Multiple Solutions Of Elliptic Systems Involving Concave-convex Nonlinearities And Sign-changing Weight Functions

This paper is devoted to study the existence and multiplicity of nontrivial nonnegative solutions for a class quasilinear elliptic systems involving concave-convex nonlinearities and sign-changing weight functions. First of all, we discuss the following elliptic systems where Ω(?) RN is a bounded domain;1≤q<p; λ>0:△pw=div(|▽w|p-2▽w;) denotes the p-Laplacian operator; a1(x), a2(x)(?) Lr(Ω) are allowed to change sign, here r>(?), p*=+∞if N≤p and p*=(?) if1<p<N; z=(u,v), F(x,z)(?) C1(Ω-×(R+)2,R+). Assume that F(x,z) satisfies the subcritical growth condition or F(x, z) is positively homogeneous of degree p*, using the Ekeland's variational principle and the mountain pass theorem, we obtain that the existence of at least two nontrivial nonnegative solutions for this elliptic systems. In addition, under the hypotheses of the nonnegativity of the weight functions, we obtain that the nontrivial nonnegative solutions of this elliptic systems are positive solutions by the maximum principle.Subsequently, we consider the following critical quasilinear elliptic systems with lower-order negative perturbations where Ω(?)RN is a bounded domain；N>p2,1<r≤(?)<q<p,入>0,μ>O；F,G,H∈C1((R+)2,R+)are homogeneous of degree P*,q and r,respectively. Under the assumption that the minimum of G and H greater than zero on the unit circle,we obtain that this systems have at least three nontrivial nonnegative solutions by using the Ekeland's variational principle and the mountain pass theorem.At last,we study the following singular and degenerate elliptic systems with critical Hardy-Sobolev exponents where Lω:=div(|x|-ap|▽ω|p-2▽ω)-μ(?) denotes the elliptic perator, N>p(a+1),λ>0,1<q<p<N,0≤μ<μ,μ△=|(?)-a)p,0≤a<(?),a≤b<a+1,p*(a,b):(?)is Hardy-Sobolev cxpo-nents,(?):(?),a1(x),a2(x)∈L(?)(Ω),b(x)∈L∞(Ω)are allowed to change sign,F∈C1((R+)2,R+)is positively homogeneous of degree p*(a,b). Under some constraints of the parameters,we obtain that the existence of at least two nontrivial nonnegative solutions for this elliptic systems by using the Ekeland:s variational principle and the mountain pass theorem. In addition,under the hypotheses of the nonnegativity of the weight functions,we obtain that the non-trivial nonnegative solutions of this elliptic systems are positive solutions by the maximum principle.