| In this dissertation, we study the generalized quasilinear Schrodinger equations with subcritical, critical, supercritical growths by variational method. Under some suitable conditions about potential function and nonlinearity, we study the existence, multiplicity and concentration of solutions.In the first chapter, we outline the background and introduce the research status and the structure of the dissertation.In the second chapter, suppose that the potential function is periodic. We employ the Nehari manifold method to obtain ground state solutions and infinitely many geometrically distinct solutions for the subcritical problem Moreover, we prove that the equation has a positive solution, a negative solution and a sequence of high energy solutions. Our result generalizes the corresponding results in Xiangdong Fang and Szulkin (2013) and Xian Wu (2014), respectively.In the third chapter, we study the existence, multiplicity and concentration of solutions for the generalized quasilinear Schrodinger equations with critical growth. At first, under the monotone condition weaken than the AR condition and usual monotone condition, we study the existence of nontrivial solutions for the following equation by variation method. Then, we obtain the existence of ground state solutions for the critical problem with a parameter ε Moreover, using category theory we prove that the number of solutions for the above equation is no less than the category of the set of global minimum points of the potential function V. At last, we give the concentration of solutions for the above equations by Moser iteration. We point out that we firstly investigate the number of solutions and the concentration of solutions for this class equations.In the last chapter, we employ cut-off techniques and Moser iteration to prove the existence of nontrivial solutions for critical or supercritical problem Our result generalizes Theorem 1.1 in Youjun Wang (2015). |
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