Existence Of Nontrivial Solutions For Quasilinear Elliptic Equations With Parameters

This paper studies the following quasilinear elliptic equations with parameters:-?u-??l(u²)l'(u²)u+V(x)u=f(u),x (?) R^N,where-y is the parameter,l(s)is some nonlinear functions,V(x)is the potential function,f(s)is the nonlinear term,N?3.For differentl(s),the equation comes from different physical phenomena and models.For example,when l(s)=s,the equation appears in various problems of plasma physics and nonlinear optics;when l(s)=(1+s)^1/2,the equation can be used to describe the self channeling effect of high power ultrashort laser pulse in plasma;when l(s)=(1-s)^1/2,the equation appears in the classical plane Heisenberg ferromagnetic spin chain.Using the variational method and perturbation method,for the above equation this paper considers the existence of nontrivial solutions in the following three cases,which improves some recent results:(1)The autonomy problems:?=±1,V(x)=?>0,f(s)=?|s|^p-2s,?>0,2<p<2N/(N-2).(2)The nonautonomous problems:?=±1,V(x)satisfies the potential condition,f(s)=?|s|^p-2s,?>0,2<p<2N/(N-2).(3)The supercritical problem:|?| is small enough,f(s)=?|s|^p-2s+|s|^q-2,?>0,p>2N/(N-2),2<q<2N/(N-2).