Lions-type theorem of the p-Laplacian and applications

Remark 1.1. The generalized version of the Berestycki-Lions conditions for the nonlinearity f is given as follows: (F2) f ∈ C(R,R), there exists C > 0 such that |sf (s)| 6 C(|s|p * α + |s|p * ) for s ∈ R; (F3) lim s→0 F(s) |s|pα = 1 and lim s→0 F(s) |s|p* = 1, where F(s) := ́ s 0 f (t)dt; and (F4) there exists an s0 ∈ R\{0} such that F(s0) ≠ 0. In view of (F1), clearly, the nonlinearity f satis es the generalized version of the Berestycki-Lions conditions (F2)-(F4). This is the “almost optimal" choice of the nonlinearity f .

As is well known, for p = , equation (Q) is a model for describing the stationary state of reaction-di usion equations in population dynamics [7]. It also arises in several other scienti c elds such as plasma physics, condensed matter physics and cosmology [6]. The existence of solution of equation (Q) has been studied extensively by modern variational methods under various hypotheses on the singular potential V and the nonlinearity f . Let us brie y recall some related results. For p = and V(|x|) = |x| α , the existence and nonexistence of solutions to equation (Q) have been studied in [3,4,15,17,20,23]. For p ≠ , the nonexistence results of equation (Q) were presented in [1,5,8,9,14,16,18] and the references therein.
For p ∈ ( , N) and V(x) = |x| p , Ghoussoub-Yuan [9] investigated the equation: −∆p u − µ |x| p |u| p− u = |u| p * − u, x ∈ R N , (1.1) where N , p ∈ ( , N) and µ ∈ , N−p p p , and established the existence of solutions to equation (1.1) by using the variational methods. Abdellaoui-Peral [1] considered the equation: and discussed the existence and nonexistence of solutions to equation (1.2) under di erent assumptions on k(x) by applying the concentration compactness principle and Pohožaev-type identity. Filippucci-Pucci-Robert [8] considered the problem: where s ∈ ( , N) and p * s = p(N−s) N−p is the Hardy-Sobolev critical exponent, and obtained the existence results of equation (1.3) by the choice of a suitable energy level for the mountain pass theorem and analysis of concentration.
Su-Wang-Willem [18] dealt with a generalized version with the singular potential: where < p < N, and V and Q satisfy (V ) V ∈ C( , ∞), V > and there exist real numbers a and a such that They attained the radial inequalities with respect to the parameters a, a , b, b , then established main results on continuous and compact embeddings and the existence of solution to equation (GQ). Badiale-Guida-Rolando [5] generalized the embedding results under di erent conditions on V and Q, and explored the existence of solution to equation (GQ) with the sub-critical and super-critical growth. If a = a = −α and b = b = in conditions (V ) and (Q ), equation (GQ) reduces to (Q). Let us introduce the result on continuous and compact embeddings described in [18].
Furthermore, the embeddings are compact if r ≠ p * α and r ≠ p * , where p * It is very natural to ask whether there exists a solution to equation (Q) with the embedding top index p * and bottom index p * α ? To the best of our knowledge, it seems that so far there is no a rmative answer in the literature.
From Proposition 1.1, the embeddings are not compact. As a result, it is di cult to prove that the Palais-Smale (minimizing) sequence is strongly convergent if we seek solutions of equation (Q) with the critical exponent. Lions [10] considered the noncompact embedding problem by the concentration-compactness principle: Only vanishing, dichotomy or tightness are possible. If one can exclude vanishing and dichotomy, then tightness occurs. It is not di cult to rule out vanishing. But sometimes it is hard to exclude the dichotomy. Therefore, it becomes interesting to ask under what conditions dichotomy cannot occur? In [11, pp. 232 As an application of Theorems 1.1 and 1.2, when the nonlinearity f satis es condition (F ), i.e. equation (Q) takes the form where λ is a constant, Sα and S are the best constants of the following inequalities [2,19]: The rest of this paper is organized as follows. In Section 2, we brie y introduce some useful notations and inequalities. In Sections 3-5, we prove Theorems 1.1-1.3, respectively.

Preliminaries
For N and p ∈ ( , N), let The following inequalities will play a crucial role in the proof of Theorem 1.3: A measurable function u : R N → R belongs to the Morrey space with the norm u M q,ϖ (R N ) , where q ∈ [ , ∞) and ϖ ∈ ( , N], if and only if |u(y)| q dy < ∞.
It follows from Lemma 2.1 and W ,p To prove the generalized version of Lions-type theorem, we need the following technical lemma.

Lemma 2.2 ([18]
). Let N , p ∈ ( , N) and α ∈ ( , p). Then the inequality Throughout this article, we will use the symbol C to denote a generic constant, possibly varying from line to line. However, special occurrences will be denoted by C ,C or the like.

Proof of Theorem 1.1
In this section, by applying the re ned Sobolev inequality with the Morrey norm and Lemma 2.2, we prove a generalized version of Lions-type theorem.
Proof of Theorem 1.1. We separate the proof into four steps.
Step 1. Note that {un} is a bounded sequence in W ,p rad (R N , α). Then, up to a subsequence, we assume According to (2.3) and Condition C, there exists a positive constant C such that for any n there holds On the other hand, from [12, pp 809] we note that {un} is bounded in W ,p rad (R N , α) and Then we have for some C > independent of n. Hence, there exists a positive constant C such that for any n there holds From this inequality, we deduce that for any n ∈ N there exist σn > and xn ∈ R N such that Step 2. We show lim n→∞ σn =σ ≠ , whereσ ∈ ( , ∞). Since N , p ∈ ( , N) and α < p, we have where ω N− is the volume of the unit sphere in R N . This is a contradiction. By the Bolzano-Weierstrass theorem, up to a subsequence, still denoted by {σn}, there existsσ ∈ [ , ∞) such that lim n→∞ σn =σ.
We now show lim n→∞ σn =σ ≠ . Suppose on the contrary that lim n→∞ σn =σ = . By using the uniform bounded- It follows from Hölder's and Sobolev's inequalities that Similarly, for each z ∈ R N we havê Covering R N by balls of radius σn, in such a way that each point of R N is contained in at most N + balls, we ndˆR where ω N− is the volume of the unit sphere in R N .
Taking n → ∞ and applying lim n→∞ σn = leads to which yields a contradiction to lim n→∞´R N |un| p * dx > given in Condition C.
Applying the embedding W ,p rad (R N , α) → D ,p rad (R N ) → L p loc (R N ), we obtain u ≢ .

Proof of Theorem 1.2
In this section, we prove that any nonnegative weak solutions of equation (Q) with α ∈ ( , p) have additional regularity properties.
Proof. Let U be a nonnegative ground state solution of equation (Q). We show thatŪ L ∈ W ,p rad (R N , α). By a straightforward calculation, we get This implies thatŪ L ∈ W ,p rad (R N , α). Note that U is a nonnegative ground state solution of equation (Q). Then SubstitutingŪ L into the above equation, we get It follows from (4.2)-(4.4) and Proposition 1.1 that We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. We divide the proof into three parts. Part (i). We rst show the L ∞ estimate of U by the following four steps.
Step 1. We claim that Takingd ∈ R + and using Hölder's inequality, we can derivê We choosed such that ˆ{ Substituting the above two inequalities into (4.1) with the choice of t = p * , we get Taking the limit as L → ∞ in the above inequality leads to Hence, we have We now choose µ := + p * −p * α p . Then we obtain p * µ ∈ p * , p * + p * −p * α p and +ˆR N |U| p * µ dx p p * (µ − ) < ∞.
Step 2. We show that From Lemma 4.1, we get For each µ ∈ [µ , µ ], we have U ∈ L p * µ (R N ) and p(µ − ) and b = p * α + pµ − p − a . Then p * b p * −a = p * + pµ − p. It follows from Young's inequality thatˆR For every x , x > , we know that We then obtain That is, Step 3. We show that For each µ ∈ [µ , µ ], we have U ∈ L p * µ (R N ) and Hence, we obtain Step 4. Iterating the above process and recalling that we have For m ∈ N, we further get .
That is, That is, For the series ∞ i= ln C µ i+ − , using the root test, we get For the series ∞ i= ln µ i+ µ i+ − , by using the ratio test, we nd Letting m → ∞ in (4.5) and (4.6), we obtain
Part (iii). Applying Theorem 1.2 (ii) and following [8,Claim 5.3], we can derive the Pohožaev-type identity Consequently, the proof is completed.

Proof of Theorem 1.3
As we see, equation (S p * α ) is variational and its solutions are the critical points of the functional de ned in W ,p rad (R N , α) by It is easy to check that c =c =c := inf

Lemma 5.1. Assume that all the conditions described in Theorem 1.3 hold. Then
We choose A straightforward calculation giveŝ It is not di cult to see thatˆ In view of p ∈ ( , This implies that wσ ∈ W ,p rad (R N , α). So we havê J(γ(t)) = J(tσ wσ). .

Perturbation Equation
Applying Theorems 1.1 and 1.2, it is easy to prove the existence of positive ground state solution of the following equation (with small ε > , see [18]): Set the energy functional of equation (S p * α +ε ) as follows: Let vε be a positive ground state solution of equation (S p * α +ε ). For all φ ∈ W ,p rad (R N , α), it follows that We then have the following lemma for equation (S p * α +ε ).
It follows from (5.6) and (5.7) that This indicates that v is a weak solution of equation (Q).