The Existence Of Ground State Solutions For Some Elliptic Equations With Critical Growth

In this thesis,we study the existence of ground state solutions for some elliptic equations with critical growth and the lack of(AR)condition via variational method.The thesis consists of four chapters:In Chapter ?,we summarize the background and research status of the related problems.Moreover,the main results of the present thesis are stated briefly.In Chapter ?,we study the following quasilinear Schrodinger equation with critical exponent:which models the self-channeling of a high-power ultra short laser in matter,where N?3,2<p<2*= 2A/N-2= and V ? C1(RN,R)is a given positive potential.Combining the change of variables and monotonicity trick,we obtain the existence of positive ground state solutions for the given equation.Our result extends the main result in[W.Huang,J.Xiang,Commun.Pure Appl.Anal.,15(2016),1309-1333]concerning the existence of positive ground state solutions with 12-4?6<p<2*.In Chapter ?,we study the following generalized quasilinear Schrodinger e-quations:-div(g2(u)?u)+g(u)g'(u)|?u|2 + V(x)u?h(u),x ?RN,where N>3,g:R?R+is an even differentiable function satisfying lin/t?+? t(t)-ta-1 =?>0 for some ?>1,h(t)?|t}a2*-2t+ |t|p-2t(2<p<a2*)does not satisfy the variant(AR)condition,and the potential V(x):RN?R is positive.By using the change of variables and Pohozaev identity,we first prove that the limiting equation admits a positive ground state solution.Under this fact,combining monotonicity trick and global compactness lemma,we obtain the existence of positive ground state solutions for the given equation.Our result extends the main result in[Y.Deng,S.Peng,S.Yan,J.Differential Equations,260(2016),1228-1262]concerning the existence of positive solutions with 2?<p<?2*.In Chapter ?,we are concerned with the following nonlinear fractional Kirch-hoff type equation:where a,b>0 are constants,? is a positive parameter,2<p<2*s,2*s =6/s-2s is the fractional Sobolev critical exponent and s ?(0,1).By using the Nehari-Pohozaev manifold and a technique in[46,48],we first establish the existence of positive ground state solutions for the limiting equation.Under suitable assumptions on V,combining monotonicity trick and global compactness lemma,we prove that the given equation admits a positive ground state solution for s ?(3/4,1).Our result extends the main result in[Z.Guo,J.Differential Equations,259(2015),2884-2902]concerning the existence of positive ground state solutions for Kirchhoff type equation with subcritical growth.