Existence And Dynamical Properties Of Solutions For Elliptic Equations Involving Critical Growth

In this thesis, we study the existence and dynamical properties of solutions for ellip-tic equations involving critical growth. It consists of four main parts:(1) the existence ofground states for nonlinear scalar field equation with critical growth;(2) singularly per-turbed problem for nonlinear elliptic equation with critical growth in bounded domain;(3) the existence and concentration of standing wave for nonlinear Schr(o|¨)dinger equationwith critical growth in RN;(4) semiclassical states for nonlinear Schr(o|¨)dinger equationwith critical frequency and growth.In the first part, we investigate the existence of ground states for nonlinear scalarfield equation involving critical growth. In1983, H. Berestycki and P. L. Lions obtainedthe existence of ground states for nonlinear scalar field equation in the subcritical case.But for the critical case, it has been left unresolved for a long time. By using the con-straint variational method and Brezis-Nirenberg technology, we resolved afrmativelythis problem. Meanwhile, we also established the mountain pass characterization and theexponential decay of ground states at infinity.In the second part, we consider the singularly perturbed problem for nonlinear ellip-tic equation with critical growth in bounded domain. In2010, with the Berestycki-Lionsconditions, J. Byeon proved the existence and concentration of single spike solutions fornonlinear elliptic equation. In this paper, by using localized deformation method, wecomplement the study of J. Byeon in the sense that, only the subcritical growth was con-sidered in that paper. On the other hand, we also obtained the exponential decay of spikesolutions at infinity.In the third part, we study the existence of standing waves for nonlinear Schr(o|¨)dingerequation with critical growth in RN. In2007, with the Berestycki-Lions conditions, J.Byeon and L. Jeanjean proved that the problem admits a single spike solution, whichexhibits concentration at local minimum of the potential, but only subcritical growth wasconsidered. In this paper, by using the penalization-type method, we extend the result ofJ. Byeon to the critical case. On the other hand, the exponential decay of solutions wasobtained.In the finally part, we discuss the Ambrosetti open problem in the critical case. By using the penalization-type method, we obtained the existence and concentration of so-lutions for nonlinear Schr(o|¨)dinger equation involving critical growth, where the potentialmay has a compact support or decay faster than|x|2at infinity.