Existence For Hénon Type Equations With The Fractional Laplacian And Asymptotic Behavior Of Solutions For Systems With Partial Hénon Weight

In this paper, we study the existence for H′enon type equations with the fractional Laplacianand the asymptotic behavior of solutions for systems with partial H′enon weight. This paper isconstituted with four chapters.In chapter1, we introduce some background and main results about H′enon type equationswith the fractional Laplacian and systems with partial H′enon weight. Furthermore, we illustratedifculties about the existence for H′enon type equations with the fractional Laplacian and asymp-totic behavior of solutions for systems with partial H′enon weight, and give the ideas to overcomethese difculties and main results of this paper.In chapter2, we consider the existence of the ground state solutions of the following problemis the critical Hardy-Sobolev exponent. We prove that problem (3) exists a ground state sol√utionprovided that N=2α and λm<λ <λm+1for some m∈Nand β>0small or N>(1+2)αand λm≤λ <λm+1for some m∈Nand β>0small, where λmis the eigenvalue of with zeroDirichlet boundary condition.In chapter3, we consider the asymptotic behavior of the ground state solution (u, v) aboutthe following systems with partial H′enon weight2is the critical Sobolev exponent. We prove that the maximum points of both components u andv concentrate at the same point on the boundary when α>0and p+q tends to2if N≥3.In chapter4, we consider the asymptotic behavior of the ground state solution (uα, vα) of (4)as αâ†'+∞. Let xα=(yα, zα), xα=(yα, zα)∈are maximum points of uα, vαrespectively,then we prove that ykα, yαâ†'y0∈B(0,1).