Existence of Nontrivial Solutions for Perturbed p-Laplacian Equation in R N with Critical Nonlinearity

and Applied Analysis 3 Proof. From u n → u in Lsloc(R N ), we have ∫ B i 󵄨󵄨󵄨󵄨un 󵄨󵄨󵄨󵄨 s 󳨀→ ∫ B i |u| s as n 󳨀→ ∞. (11) Thus, there exists n i ∈ N such that ∫ B i ( 󵄨󵄨󵄨󵄨un 󵄨󵄨󵄨󵄨 s − |u| s ) < 1 i , ∀n = n i + j, j = 1, 2, . . . . (12) In particular, for n i = n i + i, we have ∫ B i ( 󵄨󵄨󵄨󵄨 u n i 󵄨󵄨󵄨󵄨 s − |u| s ) < 1


Introduction and Main Results
In this paper, we are concerned with the existence of nontrivial solutions for the following nonlinear perturbed -Laplacian equation with critical nonlinearity: where Δ   = div(|∇| −2 ∇) is the -Laplacian operator with 1 <  < ,  ≥ 3,  * = /(−) denotes the Sobolev critical exponent, () is a nonnegative potential, () is a bounded positive function, and (, ) is a superlinear but subcritical function.
For general  > 1, most of the work (see [17][18][19] and the reference therein) dealt with (1) with  = 1, () ≡ 0 and a certain sign potential ().Liu and Zheng [20] considered the above mentioned problem with sign-changing potential and subcritical -superlinear nonlinearity.Cao et al. [21] also studied the similar problem.However, to our best knowledge, it seems that there is almost no work on the existence of semi-classical solutions to the equation in R  with critical nonlinearities.This paper will study the critical nonlinearity case in whole space.
Throughout the paper, we make the following assumption: ( 1 )  ∈ (R  ), (0) = inf ∈R  () = 0 and there exists  > 0 such that the set ]  := { ∈ R  : () < } has finite Lebesgue measure; The main tool used in the proof of Theorem 1 is variational method which was mainly developed in [8].The main difficulty in the case is to overcome the loss of the compactness of the energy functional related to (1) because of unbounded domain R  and critical nonlinearity.Although the energy functional does not satisfy the () condition, we can prove that it possesses ()  condition at some energy level .
This outline of the paper is organized as follows.In Section 2, we give the variational settings and preliminary results.In Section 3, we show that the corresponding energy functional satisfies (PS)  condition at the levels less than  0  1−/ with some  0 > 0 independent of .Furthermore, it possesses the mountain geometry structure.Section 4 is devoted to the proof of the main result.

Preliminaries
Let  =  − in (1).The equation (1) reads as In order to prove Theorem 1, we are going to prove the following result.

Necessary Lemmas
This section will show some lemmas which are important for the proof of the main result.Lemma 3. Assume that ( 1 )-( 5 ) is satisfied.For the ()  sequence {  } ⊂   for   , we get that  ≥ 0 and {  } is bounded in the space   .Proof.By direct computation and the assumptions ( 2 ) and ( 5 ), one has Together with   (  ) →  and    (  ) → 0 as  → ∞, we easily get that the (PS)  sequence is bounded in   and the energy level  ≥ 0.

By
In particular, for   = ñ + , we have Note that there exists Then This completes the proof of Lemma 4.
Let  ∈  ∞ (R + ) be a smooth function satisfying This shows that the desired conclusion holds.
By the facts that   ( Let   () := max{(), }, where  is the positive constant in the assumption ( 1 ).Since the set ]  has finite measure and where  is the best Sobolev constant of the immersion Thus The fact  * >  implies the desired conclusion.
Lemma 10.For any finite dimensional subspace  ⊂   , we have Proof.By the assumption ( 5 ), one has Since all norms in a finite-dimensional space are equivalent and  > , this implies the desired conclusion.
Lemma 8 shows that   satisfies (PS)   condition for  large enough and   small sufficiently.In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.
Define the functional It is apparent that Φ  ∈  . ( In connection with (0) = 0 and ‖∇  ‖   < , it shows that there exists Λ  > 0 such that for all  ≥ Λ  , we have
Next, we consider the energy level of the functional   below which the (PS)  condition held.