Existence Of Multiple Solutions For Two Types Of Differential Equation

Using the critical point theory,we study the existence and multiple solutions of two kinds of differential problems.This thesis is divided into five chapters.In the first chapter,we describe the background,review the recent advancements of double phase problems and Hamiltonian systems.At the end,we provide the main work and innovation of this thesis.In the second chapter,we give the basic concepts,theorems,related propositions,focus on the Musielak-Orlicz-Sobolev space for the study of double phase problems and generalized Lebesgue-Sobolev space for p(t)-Laplacian Hamiltonian systems.In the third chapter,we investigate the existence of solutions for the following double phase problems with concave-convex terms where Ω?RN is a bounded domain with Lipschitz boundary,N≥2,p*=Np/N-p,1<p<q<min{N,p*},q/p≤1+1/N,a:Ω→[0,+∞)is Lipschitz continuous.By means of the mountain pass lemma and Ekeland variational principle,we obtain that there are at least two nontrivial solutions to this problem.What’s more,if the non-linearity is odd in u,we also get infinitely many solutions by using symmetry mountain pass theorem.In the fourth chapter,we investigate the existence of infinitely many periodic solutions for the Hamiltonian systems-(|u’(t)|P(t)-2u’(t))’=▽F(t,u(t))for a.e.t∈R,where p(t)∈C(O,T;R+),p(t)=p(t+T),(?).By virtue of several auxiliary functions,we obtain a series of new super-p+ growth and asymptotic-p+growth conditions.Using the minimax methods in critical point theory,some multiplicity theorems are established,which unify and generalize some known results in the literature.Meanwhile,we also present an example to illustrate our main results are new even in the case p(t)≡p=2.