Variational Minimization Problems For A Class Of Kirchhoff Type Elliptic Equations

This paper mainly studies a class of Kirchhoff elliptic equations and relevant variational problems in RN(N?1).When the potential function satisfies V(x)?0 and it's unbounded at infinity(i.e.(?)V(x)?+?),there are many improve and perfect results on the existence of solutions(i.e.minimal elements)to this kind of variational problems.There are also some results about the asymptotic behavior of minimal elements to the nonlinear growth index,but these results only show that the blow-up point of the minimal elements falls on the bottom of the potential function(i.e.{x?RN:V(x)=0}),which means they can't reflect the influence of the specific characteristics of the bottom of the potential function to the blow-up point of the minimal elements.Recently,for the Gross-Pitaevskii(GP)equations in R2,when the bottom of the potential function is an ellipse,it has been proved that the minimal elements of their variational problems will blow up at the end points of the long axis of the ellipse of the bottom of the potential function.The main purpose of this paper is to extend the results of GP equations to a class of Kirchhoff type equations in RN.Since the energy functionals are difficult to calculate directly,we need combine the result of Kirchhoff type equations under the condition that the potential function identically equal to zero.By using the variational method and the corresponding energy analysis,we prove that the solutions of the constrained variational problems corresponding to the above Kirchhoff type equations will blow up at the end points of the long axis of the ellipse of the bottom of the potential function when the onlinear growth index approaches a certain critical index,and we also get the exact blow-up rate.The proof ideas of the main results are as follows:firstly,we studies the ellipsoidal potential function and get the minimum value of the model of the gradient and the reachable condition.We also put out two integral properties of the unique positive solution of the limit equation.Secondly,by using the reachable condition,two integral properties,and combining properly selected test functions,we establish the relationship between the energy functionals of this kind of Kirchhoff type equations and the energy functionals of the Kirchhoff type equations under the condition that the potential function identically equal to zero.Finally,we get the accurate estimate of the limit of the energy functionals and prove that the solutions of the constrained variational problems will blow up at the end points of the long axis of the ellipse of the bottom of the potential function.