The Existence Of Solutions For Singularly Perturbed Equation And The Estimation Of Critical Value

This thesis mainly studies the existence of solutions for some singularly per-turbed equations and the estimation of critical value. The thesis consists of four chapters.In the first chapter, we introduce the research background and our results.In the second chapter, we study the singularly perturbed equation The function/satisfies the standard Ambrosetti-Rabinowitz condition and suitable growth assumptions. We prove that if(?)is a domain in RN, possibly unbounded, with smooth boundary (?), then the equation has a positive solution ue. Let xe be the peak point of u, we get the estimation of the corresponding critical value, that is When (?) is a bounded domain we can get the similar conclusions as above.In the third chapter, we study the singularly perturbed equation under a weaker assumption on/ We prove the existence of a positive solution and the estimation of the corresponding critical value via a different method. The weaker assumptions on/is that the estimation of critical value isIn the last chapter, we consider the energy estimation of functional for the singularly perturbed system Assume ? and 9 satisfied appropriate assumptions,the solution of this system areυE,υE,andυE≠-υE,then the estimation of functional on Hi is...