Existence Of Solutions For Two Classes Of Quasilinear Elliptic Equations

The unbounded area and the critical exponent of the nonlinearity can lead to loss of compactness to the functional of the equation.We can’t apply the classical variational method directly.In this paper,we study the existence of solutions for the two classes quasilinear elliptic equations involving subcritical and the critical or the supercritical in the unbounded area.In chapter 2,we study a quasiliner elliptic equation of the formwhere N ≥ 3,Diu =(?)u/(?)xi,Dsai,j(x,s)=d/dsai,j(x,s),f:RN × R → R has subcritical growth with respect to u.By adding an s-operator and a potential term satisfying the corresponding conditions in the energy functional of the equation(1),then a positive solution,a negative solution and a sign-changing solution are obtained by the variational method and perturbation method.In chapter 3,we study following quasilinear elliptic equation involving critical or supercritical exponent-Δu + V(x)u-Δ(u2)u = h,(x,u)+μ|u|q-2u,X∈RN,(2)where N≥3,2<p<2·2*=4N/N-2<≤q,and μ is a positive parameter.The functions V(x)and h(x,u)satisfy some applicable assumptions respectively.The existence of a positive solution will be proved via variational methods together with truncation technique and L∞ estimate when the μ>0 is small sufficiently.