Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p, q)-Laplacian Operators

Abstract

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators: $\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$ where λ > 0 is a real parameter, Ω is a bounded domain in R ^N, with boundary ∂Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (−Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (−Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁ ^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.

Keywords

The Nehari manifold / Fractional p-Laplacian / Variational methods

References

[1] Franzina G., Palatucci G.. Fractional p-eigenvalues. Riv. Math. Univ. Parma, 2014, 5(2): 373-386

[2] Iannizzotto A., Squassina M.. Weyl-type laws for fractional p-eigenvalue problems. Asymptot. Anal., 2014, 88(4): 233-245

[3] Brown K. J., Zhang Y.. The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differential Equations, 2003, 193(2): 481-499

[4] Chen W., Deng S.. The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal. Real World Appl., 2016, 27: 80-92

[5] Zhang G., Liu X., Liu S.. Remarks on a class of quasilinear elliptic systems involving the (p, q)-Laplacian. Electron. J. Differential Equations, 2005

[6] Goyal S., Sreenadh K.. Existence of multiple solutions of p-fractional Laplace operator with signchanging weight function. Adv. Nonlinear Anal., 2015, 4(1): 37-58

[7] Fiscella A., Pucci P., Saldi S.. Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators. Nonlinear Anal., 2017, 158(2): 109-131

[8] Di Castro A., Kuusi T., Palatucci G.. Local behavior of fractional p-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2016, 33(5): 1279-1299

[9] Di Nezza E., Palatucci G., Valdinoci E.. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 2012, 136(5): 521-573

[10] Grisvard P.. Elliptic Problems in Nonsmooth Domains, 2011, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM)

[11] Pucci P., Xiang M., Zhang B.. Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in R^N. Calc. Var. Partial Differential Equations, 2015, 54(3): 2785-2806

[12] Drabek P., Stavrakakis N. M., Zographopoulos N. B.. Multiple nonsemitrivial solutions for quasilinear elliptic systems. Differential Integral Equations, 2003, 16(12): 1519-1531

[13] Amghibech S.. On the discrete version of Picone’s identity. Discrete Appl. Math., 2008, 156(1): 1-10

[14] Mosconi S., Squassina M.. Nonlocal problems at nearly critical growth. Nonlinear Anal., 2016, 136: 84-101

[15] Del Pezzo L. M., Quaas A.. A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian. J. Differential Equations, 2017, 263(1): 765-778