A Study On Solutions For Elliptic Problems With Strong Nonlinearity

This thesis considers some kinds of nonlinear elliptic problems with strong nonlinearity. The thesis consists of six chapters:Chapter 1 is an introduction to this thesis. It includes the background and all the problems will be studied in this dissertation as well as the main results we have obtained.In Chapter 2, we recall some mathematical preliminaries including funda-mental inequalities and some important facts in nonlinear analysis and elliptic problems which will be used in the later chapters.In Chapter 3, we study a kind of quasilinear Schrodinger equation where Ω (?) RN(N ≥ 3) is a smooth bounded domain. The first section of this chapter is concerned with the case f(x,u) is made up of a convex term with any growth and a singular term in a bounded smooth domain. Multiplicity results are obtained by critical point theory together with truncation arguments and the method of upper and lower solutions. In the second part of this chapter, using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for our problem. Our main results in the second section of this chapter can be viewed as a partial extension of the results of Zhang et al. in [122] and Zhou and Wu in [124] concerning the the existence of solutions to our problem in the case of p = 2 and a recent result of Liu and Zhao in [93] two solutions are obtained for our problem.In Chapter 4, we investigate the existence of positive solutions to a class of nonlocal boundary value problem of the p-Kirchhoff type where Ω C RN(N> 3) is a bounded smooth domain, M, f and g are continu-ous functions. The existence of a positive solution is stated through an iterative method based on Mountain pass theorem.In Chapter 5, we consider the existence of multiple solutions of the following problem: where Ω (?) RN(N> 3) is a bounded domain with smooth boundary aΩ,0 ＜ q < 1 < s < 2* - 1 ≤ p, 2* :=(2N)/N-2, λ and μ are nonnegative parameters. By using variational methods, truncation and Moser iteration techniques, we show that if the parameters λ and μ, are small enough, then the problem has at least two positive solutions.In Chapter 6, we establish the regularity of minimizer for free boundary prob-lem in Orlicz-Sobolev spaces. We obtain the uniqueness and C1,α- regularity of minimizer by dealing with the equivalence of minimizer and strong solution, which implies non-degeneracy of minimizer near the free boundary.