Existence Of Solutions For Schr (?) Dinger - Poisson System And Its Related Problems

In this thesis, by using variational methods, we mainly study the existence of nontrivial solutions, nodal solutions and multiplicity results of a class Schrodinger-Poisson system and its related problems, furthermore, also, the properties of the solutions are discussed. Our results obtained in the thesis are included in the following five parts.In Chapter2. we study the following Schrodinger equation with indefinite potential-△u+V(x)u=Q(x)f(u), x∈RN, where V(x) is sign-changing, infx∈RN,Q(x)>0, the nonlinearity f(u) is superlinear satisfying some weak monotonicity conditions. By the approximation of bounded domains, we construct a special minimizing sequence constrained on the Nehari manifold, and then, together with the concentration compactness principle and the estimation on the energy of functional, we prove the existence of least energy nodal solutions of the above equation.In Chapter3, we study the following Schrodinger-Poisson system where p∈(4,6), A>0is a parameter, infx∈RN3V(x)>0and K. Q satisfy some proper assumptions. Actually, the system can be seen as the Schrodinger equation studied in Chapter2in R3coupled with a nonlocal term (which given by the second equation of the system). After carefully analyzing the interaction of the nonlocal term on the Nehari manifold, by using the variational methods combining with the estimation on the energy of functional, we prove the existence of least energy nodal solutions provided λ>0small. Moreover, the asymptotical behaviors of the nodal solutions about λ and the "energy doubling" properties are also considered.In Chapter4, we study the following Schrodinger-Poisson system with radial potentials vanishing at infinitywhere λ>0is a parameter, V～|x|-α, Q～|x|-β(α>0, β>0), f(u) is asymptotically linear at infinity. By using the cut-off technique and the variational methods, we prove the existence and multiplicity of positive solutions for the system when λ is small and α,β belong to different ranges. We also study the asymptotical behaviors of the solutions as λ→0+and the exponential decay of the solutions as|x|→∞.In Chapter5, we study the following special Schrodinger-Poisson systemwhere ω∈R is a parameter and p∈(2,6). By using the standard scaling arguments, we prove the existence of normalized solution provided p e (2,3), that is, for each p small, the system has the solution (ωρ,uρ) with‖uρ‖2=ρ, which gives the affirmative answer to the open problem proposed by Bellazzini and Siciliano. Moreover, the asymptotical behaviors of the normalized solution about p are also studied.In Chapter6, we study the following Kirchhoff equationwhere p∈(2,6), λ>0is a parameter and V(x) satisfies some suitable assumptions. By using the the method of Nehari manifold and the variational methods, we obtain that the existence and non-existence results when λ and p belong to different ranges. Moreover we also get the "energy doubling" properties of the least energy nodal solutions.