The Existence Of Nontrivial Solutions For A Class Of P-harmonic Elliptic Equations

In this paper, we consider the existence of nontrivial solutions for p-harmonic problem:where m > 0, f(x,u)/|u|p-2u tends to a positive constant as u→+∞. In this case, f(x,u) does not satisfy the following Ambrosetti-Rabinowitz type condition, that is, for someθ> 0, and for all (x, s)∈R^N×R, x∈Ω, the inequalityis no longer true, where F(x, s) = (?)₀^s f(x,t)dt. It was known, this condition is very important in applying Mountain Pass Theorem. Now, by a variant version of Mountain Pass Theorem, we prove that there exists a nontrivial solution to the above problem. Furthermore, if f(x,u) = f(u), the existence of a ground state to this problem is also proved by using artificial constraint method.