Existence Of Ground State Solutions To Several Kinds Of Schr(?)dinger Equations

Schr （?）dinger equation is the core equation of quantum mechanics,which reveals the basic laws of material movement in the micro-physical world,and has great significance in the history of physics.It is known as one of the “Ten Classical Formulas” and is the most widely used and influential formula in the world of atomic physics literature.The Choquard equation,magnetic Schr （?）dinger equation,Schr （?）dinger-Poisson system and Schr （?）dingerKirchhoff equation studied in this paper are derived from Schr （?）dinger equation,which have profound physical background and wide application value.This doctoral dissertation mainly studies the existence of ground state solutions of the above equations by variational method,introduces some new analytical skills and research strategies,and improves and extends the related works in the existing literature.The main results of this paper are as follows:In Chapter 1,we introduce the physical background,research significance,source,research status,main work,innovation and related preparatory knowledge of the studied problem.In Chapter 2,we study the logarithmic Choquard equation（?）By introducing some mathematical strategies and analytical skills,we prove the existence of nontrival solutions of mountain-pass type,least energy solutions and ground state solutions,which improves and extends the existing related works.In Chapter 3,we investigate the following nonlinear magnetic Schr （?）dinger equation:（?）where V is the electric potential and A is the magnetic potential.We prove the existence of ground state solutions both in the indefinite case with subcritical exponential growth and in the definite case with critical exponential growth.By using subtle estimates and establishing a new energy estimate inequality in complex field,we overcome the difficulty brought from the presence of magnetic field and weaken the Ambrosetti-Rabinowitz type condition and the strict monotonicity condition,which are commonly used in the indefinite case.Furthermore,in the definite case,we introduce a Moser type function involving magnetic potential and some new analytical techniques,which can also be applied to related magnetic elliptic equations.Our results extend and complement the present ones in the literature.In Chapter 4,we are concerned with a planar Schr （?）dinger-Poisson system involving Stein-Weiss nonlinearity（?）and its degenerate case （?） where β≥0,0<μ<2,2β+μ<2,V∈C（R²,R） and f is of exponential critical growth.By combining variational methods,Stein-Weiss inequality and some delicate analysis,we derive the existence of ground state solution for the first system.Under some mild assumptions,we introduce the Pohozaev identity of the equivalent equation of the second system and use Jeanjean’s monotonicity method to achieve the existence of nontrivial solution for the second system.In Chapter 5,we are concerned with the following Schr （?）dinger-Kirchhoff equation:（?）where a,b are positive constants,V∈C（R²,（0,+∞））is a trapping potential,and f has critical exponential growth of Trudinger-Moser type.By developing some new analytical approaches and techniques,we prove the existence of nontrivial solutions and least energy solutions.Without any monotonicity conditions on f,we also give the mountain pass characterization of the least energy solution by constructing a fine path.In particular,we remove the common restriction on lim inf_t→+∞（tf（t））/e^α₀t2,which is crucial in the literature to overcome the loss of the compactness caused by the critical exponential nonlinearity.Our approach could be extended to other classes of critical exponential growth problems with trapping potentials.