The Existence Of Nontrivial Solutions For Biharmonic Quasilinear Elliptic Equations

In this paper,we consider the following quasilinear fourth-order elliptic equations(?)(0.1)where Δ2=Δ(Δ)is the biharmonic operator,V∈(RN,R)is a positive continuous function,4<q<2*,2*=2N/N-4 for 4<N≤6;and 2*=+∞ for N≤4.We consider several types of potentials V(x),and then we study the existence of nontrivial solutions for the quasilinear elliptic equations.When 0<V0≤V(x)≤V∞,(?)V(x)=V∞ and V(x)(?)V∞,we prove the existence of ground state solutions to the equations(0.1)by Mountain Pass Lemma.In addition,if 0<V0≤V(x)≤V∞,(?)V(x)=∞,we also get the existence of ground state solutions for(0.1),and then prove that the equations(0.1)have an infinite number of solutions by Fountain Theorem.