Existence Of Minimizers To Nonlinear Kirchhoff Constrained Variational Problem

The Kirchhoff equations are used to describe the free vibration model of elastic strings.Due to the existence of non-local terms,many interesting variational problems arise,which have attracted extensive attention from researchers in related fields.Thesis mainly studies the existence of minimizers for nonlinear Kirchhoff constrained variational problems.The related research work and innovative results are as follows:Firstly,we consider the existence and non-existence of minimizers of Kirchhoff constrained variational problems in the combined nonlinear case of constrained critical exponents and low-order perturbation terms.Specifically,for the Kirchhoff constrained variational problem,when the nonlinear term contains only one power term and the exponent is constrained critical,it can be known from the existing literature that there is no minimizer.In the case of constrained critical exponents,by adding a low-order perturbation term,we considered the influence of exponents and coefficients of the perturbation term on the existence of minimizers of the Kirchhoff-type constrained variational problem with constrained critical exponents via using scaling techniques,the concentration-compactness lemma and the Pohozaev identity.Through the refined energy estimates,we discussed the limit behavior of the minimal energy and minimizers when the exponents of perturbation term tend to constrained critical exponents.Furthermore,it is proved that the minimizer of the constrained variational problem is the ground state of the corresponding Kirchhoff equation under certain conditions.Secondly,thesis uses constrained variation theory and symmetric decreasing rearrangement technique to consider the existence and non-existence of Schwarz symmetric minimizers for a class of Kirchhoff constrained variational problems in the case of general nonlinear terms.A detailed and clear division is obtained for the relationship between the exponents of the nonlinear term and the parameters in the constrained problem and the existence of the Schwarz symmetric minimizers of the minimization problem.In general,thesis considers the existence of minimizers to Kirchhoff constrained variational problems in the combined nonlinear case where the general nonlinear term satisfies constrained subcritical growth or constrained critical growth plus subcritical growth.The conclusion of thesis enriches the results of the existence and limit behavior of the minimizer of the Kirchhoff constrained variational problem,which provides upfront preparation for the discussion of Kirchhoff constrained variational problem of other combinatorial nonlinearities or general nonlinearities satisfying the critical or supercritical case.