Ground-state Solution Of The P-Laplace Equation On The Costa-type Non-P Times Condition

In this paper, we consider p-Laplacian equation in RN with a potential, and do not assume Ambrosetti-Rabinowitz condition is satisfied. We mainly use the Mountain-pass theorem, Lion’s lemma and other critical point theory to prove the existence of ground states. The result is applied to find homoclinic type solutions of Hamiltonian systems as well as second and fourth order elliptic partial differential equations.In this paper,we will study the p-Laplacian equationThe study of elliptic problems of the form (1) in the semilinear case p=2, has been motivated in part by studying standing waves for the nonlinear Schrodinger equationwhen p∈(1,N),we refer to the classical works in ([1]-[8]), which are concerning with the applications of form (1), such as the superlinear condition of A.Ambrosett and P.H.Rabinowitz([8]).(AR) is usually assumed to be satisfied. The role of (AR) is to ensure the boundness of the Palais-Smale(Ps) sequences of functionals.The energy functional Φ in our paper isIt’s well known that, the critical points of Φ(u) are the classical solutions of form (1).In this paper, we assume that F(x,u) satisfies, u·Fu(x,u)>pF(x,u),(?)x∈RN,u∈R\{0} and for some M,v,δ>0u9Fu (x,u)-pF(x,u)≥M|u|v>0,(?)x∈RN,|u|≥δ,(No)(No) is calld Costa type non-p times condition.Our main ideas in this paper are from ([10]), we also notice that, in ([8]), F satisfies (?)θ≥1s.t. θf(x, u)≥f(x, su)((x, u)∈RK×R, s∈[0,1]), where f(x,u)=Fu(x,u)u-pF(x,u). Our methods are completely different from those in ([8]).