Positive Solution To P-Laplacian Type Scalar Field Equation In R～N With Nonlinearity Asymptotic To U～(p-1) At Infinity

In this paper,we consider the following elliptic 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,(?)θ＞0,0≤F(x,u)(?)integral from n=0 to m (f(x,s)ds)≤1/p+θf(x,u)u,for all (x,u)∈R^N×R.which is important in applying Mountain Pass Theorem. By a variant version ofMountain Pass Theorem, we prove that there exists a positive solution to thepresent problem. Furthermore, if f(x,u)≡f(u), the existence of a ground stateto the above problem is also proved by using artificial constraint method.This paper is made up of four parts.Part one introduces the elliptic equation mentioned above and its back-ground. Furthermore, we state our main results and the key points of the proofand compare them with the results obtained before by the others.Part two discusses some preliminary results which will be used frequently inthe following text. e.g. Pohozaev type identity for the P-Laplacian, VanishingLemma,the Ekeland's variational principle on Finsler manifold and so on.Part three gives the proof of Theorem 1.1. At first, we get the fact∧≠φ(∧is defined by (1.7)) by Lemma 3.1 and we get the result J^∞＞0(J^∞is defined by(1.8)) by Lemma 3. 2. In fact,we use Lemma 2.2 which is important in provingLemma 3.2. Then, we divide our proof into two steps.Step 1: The special (P. S) sequence {u_n} is bounded in W^1,P(R^N). (There issuch a sequence {u_n} by Lemma 2. 8).Step 2: J^∞is achieved by some u_o∈W^1,P(R^N) and u⁰＞0. Part four finishes the proof of Theorem 1.2. We adopt the version of Moun-tain Pass Theorem (Proposition 4. 1) to prove it. Of course,we prove that thefunctional defined by (1.2) satisfies the conditions of Proposition 4. 1. Then,wegive the proof of Theorem 1.2 by three steps. Finally,we state and verify theTheorem 1.3,that is, if f(x,t) satisfies the condition: (?)／(?)(f(x,t)t) is continuous with respect to t∈R¹besides all the conditions (C1—C6),we can get more precise result than Theo-rem 1.2.