Multiplicity of solutions for a class of fractional Emphasis Type="Italic"p/Emphasis-Kirchhoff system with sign-changing weight functions

摘要

In this paper, we investigate the fractional p-Kirchhoff -type system: $$egin{aligned} extstyleegin{cases} M (int_{{ mathbb {R} }^{2N}}rac{ert u(x)-u(y) ert ^{p}}{ert x-y ert ^{N+ps}},dx,dy )(- Delta )^{s}_{p}u=mu g(x)ert u ert ^{eta -2}u+rac{a}{a+b}h(x)ert u ert ^{a-2}uert v ert ^{b},&mbox{in } Omega , M (int_{{ mathbb {R} }^{2N}}rac{ert v(x)-v(y) ert ^{p}}{ert x-y ert ^{N+ps}},dx,dy )(- Delta )^{s}_{p}v=sigma f(x)ert v ert ^{eta -2}v+rac{b}{a+b}h(x)ert v ert ^{b-2}vert u ert ^{a},&mbox{in } Omega , u=v=0,&mbox{in } { mathbb {R} }^{N}setminus Omega , end{cases}displaystyle end{aligned}$$ where (Omega subset mathbb{R}^{N}) is a smooth bounded domain, ((-Delta )^{s}_{p}) is the fractional p-Laplacian operator with (0 s1p) and (ps N ). (a1), (b1) satisfy (2 a+b p_{s}^{*}). (1eta p_{s}^{*}), (p_{s}^{*}=rac{Np}{N-ps}) is the fractional critical exponent. ??, ?? are two real parameters. (M(t)=k+lambda t^{au }), (k0), ??, (au geq 0), (au =0) if and only if (lambda =0). The weight functions g, f, h change sign in ?? and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that (2 a+b pleq p(au +1)eta p_{s}^{*}) and ((mu ,sigma )) belongs to a certain subset of (mathbb {R} ^{2}). Also, by using the mountain pass theorem, we prove that there exist (lambda _{1}geq lambda_{0}) such that the system admits at least a nontrivial solution for (lambda in (0,lambda_{0})) and no nontrivial solution for (lambda lambda_{1}) under the assumptions (mu =sigma =0) and (p a+bmin {p(au +1),p_{s}^{*}}).KeywordsFractional p-Kirchhoff system??Multiplicity??Sign-changing weight functions??Nehari manifold??Mountain pass theorem??MSC35R11??35A15??35J60??1 IntroductionIn this paper, we investigate the multiplicity of solutions to the following fractional elliptic system: $$egin{aligned} extstyleegin{cases} M (int_{{ mathbb {R} }^{2N}}rac{ert u(x)-u(y) ert ^{p}}{ert x-y ert ^{N+ps}},dx,dy )(-Delta )^{s}_{p}u=mu g(x)ert u ert ^{eta -2}u+rac{a}{a+b}h(x)ert u ert ^{a-2}uert v ert ^{b } &mbox{in } Omega , M (int_{{ mathbb {R} }^{2N}}rac{ert v(x)-v(y) ert ^{p}}{ert x-y ert ^{N+ps}},dx,dy )(-Delta )^{s}_{p}v=sigma f(x)ert v ert ^{eta -2}v+rac{b}{a+b}h(x)ert v ert ^{b-2}vert u ert ^{a } &mbox{in } Omega , u=v=0&mbox{in } { mathbb {R} }^{N}setminus Omega , end{cases}displaystyle end{aligned}$$ (1.1) where (Omega subset mathbb{R}^{N}) is a smooth bounded domain, (0 s1p) and (ps N). (a1), (b1) satisfy (2 a+b p^{st }_{s}). (1eta p_{s}^{*}), (p_{s}^{*}=rac{Np}{N-ps}) is the fractional critical exponent. ??, ?? are two real parameters. (M(t)=k+lambda t^{au }), (k0), ??, (au geq 0), (au =0) if and only if (lambda =0). The weight functions g, f, h change sign in ?? and satisfy further assumption which will be given later. ((-Delta )^{s}_{p}) is the fractional p-Laplacian operator defined on smooth functions by $$ (-Delta )^{s}_{p}m(x)=2lim_{arepsilon ightarrow 0^{+}} int_{{ mathbb {R} }^{N}setminus B_{arepsilon }(x)} rac{ert m(x)-m(y) ert ^{p-2}(m(x)-m(y))}{ert x-y ert ^{N+ps}},dy,quad xin { mathbb {R} } ^{N}. $$Problem (1.1) is related to the stationary analogue of the following Kirchhoff model: $$ ho u_{tt}- iggl(rac{p_{0}}{h}+rac{E}{2L} int_{0}^{L}u_{x}^{2},dx iggr)u _{xx}=0 $$ which was proposed by Kirchhoff in 1883 as a generalization of the well-known Da??Alembert wave equation for free vibrations of elastic strings, where ??, (p_{0}), h, E, L are constants which represent some physical meanings, respectively. Indeed, Kirchhoffa??s model takes into account the changes in length of the string produced by transverse vibrations. In particular, Kirchhoffa??s equation models several physical and biological systems, we refer to [1] for more details. The Kirchhoff type equation and system have attracted attention and have been discussed by many authors, we refer to [9, 14, 21, 22, 23, 29] and the references therein.Up to now, a great attention has been paid to the study of the fractional Laplacian equation and system, see, for example, [4, 5, 10, 11, 13, 18, 22, 23, 29, 30, 33]. In particular, the fractional and nonlocal operators of elliptic type arise in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastic stabilization of L??vy processes, see [2, 8]. The literature on fractional nonlocal operators and their applications is very interesting and quite large, see, fo