The Existence Of Solutions For Several Classes Of Nonlinear Elliptic Equations

Nonlinear functional analysis is a relatively new field. It is based on all sorts of nonlinear problems which have their roots in mathematics, physis, chemistry, astronomy, biology, control theory, engineering, economics and so on. The idea of nonlinear functional analysis is to solve specific nonlinear problems through establishing various abstract theories. It has a wide application in nonlinear differential equations, computational mathematics, dynamical systems, control theory, optimization theory, etc. Nonlinear functional analysis consists of partial order theory, topological degree theory, critical point theory and so on. Many mathematicians have made great contributations to it, for example, E. Rothe, M.A. Krasnosel'skii, L. Lusternick, L. Schnirelman, H. Amann, L. Nirenberg, I. Ekeland, A. Ambrosetti, P.H. Rabinowitz, M. Willem, M. Struwe, H. Brezis, J. Mawhin. Many famous mathematicians in China, for example, Kung-Ching Chang, Dajun Guo, Wenyuan Chen, Shujie Li, Jingxian Sun, Zhaoli Liu, Wen-ming Zou have good works in nonlinear functional analysis also(See [1-5], [71], [78], [82], [91], [97-98]).The variational method is an important theory of nonlinear functional anal-ysis. The basic idea is to regard the solution of an equation as the critical point of the corresponding functional. In the classical theory, people paid much attention to search for the extreme value of the functional. We should point at that for a number of functionals, extremal values do not exist. On the other hand, people hope to find all critical points of the functional. Thus, variational calculus in the large arise. The minimax principle is an important principle in variational meth-ods. From the minimax principle, people got a number of minimax theorems, which play an important role in searching for the critical value of the functional. Minimax theorems can be used to deal with functionals which do not have ex-tremal values. Besides, the Lusternik-Schnirelmann theory and the Morse theory are also important theories of variational methods.The present paper investigates the existence of solutions and multiplicity for several classes of nonlinear elliptic equations by using variational methods.In Chapter 1, we introduce the Sobolev space and give the corresponding embedding theorems. We also recall some compact conditions.In Chapter 2, we investigate the existence of multiple solutions for the fol-lowing biharmonic problem: whereΔ2 is the biharmonic operator, a is a real parameter,Ω(?)RN is a bounded domain with smooth boundary (?)Ω, N≥5, f(x,u) and g(x,u) satisfy some additional conditions. By the variant Fountain Theorem established in [96], we obtain the existence of infinitely many high energy solutions both in the case a≥λ1 and a<λ1, where Ai is the first eigenvalue of -Δin H01(Ω).In Chapter 3, we are concerned with the multiplicity of solutions for the following biharmonic problem: whereΔ2 is the biharmonic operator,Ω(?)Rs is a bounded domain with smooth boundary (?)Ωand s∈N. a<λ1,λis a real parameter,1< q< 2, f(x,u) satisfies some additional assumptions. The solutions of the problem are obtained from the versions of mountain pass lemma and linking theorem.In Chapter 4, we study the following problem of a singular elliptic system with critical exponents: whereΩis a bounded domain in RN (N≥3) with smooth boundary (?)Ωand the origin 0∈Ωλ>0, 11,α+β=2*:=2N/(N-2),0< s1,s2<2 and 2*(s1):=(2(N-s1))/(N-2), 2*(s2):=(2(N-s2))/(N-2).It is well-known that 2* is the critical Sobolev exponent and 2*(s1), 2*(s2) are the critical Hardy-Sobolev exponents. The coefficient f is nonnegative and continuous onΩsatisfying some additional conditions. 0≤μ<μ:= ((N-2)/2)2 , andμis the best constant of the Hardy inequality. By using variational methods, we investigate multiple positive solutions of the problem which is based on the argument of the compactness. We obtain the existence of a positive solution by the Ekeland variational principle and use the Mountain Pass Theorem to find the second positive solution.In Chapter 5, we study the following problem of a singular elliptic system with multiple critical exponents: whereΔpu = div(|▽u|p-2▽u) is the p-Laplacian of u, 2≤p1,β>1,α+β=p*:=(Np)/(N-p),0