Research On The Fourth Order Semilinear Elliptic Equation With Weight

This thesis is concerned with the existence, multiplicity and the asymptotic behavior of solutions for the following fourth order semilinear elliptic equationΔ2u=|x|a|u|p-2u inΩ(1) either with Navier boundary conditions u=Δu=0 on (?)Ω(2) or with Di (?)Ωrichlet boundary conditions u=|▽u|=0 on (?)Ω, (3) whereΩ(?) RN is a bounded smooth domain, a is a nonnegative parameter and p> 2.First, Chapter One is devoted to the primitive introduction of the background and study situation, the list of the main work in this thesis and the existence of radial solutions and non-existence for the corresponding problem in the unit ball.Next, we mainly consider the Problem (1) with critical Sobolev exponent under Navier boundary conditions (2). In Chapter Two, the existence of nonradial positive solutions to the problem (α> 0) in the unit ball is investigated. We choose some suitable subgroup of O(N), and consider the minimization problem restricted to the subsets of H2(Ω)∩H01 (Ω) invariant under the action of this subgroup. Along the line of Brezis-Nirenberg [13], and by comparing the energy of solutions, we prove the existence of nonradial positive solutions to the problem when N≥6 and a is large enough. More-over, in Chapter Three, we study the problem in the more general domain. We consider an appropriate minimax problem. By constructing a flow associated with corresponding functional when a is small enough, we prove the existence of positive solutions to the problem.Problem (1) with subcritical Sobolev exponent under Dirichlet boundary conditions (3) is investigated in the next two chapters. In Chapter Four, we assume thatΩis a unit ball in RN, and we study the positive solutions. First, by comparing the energy of solutions, a symmetry breaking result with respect to the parameter a is given. Then, by means of concentration-compactness principle and Blow-up analysis, we study the asymptotic behavior of the least energy solution and find that the solution concentrates at precisely one point of the boundary (?)Ωwhen p tends to 2N/(N-4). Furthermore, in order to investigate the effect of domain on the solution, we assume thatΩis an annulus in RN and change the weight function accordingly. In Chapter Five, by concentration-compactness principle and linking argument, we prove that for a large enough and p sufficiently close to 2N/(N-4), the problem admits at least four solutions, and three of them is nonradial.