Nonlinear Problems With Magnetic Field Or Double Phase

This dissertation mainly studies the nonlinear magnetic field and double phase problems which are of great significance in the fields of math-ematics and physics.Using the tools of calculus of variations（critical point theory）,algebraic topology（category theory）and nonlinear analysis,and com-bining with the techniques of truncation,perturbation and comparison,it estab-lishes the properties of the existence,nonexistence,multiplicity,concentration,asymptotic behavior and bifurcation of the solutions of the related problems.This dissertation first summarizes the research background and signifi-cance,research status and sources of the problems,and briefly describes the relevant preparatory knowledge.Secondly,under the suitable assumptions,the existence,nonexistence,multiplicity,asymptotic behavior and bifurcation of solutions for several kinds of local problems driven by magnetic Laplace operator are studied respectively by using variational methods.Then,under some natural assumptions,the existence of infinite solutions to the Kirchhoff problem with magnetic potential is proved by using the sym-metric mountain pass theorem.The existence of solutions for magnetic system-s with Stein-Weiss convolution nonlinearity is obtained by applying mountain pass theorem,and the Brezis-Lieb type lemma with Stein-Weiss convolution reaction term is obtained.Finally,we study the following fractional double phase problems with non-equilibrium growth and critical or supercritical nonlinear terms in R^N:（?）whereis a positive parameter,（?） is the fractional t-Laplace operator,（?） and （?） are continuous functions.The concentration and multiplicity of solutions to this problem are proved by using the methods of topology,variation and analysis.