^{\theta -1},\;t\ge 0\) with \( \theta >1, a\ge 0\) and \(b>0\). The exponents satisfy \(1<\gamma<2<{2\theta<p<2^*_{s}=2N/(N-2s)}\) (when \(a\ne 0\)) and \(2<\gamma<2\theta<p<2^*_{s}\) (when \(a=0\)). The parameter \(\lambda \) involved in the problem is real and positive. The problem under consideration has nonlocal behaviour due to the presence of nonlocal fractional Laplacian operator as well as the nonlocal Kirchhoff term \(M(\Vert u\Vert ^2_X)\), where \(\Vert u\Vert ^{2}_{X}=\iint _{\mathbb R^{2N}} \frac{|u(x)-u(y)|^2}{\left| x-y\right| ^{N+2s}}dxdy\). The weight functions \(f, g:\varOmega \rightarrow \mathbb R\) are continuous, f is positive while g is allowed to change sign. In this paper an extremal value of the parameter, a threshold to apply Nehari manifold method, is characterized variationally for both degenerate and non-degenerate Kirchhoff cases to show an existence of at least two positive solutions even when \(\lambda \) crosses the extremal parameter value by executing fine analysis based on fibering maps and Nehari manifold.
Similar content being viewed by others
References
Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994)
Ambrosetti, A., Garcia-Azorero, J., Peral, T.: Existence and multiplicity results for some nonlinear elliptic equations. A survey. Rend. Mat. Appl. 20, 167–198 (2000)
Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)
Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré C Anal. Non Linéaire 32(4), 875–900 (2015)
Binlin, Z., Fiscella, A., Liang, S.: Infinitely many solutions for critical degenerate Kirchhoff type equations involving the fractional \(p-\)Laplacian. Appl. Math. Optim. 80, 63–80 (2019)
Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. J. Differential Equations 193(2), 481–499 (2003)
Chen, C., Kuo, Y., Wu, T.: The Nehari manfiold for Kirchhoff problem involving sign-changing weight functions. J. Differential Equations 250(4), 1876–1908 (2011)
Fiscella, A., Mishra, P.K.: The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. Nonlinear Anal. 186, 6–32 (2019)
Fiscella, A., Pucci, P.: Degenerate Kirchhoff \((p, q)\)-fractional systems with critical nonlinearities. Fract. Calc. Appl. Anal. 23, 723–752 (2020). https://doi.org/10.1515/fca-2020-0036
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
Goyal, S., Sreenadh, K.: Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. 125, 545–558 (2015)
Goyal, S., Sreenadh, K.: Existence of multiple solutions of \(p-\)fractional Laplace operator with sign-changing weight function. Adv. Nonlinear Anal. 4(1), 37–58 (2015)
Pokhozhaev, S.I.: The fibration method for solving nonlinear boundary value problems. Tr. Mat. Inst. Steklova 192, 146–163 (1990)
Il’yasov, Y.: On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient. Topol. Methods Non- linear Anal. 49(2), 683–714 (2017)
Il’yasov, Y., Silva, K.: On the branch of positive solution for \(p-\)Laplacian problem at the extreme value of the Nehari manifold method. Proc. Amer. Math. Soc. 146(7), 2924–2935 (2018)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Nehari, Z.: Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105, 141–175 (1961)
Nehari, Z.: On a class of nonlinear second-order differential equations. Trans. Amer. Math. Soc. 95, 101–123 (1960)
Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev space. Bull. Sci. Math. 136(5), 521–573 (2012)
do Ó, J.M., He, X., Mishra, P.K.: Fractional Kirchhoff problem with critical indefinite nonlinearity. Math. Nachr. 292, 615–632 (2019)
Pucci, P., Rǎdulescu, V.D.: Progress in nonlinear Kirchhoff problems [Editorial]. Nonlinear Anal. 186, 1–5 (2019)
Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \({\mathbb{R} }^N\) involving nonlocal operators. Rev. Mat. Iberoam. 32(1), 1–22 (2016)
Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional \(p-\)Kirchhoff equations. Adv. Nonlinear Anal. 5(1), 27–55 (2016)
Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pures Appl. 101(3), 275–302 (2014)
Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Amer. Math. Soc. 367(1), 67–102 (2015)
Silva, K., Macedo, A.: Local minimizer over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity. J. Differential Equations 265(5), 1894–1921 (2018)
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60(1), 67–112 (2007)
Wu, T.F.: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math. 39(3), 995–1012 (2009)
Wu, T.F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318(1), 253–270 (2006)
Xiang, M.Q., Bisci, G.M., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p-\)Laplacian. Nonlinearity 29(2), 357–374 (2016)
Acknowledgements
The research of the first author is supported by Science and Engineering Research Board, Govt. of India, grant SRG/2021/001076.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Mishra, P.K., Tripathi, V.M. Nehari manifold approach for fractional Kirchhoff problems with extremal value of the parameter. Fract Calc Appl Anal 27, 919–943 (2024). https://doi.org/10.1007/s13540-024-00261-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13540-024-00261-9