Existence And Asymptotic Behavior Of Sign-changing Solutions For Nonlinear Elliptic Equations With Nonlocal Term

In this paper, we mainly study the existence and asymptotic behavior of sign-changing solutions for some nonlocal elliptic equations, including the Kirchhoff-type problem, the nonlinear Schrodinger-Poisson system and the fractional Laplacian elliptic equations.The thesis consists of five chapters:In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In Chapter Two, the existence and asymptotic behavior of nodal solutions to the following Kirchhoff problem where V(x) is a smooth function, a,b are positive constants. Because the so-called nonlocal term b((?)Ω|â–½n|2dx)Δu is involving in the equation, the variational func-tional of the equation has totally different properties from the case of b=0. Un-der certain assumptions on f(x,t), we prove that, for any give positive integer k, the problem (E1) has a sign-changing solution ukb, which changes signs exact k times. Moreover, the energy of ukb. is strictly increasing in k, and for any sequence {bn}â†'0+(nâ†'+∞), there is a subsequence {bns}, such that ukbns converges in H1(R3) to uk as sâ†'∞, where ωk also changes signs exact k times and solves the following equationIn Chapter Three, We investigate the existence of least energy sign-changing solutions of the following Kirchhoff type problem where a,b are positive constants, Ω be a bounded domain in RN, N=1,2,3, with a smooth boundary (?)Ω. Combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution ub.Moreover, we show that the energy of ub is strictly larger than twice of the ground state energy. Finally, we regard b as a parameter and give a convergence property of ub as b(?)0.In Chapter Four, we study the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrodinger-Poisson system where V(x) is a smooth function, A is a positive parameter. Because the so-called nonlocal term λφu(x)u is involving in the equation, the variational functional of the equation has totally different properties from the case of λ=0. Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution uλ. More-over, we show that any sign-changing solution of the problem has an energy exceed-ing twice the least energy, and for any sequence {λn}â†'0+(nâ†'∞), there is a subsequence {λnk}, such that uλnk converges in H1(R3) to u0 as kâ†'∞, where u0 is a sign-changing solution of the following equationIn Chapter Five, we study the existence of sign-changing solutions for fractional elliptic equations of the form where s ∈(0,1), N> 2s and Ω (?) RN is a bounded smooth domain. We prove, via variational method combining invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Add some other condition on f(x,u), we show that the problem has a least energy sign-changing solution with its energy exceeding twice the least energy. Moreover, if f(x,u) is odd in u, we obtain an unbounded sequence of sign-changing solutions.