On the existence of solutions to a fractional (p, q)-Laplacian system on bounded domains
Tóm tắt
We study the existence of solutions for the fractional (p, q)-laplacian system
$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )_p^su &{} =\lambda b(x)|u|^{\gamma -2}u +\displaystyle \frac{\alpha }{\alpha +\beta } a(x)|u|^{\alpha -2}u |v |^\beta &{}in&{} \Omega , \\ (-\Delta )_q^lv &{} =\nu c(x)|v|^{\gamma -2}v + \displaystyle \frac{\beta }{\alpha +\beta }a(x)|u|^\alpha |v |^{\beta -2} v &{} in &{} \Omega , \\ v= u &{} =0 &{} in &{} {\mathbb {R}}^N\setminus \Omega . \end{array}\right. \end{aligned}$$
where
$$\Omega $$
is a bounded set of
$${\mathbb {R}}^N$$
with
$$C^1 $$
-boundary
$$\partial \Omega $$
,
$$(-\Delta )_p^s$$
and
$$(-\Delta )_q^l$$
are respectively the s-fractional p-laplacian and the l-fractional q-laplacian operators for
$$l,s\in (0,1)$$
,
$$\lambda $$
and
$$\nu $$
are real parameters,
$$a,b,c:\Omega \rightarrow \Omega $$
are appropriate functions and
$$\alpha ,\beta ,p$$
and q are reals satisfying adequate hypotheses.
Tài liệu tham khảo
Caffarelli, L.A.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symp. 7, 37–52 (2012)
Chen, J., Cheng, B., Tang, X.: New existence of multiple solutions for nonhomogeneous Schrödinger-Kirchhoff problems involving the fractional \(p\)-Laplacian with sign-changing potential. Rev. Real Acad. Cien. Exact., Fís. Nat. Ser. A. Mat. 1–24 (2016)
Cheng, K., Gao, Q.: Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in \({\mathbb{R}}^{N}\). arXiv:1701.03862v1
Chen, W., Deng, S.: The Nehari manifold for a fractional \(p\)-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal. Real World Appl. 27, 80–92 (2016)
Goyal, S., Sreenadh, K.: A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and signchanging weight function. Proc. Indian Acad. Sci. (Math. Sci.) 125(4), 545–558 (2015)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Pohozaev, S.I.: On an approach to nonlinear equations. Doklady Acad. Sci. USSR. 247, 1327–1331 (1979)
Pohozaev, S.I.: On the method of fibering a solution in nonlinear boundary value problems. Proc. Stekl. Ins. Math. 192, 157–173 (1990)
Pucci, P., Xiang, M.Q., Zhang, B.L.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \({\mathbb{R}}^N\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)
Xiang, Q.M., Zhang, B.L., Ferrara, M.: Existence of solutions for Kirchhoff type problem involving the nonlocal fractional p-Laplacian. J. Math. Anal. Appl. 424, 1021–1041 (2015)
Wang, L., Zhang, B.: Infinitely many solutions for Schrodinger-Kirchhoff type equations involving the fractional \(p\)-Laplacian and critical exponent. Electron. J. Differ. Equ. 339, 18 pages (2016)
Zhang, L., Chen, Y.: Infinitely many solutions for sublinear indefinite nonlocal elliptic equations perturbed from symmetry. Nonlinear Anal. Theory Methods Appl. Ser. A Theory Methods. 151, 126–144 (2017)
Zhen, M., Zhang, B.: The Nehari manifold for fractional \(p\)-Laplacian system involving concave-convex nonlinearities and sign-changing weight functions. Complex Var. Ellipt. Equ. 66(10), 1731–1754 (2021)