Contents 《博士后文库》序言 Preface Chapter 1 Schr.dinger-Poisson Equations 1 1.1 Schr.dinger-Poisson equations with sign-changing potential 1 1.1.1 Introduction and main results 1 1.1.2 Variational setting and compactness condition 4 1.1.3 Proofs of main results 15 1.2 Nonlinear Schr.dinger-Poisson equations with sublinear case 21 1.2.1 Introduction and main results 21 1.2.2 Variation set and proofs of main results 24 1.3 Nonlinear term involving a combination of concave and convex terms.27 1.3.1 Existence of solution.27 1.3.2 The multiplicity of solutions.31 1.3.3 The proof of Theorem 1.6 35 Chapter 2 Klein-Gordon-Maxwell System 37 2.1 Two solutions for nonhomogeneous Klein-Gordon-Maxwell system 37 2.1.1 Introduction and main results 37 2.1.2 Variational setting and compactness condition 41 2.1.3 Proofs of main results 54 2.2 The primitive of the nonlinearity f is of 2-superlinear growth at infinity 61 2.2.1 Introduction and main results 61 2.2.2 Variational setting and compactness condition 64 2.2.3 Proofs of main results 70 2.3 Proofs of results 71 2.3.1 The proof of Theorem 2.5 74 2.3.2 The proof of Theorem 2.6 79 2.4 Ground state solutions for critical Klein-Gordon-Maxwell equations 85 2.4.1 Introduction and main results 85 2.4.2 Variational setting and preliminaries 86 2.4.3 The Nehari manifold N 89 2.4.4 Proofs of main results 92 Chapter 3 Klein-Gordon Equation Coupled with Born-Infeld Theory 96 3.1 Introduction and main results.96 3.2 Variational setting and compactness condition 100 3.3 Proofs of main results 106 Chapter 4 Localized Nodal Solutions for Kirchhoff Equations 108 4.1 Introduction and main results.108 4.2 Variational setting and compactness condition 113 4.3 Existence of multiple sign-changing critical points of Γ" 118 4.4 The proof of Theorem 4.1 121 4.5 Proof of Proposition 4.2 134 Chapter 5 Infinitely Many Sign-Changing Solutions 138 5.1 Sign-changing solutions for an elliptic equation involving critical Sobolev and Hardy-Sobolev exponent 138 5.1.1 Introduction and main results 138 5.1.2 Preliminaries 142 5.1.3 Auxiliary operator and invariant subsets of descending flow. 144 5.1.4 The proof of Theorem 5.1146 5.2 Infinitely many sign-changing solutions for an elliptic equation involving double critical Hardy-Sobolev-Maz’ya terms.152 5.2.1 Introduction and main results 152 5.2.2 Preliminaries 156 5.2.3 Auxiliary operator and invariant subsets of descending flow. 158 5.2.4 The proof of Theorem 5.3 160 5.3 Sign-changing solutions for Hardy-Sobolev-Maz’ya equation involving critical growth 166 5.3.1 Introduction and main results 166 5.3.2 Preliminaries 170 5.3.3 Auxiliary operator and invariant subsets of descending flow. 173 5.3.4 The proof of Theorem 5.5 175 Chapter 6 Multiple Solutions for Nonhomogeneous Choquard Equations 182 6.1 Introduction and main results.182 6.2 Variational setting and compactness condition 185 6.3 Local minimum solution 192 6.4 The proof of Theorem 6.2 202 References 218 Index 232 编后记 233
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