Topological Methods in Nonlinear Analysis

Sign changing solutions of $p$-fractional equations with concave-convex nonlinearities

Mousomi Bhakta and Debangana Mukherjee

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Abstract

We study the existence of sign changing solutions to the following $p$-fractional problem with concave-critical nonlinearities: \begin{alignat*}2 (-\Delta)^s_pu &= \mu |u|^{q-1}u + |u|^{p^*_s-2}u &\quad&\mbox{in }\Omega,\\ u&=0&\quad&\mbox{in } \mathbb{R}^N\setminus\Omega, \end{alignat*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0< q< p-1$, $\mu\in\mathbb{R}^+$ and $p_s^*={Np}/({N-ps})$. $\Omega$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary, $N> ps$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 2 (2018), 511-544.

Dates
First available in Project Euclid: 18 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1516244427

Digital Object Identifier
doi:10.12775/TMNA.2017.052

Mathematical Reviews number (MathSciNet)
MR3829042

Zentralblatt MATH identifier
06738554

Citation

Bhakta, Mousomi; Mukherjee, Debangana. Sign changing solutions of $p$-fractional equations with concave-convex nonlinearities. Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 511--544. doi:10.12775/TMNA.2017.052. https://projecteuclid.org/euclid.tmna/1516244427


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References

  • A. Ambrosetti, G.J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.
  • A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.
  • B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 4, 875–900.
  • T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3555–3561.
  • M. Bhakta and D. Mukherjee, Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities, Differential Integral Equations 30 (2017), no. 5–6, 387–422.
  • C. Brandle, E. Colorado, A. Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71.
  • L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), no. 3, 769–799.
  • L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound. 16 (2014), no. 3, 419–458.
  • L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremal functions for the fractional Sobolev inequality, Calc. Var. Partial Differential Equations 55 (2016), 1–32.
  • F. Charro, E. Colorado and I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side, J. Differential Equationns 246 (2009), 4221–4248.
  • J. Chen, Some further results on a semilinear equation with concave-convex nonlinearity, Nonlinear Anal. 62 (2005), no. 1, 71–87
  • W. Chen and S. Deng, The Nehari manifold for non-local elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys. 66 (2015), no. 4, 1387–1400.
  • W. Chen and M. Squassina, Nonlocal systems with critical concave-convex powers, Adv. Nonlinear Stud. 16 (2016), 821–842.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
  • P. Drábek and S. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 703–726.
  • S. Goyal and K. Sreenadh, Nehari manifold for non-local elliptic operator with concave-convex non-linearities and signchanging weight function, Proc. Indian Acad. Sci. 125 (2015), 545–558.
  • S. Goyal and K. Sreenadh, Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal. 4 (2015), 37–58.
  • A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), 101–125.
  • E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795–826.
  • S. Mosconi, K. Perera, M. Squassina and Y. Yang, The Brezis–Nirenberg problem for the fractional $p$-Laplacian, Calc. Var. Partial Differential Equations 55 (2016), no. 4, paper No. 105, 25 pp.
  • K. Perera, M. Squassina and Y. Yang, Bifurcation results for critical growth fractional $p$-Laplacian problems, Math. Nachr. 289 (2016), 332–342.
  • R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.
  • R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), no. 1, 67–102.
  • M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 705–717.