R^N中带冲突非线性项的拟线性椭圆方程的多解性

Multiplicity of Solutions for a Quasilinear Elliptic Equation with Conflicting Nonlinearities on R^N

考虑 RN中含正参数 μ的拟线性椭圆方程- div(| u| p -2 u) + | u| p-2 u=q(x) | u| α-2 u-μr(x) | u| β-2 u,u∈ W1,p(RN) ,其中 :1<p<α<β<p* ,q∈ L∞ (RN)∩ L ββ-α(RN) ,q(x) >0 ,r∈ L∞ (RN) ,r(x)≥ d>0 .证明了当 μ充分大时该方程无非零解 ,而当μ充分小时该方程有足够多的分别具有正能量与负能量的解 .

Consider the quasilinear elliptic equation on R N with a positive parameter μ of form -div(|u| p-2u)+|u| p-2u=q(x)|u| α-2u-μr(x)|u| β-2u,u∈W 1,p(R N), where 1<p<α<β<p *,q∈L ∞(R N)∩L ββ-α(R N),q(x)>0,r∈L ∞(R N),r(x)≥d>0.We prove that the equation has no nontrivial solutions when μ is large enough,and for every positive integer k,there exists μ k>0 such that for μ∈(0,μ k) the equation has at least two kinds of solutions,{±u i∶i=1,2,...,k} and {±v i∶i=1,2,...,k}, at negative and positive energies respectively.