| Abstract:This doctoral dissertation mainly studies three types of variable exponential problems:Schr?dinger-Poisson problem,p(x)-Laplacian-Like problem and Schr?dinger-Kirchhoff problem.It has a wide applications in electromagnetic fields,capillary phenomena,non-Newtonian mechanics and other problems.Since the variable exponential elliptic operator is not homogeneous,this makes some conclusions that the classical function space holds cannot be extended to the variable exponential space accordingly,which brings difficulties to the research.For instance,the translation technique of Schr?dingerPoisson systems with variable exponents fails,as compared with constant exponents.Furthermore,the equation no longer holds in a point-wise sense due to the appearance of nonlocal Poisson terms and Kirchhoff type functions of the form M(·).This is another hurdle.The main contents of the full text are as follows:First,we studies the Schr?dinger-Poisson problem.By weakening the usual monotonic condition,a new estimate suitable for the variable exponential Schr?dinger-Poisson problem is established and overcome the difficulty of Palias-Samle sequence noncompact,because of the lack of a global AmbrosettiRabinowitz condition.The existence of the ground state solution for this problem is proved by the non-Nehari manifold method.Secondly,we studies the p(x)-Laplacian-Like problem on the bounded region Ω?RN.By defining its energy functional and proving that its derivative is a mapping of the type(S+),we overcome the difficulties due to the non-local term M(u)appears,which leads to weak convergence(?)in W1,p(x)(Ω)cannot be directly derived M(un)→ M(u).And the existence of its non-trivial solution is proved by the mountain pass theorem which contains the Ceremi condition.Next,we studies the Schr?dinger-Kirchhoff problem.By making full use of the properties for the Kirchhoff function M,we overcome the difficulties due to the lack of compactness in the full space RN and the lack of translation invariance in variable exponential space.Thus,the existence of its nontrivial solution is proved.As one of the applications of the above three equations,we studies the plane nonautonomous Kirchhoff problem when p(x)≡ 2.By introducing a new estimation method to accurately estimate the level value of its minimax energy functional.Because there are double difficulties arising from the appearance of nonlocal terms ‖▽u‖22 and the nonlinearity which is of critical growth of(MT)τin whole Euclidean space R2.Thus,the compactness of the Cerami sequence is restored and the existence of its ground state solution is proved.Finally,as the second application of the above three equations,we studies the plane Choquard problem.By accurately estimating the mountain pass level value of its limit equation,we used the idea of approximation to overcome the difficulty,due to its potential function is not periodic and leads to the non-local convolution term is not weak sequence convergence.Furthermore,since the weak limit of the minimization sequence of the Nehari manifold may be trivial.By applying the(MT)τ inequality,the weak limit is proved to be non-trivial and so we got the existence of the minimum energy solution. |
