The Existence Of Nontrivial Solutions To Nonlinear Elliptic Equation Of P-q-Laplacian Type On R~N

In this paper , we study the following nonlinear elliptic equation of p - q -Laplacian type on R^N:where 1 < q≤p < N , and△_su = div(|▽u|^s-2 |▽u) is the s - Laplacian of u .We prove that under suitable conditions on f(x, t) , if g(x)≡0 and a(x)≡m > 0 , b(x)≡n > 0 for some constant m,n , then the problem (*) has at least one nontrivial weak solution (see Theorem 1.12), generalizing a similar result for p-Laplace type equation in [36]. Also , we proved that under essentially the same assumptions on f(x, t) as that in Theorem 1.12 , there is a constant C > 0 , such that if‖g‖_*< C , then the problem (*) possesses at least two nontrivial weak solutions (see Theorem 1.15) , generalizing a similar result in [9] for p - Laplacian type equation . Since our assumptions on f(x,t) are weaker than that in [9] , Theorem 1.15 is better than the main result in [9] , even if p = q.