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:

1. f ∈ C(ℝ, ℝ), there exists C > 0 such that |sf(s)|⩽C(|s|pα∗+|s|p∗) for s ∈ ℝ;

2. lims→0F(s)|s|pα∗=1 and lims→0F(s)|s|p∗=1, where F(s):=∫0sf(t)dt ; and

3. there exists an s₀ ∈ ℝ ∖ {0} such that F(s₀) ≠ 0.

In view of (F₁), clearly, the nonlinearity f satisfies the generalized version of the Berestycki-Lions conditions (F₂)-(F₄). This is the “almost optimal” choice of the nonlinearity f.

Proposition 1.1

Suppose that N ⩾ 3 and p ∈ (1, N). Then we have

Furthermore, the embeddings are compact if r ≠ pα∗ and r ≠ p^*, where pα∗=p2(N−1)+pαp(N−1)−α(p−1)=p+p2αp(N−1)−α(p−1),p∗=NpN−p, and Wrad1,p(RN,α):=Drad1,p(RN)∩Lp(RN,α) is the set of radial functions in W^1,p(ℝ^N, α).

Proposition 1.2

(Lions Theorem) Let N ⩾ 3 and p ∈ (1, N). Let {u_n} ⊂ W^1,p(ℝ^N) be any bounded sequence satisfying

(Condition A) limn→∞∫RN|un|rdx>0 for r ∈ (p, p^*).

Then there exists {y_n} ⊂ ℝ^N such that the sequence ū_n(x) = u_n(x + y_n) converges weakly and a.e. to ū ≢ 0 in Llocr (ℝ^N).

Proposition 1.3

(Lions-type theorem I) Let N ⩾ 3 and p ∈ (1, N). Let {u_n} ⊂ Wrad1,p (ℝ^N, α) be any bounded sequence satisfying

(Condition B) limn→∞∫RN|un|rdx>0, where

Then the sequence {u_n} converges weakly and a.e. to u ≢ 0 in Llocr (ℝ^N).

Theorem 1.1

Suppose that N ⩾ 3, p ∈ (1, N) and α ∈ (0, p). Let {u_n} ⊂ Wrad1,p (ℝ^N, α) be any bounded sequence satisfying

(Condition C) limn→∞∫RN|un|p∗dx>0 and limn→∞∫RN|un|pα∗dx>0.

Then the sequence {u_n} converges weakly and a.e. to u ≢ 0 in Llocp (ℝ^N).

Theorem 1.2

Suppose that N ⩾ 3, p ∈ (1, N), α ∈ (0, p) and condition (F₂) holds. If U is a nonnegative weak solution of equation (𝓠), then the following statements are true.

1. U ∈ L^r(ℝ^N) for r ∈ [ pα∗ , ∞].

2. U is a positive solution.

3. U satisfies the Pohožaev-type identity:

   N−pp∫RN|∇U|pdx+N−αp∫RNA|x|α|U|pdx=N∫RNF(U)dx.

Theorem 1.3

Assume that N ⩾ 3, p ∈ (1, N ), α ∈ (0, p), q ∈ p(N−p+α)N−p,p∗ and condition (F₁) holds. Suppose that

where λ is a constant, S_α and S are the best constants of the following inequalities [2, 19]:

and

Then equation ( Spα∗ ) has a positive ground state solution u ∈ Wrad1,p (ℝ^N, α).

Lemma 2.1

([12], Refined Sobolev inequality with the Morrey norm) For N ⩾ 3 and p ∈ (1, N), there exists C > 0 such that for ι and ϑ satisfying pp∗⩽ι<1 and 1 ⩽ ϑ < p^*, we have

for u ∈ D^1,p(ℝ^N).

Lemma 2.2

([18]). Let N ⩾ 3, p ∈ (1, N) and α ∈ (0, p). Then the inequality

holds for u ∈ Wrad1,p (ℝ^N, α).

Proof of Theorem 1.1

We separate the proof into four steps.

1. Note that {u_n} is a bounded sequence in Wrad1,p (ℝ^N, α). Then, up to a subsequence, we assume

   un⇀u in Wrad1,p(RN,α), un→u a.e. in RN,un→u in Llocp(RN).

   According to (2.3) and Condition C, there exists a positive constant C such that for any n there holds

   ∥un∥Mp,N−p(RN)⩾C>0.

   On the other hand, from [12, pp 809] we note that {u_n} is bounded in Wrad1,p (ℝ^N, α) and

   Wrad1,p(RN,α)↪Drad1,p(RN)↪Lp∗(RN)↪Mp,N−p(RN).

   Then we have

   ∥un∥Mp,N−p(RN)⩽C

   for some C > 0 independent of n. Hence, there exists a positive constant C₀ such that for any n there holds

   C0⩽∥un∥Mp,N−p(RN)⩽C0−1.

   From this inequality, we deduce that for any n ∈ ℕ there exist σ_n > 0 and x_n ∈ ℝ^N such that

   σn−p∫B(xn,σn)|un(y)|pdy⩾∥un∥Mp,N−p(RN)p−C2n⩾C1>0. (3.1)

2. We show limn→∞ σ_n = σ̄ ≠ 0, where σ̄ ∈ (0, ∞).

   It suffices to show that limn→∞ σ_n ≠ ∞. Otherwise, we suppose limn→∞ σ_n = ∞. In view of the boundedness of {u_n} and Condition C, we get

   0<limn→∞∫RN|un|pα∗dy⩽C.

   Since N ⩾ 3, p ∈ (1, N) and α < p, we have

   −p+N1−ppα∗<0⇔pα∗<p∗. (3.2)

   It follows from (3.1)-(3.2) that

   0<C1⩽σn−p∫B(xn,σn)|un(y)|pdy⩽σn−p∫B(0,σn)dypα∗−ppα∗∫B(xn,σn)|un(y)|pα∗dyppα∗=σn−pωN−1NσnNpα∗−ppα∗∫B(xn,σn)|un(y)|pα∗dyppα∗⩽σn−p+N(1−ppα∗)ωN−1Npα∗−ppα∗∫RN|un(y)|pα∗dyppα∗⩽Cσn−p+N(1−ppα∗)→0, as n→∞,

   where ω_N–1 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 {σ_n}, there exists σ̄ ∈ [0, ∞) such that

   limn→∞σn=σ¯.

   We now show limn→∞ σ_n = σ̄ ≠ 0. Suppose on the contrary that limn→∞ σ_n = σ̄ = 0. By using the uniform boundedness of {u_n} in Wrad1,p (ℝ^N, α), we have

   Climn→∞∫RN|un|p∗dypp∗⩽limn→∞∥un∥Wrad1,p(RN,α)p⩽C¯.

   It follows from Hölder’s and Sobolev’s inequalities that

   ∫B(0,σn)|un|pα∗dy⩽∫B(0,σn)dyp∗−pα∗p∗∫B(0,σn)|un|p∗dypα∗p∗⩽S−pα∗p∫B(0,σn)dyp∗−pα∗p∗∫RN|∇un|pdypα∗−pp∫B(0,σn)|∇un|pdy⩽CS−pα∗p∫B(0,σn)dyp∗−pα∗p∗∫B(0,σn)|∇un|pdy.

   Similarly, for each z ∈ ℝ^N we have

   ∫B(z,σn)|un|pα∗dy⩽CS−pα∗p∫B(0,σn)dyp∗−pα∗p∗∫B(z,σn)|∇un|pdy.

   Covering ℝ^N by balls of radius σ_n, in such a way that each point of ℝ^N is contained in at most N + 1 balls, we find

   ∫RN|un|pα∗dy⩽C(N+1)S−pα∗p∫B(0,σn)dyp∗−pα∗p∗∫RN|∇un|pdy⩽C(N+1)S−pα∗p∫B(0,σn)dyp∗−pα∗p∗=C(N+1)S−pα∗pωN−1N1−pα∗p∗σnN(1−pα∗p∗),

   where ω_N–1 is the volume of the unit sphere in ℝ^N.

   Taking n → ∞ and applying limn→∞ σ_n = 0 leads to

   limn→∞∫RN|un|pα∗dy⩽C(N+1)S−pα∗pωN−1N1−pα∗p∗limn→∞σnN(1−pα∗p∗)=0,

   which yields a contradiction to limn→∞ ∫_ℝ^N |u_n|^p* dx > 0 given in Condition C.

   Using limn→∞ σ_n = σ̄ ≠ 0, up to a subsequence, we have σn∈(σ¯2,2σ¯) and

   2pσ¯p∫B(xn,σn)|un(y)|pdy⩾C1>0,

   which gives

   ∫B(xn,σn)|un(y)|2dy⩾C1σ¯p2p>0. (3.3)

3. We show that {x_n} is a bounded sequence.

   By way of contradiction, we can assume |x_n| → ∞ as n → ∞. According to Lemma 2.2, we have

   sup|x|>0|un(x)|⩽C|x|p(N−1)−α(p−1)p2∥un∥Wrad1,p(RN,α)⩽C|x|p(N−1)−α(p−1)p2.

   For any C1σ¯p2p|B(0,2σ¯)|1p>ε>0, there exists an M > 0 such that for any n > M there holds

   |un(x)|⩽C||xn|−σn|p(N−1)−α(p−1)p2⩽ε, x∈Bc(0,||xn|−σn|).

   From B(x_n, σ_n) ⊂ B^c(0, ||x_n| – σ_n|), it follows that

   ∫B(xn,σn)|un(y)|pdy⩽εp∫B(xn,σn)dy=εp|B(xn,σn)|=εp|B(0,σn)|⩽εp|B(0,2σ¯)|<C1σ¯p2p.

4. Note that {x_n} is bounded. There exists 0 < C̃ < ∞ such that 0 ⩽ |x_n| < C̃. In view of limn→∞ σ_n = σ̄ ≠ 0, up to a subsequence, we have

   B(xn,σn)⊂B(0,|xn|+σn)⊂B(0,C~+2σ¯).

   From (3.1), we deduce that

   0<C1σ¯p=C1limn→∞σnp⩽limn→∞σnplimn→∞σn−p∫B(xn,σn)|un|pdy=limn→∞∫B(xn,σn)|un|pdy⩽limn→∞∫B(0,C~+2σ¯)|un|pdy.

   Applying the embedding Wrad1,p (ℝ^N, α) ↪ Drad1,p (ℝ^N) ↪ Llocp (ℝ^N), we obtain u ≢ 0.□

Lemma 4.1

Assume that all the conditions described in Theorem 1.2 hold. For each L > 2, define

Set Ū_L = UULp(μ−1) , where μ > 1. Let C̃_t be the best embedding constant from Wrad1,p (ℝ^N, α) to L^t(ℝ^N) for t ∈ [ pα∗ , p^*]. Then we have

Proof

Let U be a nonnegative ground state solution of equation (𝓠). We show that Ū_L ∈ Wrad1,p (ℝ^N, α). By a straightforward calculation, we get

and

This implies that Ū_L ∈ Wrad1,p (ℝ^N, α).

Note that U is a nonnegative ground state solution of equation (𝓠). Then

Substituting Ū_L into the above equation, we get

Since

it follows that

Note that