Existence Of Solutions For Several Semi-linear Elliptic Equations (Systems) On The Whole Space

In this dissertation, firstly, we study the multiplicity of solutions and the existence of ground state solutions for Schrodinger equations and Kirchhoff equations involving concave-convex nonlinearities in the whole space. Secondly, we consider the existence of prescribed L2-solutions for Kirchhoff systems and Schrodinger systems with saturable nonlinearities. The main results are as follows:In Chapter 1, we review the proceeding of variational method, introduce the relevant critical point theory, summarize our main work and state the innovation.In Chapter 2, we consider Schrodinger equations with a bounded potential function and concave-convex nonlinearities. With the method of the decomposition of Nehari manifold, we obtain the multiplicity of solutions and the existence of ground state solutions for Schrodinger equations. These results generalize the present results which are about Schrodinger equations with constant potential and concave-convex nonlinearities.In Chapter 3, we are concerned with Kirchhoff type problems with concave and convex nonlinearities. The super-triple power concave term, and the super-linear power concave term and the general concave term are considered in the whole space, respectively. Under suitable conditions, we prove that the Kirchhoff problem has at least two solutions.In Chapter 4, we are devoted to considering Kirchhoff type systems. Under different type of potentials, we respectively consider L2-subcritical case and L2-critical case, and prove the existence of solutions with prescribed L2-norm. These results generalize the present result-s which are about the solutions with prescribed L2-norm for the single Kirchhoff equation.Moreover, we consider the existence of solution with prescribed L2-norm for Kirchhoff system with negative coupling constant.In Chapter 5, under the general constraint, we consider Schrodinger systems with sat-urable nonlinearities. With the help of the properties of prescribed L2-solutions for the scalar equation, we can exclude the semi-trivial solutions and obtain the existence of solutions with prescribed L2-norm. Moreover, our results can be extended to the systems with more than two components and applied to the case of square-root nonlinearities.In Chapter 6, we summarize the main results and make a plan for our future research.