非线性Kirchhoff型椭圆方程的最低能量解

Least Energy Solution for Nonlinear Kirchhoff Type Elliptic Equation

该文讨论以下非线性Kirchhoff型椭圆方程非平凡解和非负最低能量解的存在性■其中p∈(3,5), a,b> 0, V∈C(R^3,R^+)并且■V(x)=∞.通过变分方法,该文首先证明了对于任何b> 0,存在δ(b)> 0,使得当μ_1≤μ<μ1+δ(b)时,方程(0.1)有非平凡解.其次,进一步证明了存在δ_1(b)∈(0,δ(b)),当μ_1<μ<μ_1+δ_1(b)时,方程(0.1)有非负的最低能量解,这里μ_1是Schrodinger算子-△+V的第一特征值.最后利用对称山路引理证明了对任意的μ∈R,方程(0.1)存在无穷多个非平凡解.

In this paper, we study the existence of nontrivial solution and nonnegative least energy solution for the following nonlinear Kirchhoff type elliptic equation{-(a+b∫R^3∣▽u∣^2dx)△u+V(x)u=μu+∣u∣^p-1u,x∈R^3,u∈H^1(R^3),x∈R^3(0.1) where p∈(3,5), a,b>0, V∈C(R3,R+)and lim|x|→+∞V(x)=∞. By using variational methods, firstly we prove that for any b>0, there exists δ(b)>0such that problem (0.1) with μ1≤μ<μ1+δ(b)has a nontrivial solution, where μ1 denotes the first eigenvalue of the Schrodinger operator-△+V. Secondly, we show that there exists δ1(b)∈(0,δ(b))such that problem (0.1) with μ1<μ<μ1+δ1(b)has a nonnegative least energy solution. Finally, by using the symmetric Mountain Pass lemma we prove that problem (0.1)has infinitely many nontrivial solutions for any μ∈R.