We are concerned with a nonlinear elliptic equation,involving a Kirchhoff type nonlocal term and a potential V(x),onℝ3.As is well known that,even in,H_(r)^(1)(R^(3)),the nonlinear term is a pure power form of∣u∣^(p−...We are concerned with a nonlinear elliptic equation,involving a Kirchhoff type nonlocal term and a potential V(x),onℝ3.As is well known that,even in,H_(r)^(1)(R^(3)),the nonlinear term is a pure power form of∣u∣^(p−1)u and V(x)≡1,it seems very difficult to apply the mountain-pass theorem to get a solution(i.e.,mountain-pass solution)to this kind of equation for all p∈(1,5),due to the difficulty of verifying the boundedness of the PalaisSmale sequence obtained by the mountain-pass theorem when p∈(1,3).In this paper,we find a new strategy to overcome this difficulty,and then get a mountain-pass solution to the equation for all p∈(1,5)and for both V(x)being constant and nonconstant.Also,we find a possibly optimal condition on V(x).展开更多
该文主要考虑一类在R^(3)上带有Kirchhoff型非局部项的非线性椭圆方程−(a+b∫_(R)^(3)|∇u|^(2))Δu+V(x)u=Q(x)|u|^(p−1)u,x∈R^(3),(0.1)其中a,b>0是常数,p∈(1,5),V(x)和Q(x)均为L^(∞)(R^(3))函数.由于非局部项的出现,若按经典的...该文主要考虑一类在R^(3)上带有Kirchhoff型非局部项的非线性椭圆方程−(a+b∫_(R)^(3)|∇u|^(2))Δu+V(x)u=Q(x)|u|^(p−1)u,x∈R^(3),(0.1)其中a,b>0是常数,p∈(1,5),V(x)和Q(x)均为L^(∞)(R^(3))函数.由于非局部项的出现,若按经典的思路来应用山路引理得到这类方程的解(即山路解),必须要求3≤p<5.当p∈(1,3)时,应用山路引理的困难在于无法验证(PS)序列的有界性.为克服该困难,文献[Acta Math Sci,2025,45B(2):385-400]通过引入新的技巧证明了方程(0.1)在Q(x)≡1时对p∈(1,5)有山路解,并讨论了山路解与基态解的关系.该文拟在克服V(x)和Q(x)的相互影响下,将文献[Acta Math Sci,2025,45B(2):385-400]中的结果推广到Q(x)■1的一般情形.展开更多
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文摘We are concerned with a nonlinear elliptic equation,involving a Kirchhoff type nonlocal term and a potential V(x),onℝ3.As is well known that,even in,H_(r)^(1)(R^(3)),the nonlinear term is a pure power form of∣u∣^(p−1)u and V(x)≡1,it seems very difficult to apply the mountain-pass theorem to get a solution(i.e.,mountain-pass solution)to this kind of equation for all p∈(1,5),due to the difficulty of verifying the boundedness of the PalaisSmale sequence obtained by the mountain-pass theorem when p∈(1,3).In this paper,we find a new strategy to overcome this difficulty,and then get a mountain-pass solution to the equation for all p∈(1,5)and for both V(x)being constant and nonconstant.Also,we find a possibly optimal condition on V(x).
文摘该文主要考虑一类在R^(3)上带有Kirchhoff型非局部项的非线性椭圆方程−(a+b∫_(R)^(3)|∇u|^(2))Δu+V(x)u=Q(x)|u|^(p−1)u,x∈R^(3),(0.1)其中a,b>0是常数,p∈(1,5),V(x)和Q(x)均为L^(∞)(R^(3))函数.由于非局部项的出现,若按经典的思路来应用山路引理得到这类方程的解(即山路解),必须要求3≤p<5.当p∈(1,3)时,应用山路引理的困难在于无法验证(PS)序列的有界性.为克服该困难,文献[Acta Math Sci,2025,45B(2):385-400]通过引入新的技巧证明了方程(0.1)在Q(x)≡1时对p∈(1,5)有山路解,并讨论了山路解与基态解的关系.该文拟在克服V(x)和Q(x)的相互影响下,将文献[Acta Math Sci,2025,45B(2):385-400]中的结果推广到Q(x)■1的一般情形.
