Positive Ground State Solutions For A Nonlinear Kirchhoff-type Equation In R~3

In this thesis, we consider positive ground state solution of the following non-linear Kirchhoff-type equation where b is a positive constant, 2 < p < 6.Firstly, we study the equation , assume that the potential V(x) = V∞, using the variational methods, we deal with the existence of positive state solutions for Degenerate Kirchhoff-type problem with nonlinear team in R3. So we obtained the following theorem:Theorem 1 Assume b, V∞ are positive constants, and 2 < p < 6, so equation (0.0.2) has a positive state solution in H1(R3).Secondly, we study the equation (0.0.1), assume that the potential V verifies the following hypotheses:(V1)V∈C(R3,R) is weakly differentiate and satisfies ((?)V(x),x) E L∞(R3) (?) L2/3(R3) and Vp-2/2V(x)-((?)V(x)x)≥0,a.e.x∈R3{V2) for almost every x ∈ R3, 0 < inf V(x)≤< V(x)≤ lim V(x)=V∞< +∞.So we obtained the following theorem:Theorem 2 Assume b is a positive constant,2<p< 6 and V verifies (Vi)-(V2), so equation (0.0.1) has a positive state solution in H1(R3).