Fractional Choquard Equations with Confining Potential With or Without Subcritical Perturbations

Remark 2.1.

If inf⁡V>0, then HVs⁢(ℝN)↪Hs⁢(ℝN), that is, there exists C>0 such that

Indeed, we have

so we have the continuous embedding.

Proposition 2.2.

Let N>2⁢s and γ∈[0,+∞). If V∈C⁢(RN), V⁢(x)≥V0>0 and

then there exists a constant C>0 such that

that is, the embedding HVs⁢(RN)↪L2⁢(RN;|x|γ⁢d⁢x) is continuous.

If, in addition,

then the corresponding embedding is compact. Moreover, when γ=0, the embedding HVs⁢(RN)↪Lq⁢(RN) is compact for every q∈(2,2*).

Proof of Proposition 2.2.

Since (2.2) holds, we can take λ∈(0,+∞) such that

Then there exists k>0 sufficiently large so that, if x∈ℝN∖B⁢(0,k), we have V⁢(x)≥λ⁢|x|γ. So, multiplying by u2 and integrating in ℝN∖B⁢(0,k), we obtain

Then, since γ≥0 and V⁢(x)≥V0>0, we can write

so (2.3) holds.

As for the compactness, if (2.4) holds, let ℱ⊂HVs⁢(ℝN) be a bounded set, and take a sequence (vn)n⊂ℱ. Up to a subsequence, we may assume that vn⇀v in HVs⁢(ℝN) as n→∞, so we want to prove that vn→v in L2⁢(ℝN;|x|γ⁢d⁢x) as n→∞. Of course, we can assume that v≡0. Since vn is bounded in HVs⁢(ℝN); by assumption (2.4), for every ε>0, there exists R>0 such that

Since vn∈HVs⁢(ℝN) for all n∈ℕ, we have vn∈HVs⁢(B⁢(0,R)) and vn⇀0 in HVs⁢(B⁢(0,R)) as n→∞. By the fractional Rellich–Kondrakov theorem, HVs⁢(B⁢(0,R)) is compactly embedded in L2⁢(B⁢(0,R)), so it follows that vn→0 in L2⁢(B⁢(0,R)) as n→∞. Since γ≥0, the space L2⁢(B⁢(0,R)) is naturally embedded in the weighted space L2⁢(B⁢(0,R);|x|γ⁢d⁢x), so vn→0 in L2⁢(B⁢(0,R);|x|γ⁢d⁢x). Therefore, there exists N1>0 such that, for every n≥N1, we have

Then, for n≥N1, by (2.5) and (2.6), we obtain

This proves that ∫ℝN|x|γ⁢vn2⁢d⁢x→0 as n→∞, that is, vn→0 in L2⁢(ℝN;|x|γ⁢d⁢x), so the desired embedding is compact.

To conclude the proof, in the case γ=0, we use some interpolation to show that the previous sequence (vn)n in HVs⁢(ℝN) is compact in Lq⁢(ℝN) with q∈(2,2*). To this purpose, take q¯=2⁢NN-2⁢s if N>2⁢s; then there exists β∈(0,1) such that

Using the interpolation inequality, we have ∥vn∥Lq≤C1⁢∥vn∥L2β⁢∥vn∥Lq¯1-β. By the fractional Sobolev inequality (see [6]), we know that ∥vn∥Lq¯≤C⁢∥vn∥Hs, so

By (2.1), we have

In the end, we have shown that

Since we have just proved that vn→0 in L2⁢(ℝN) as n→∞, (vn)n being bounded in HVs⁢(ℝN), we get

This concludes the proof. ∎

Remark 2.3.

Although we are not interested in the case N≤2⁢s, we notice that the previous result holds true also in this situation with minor adaptations.

Theorem 2.4 (Hardy–Littlewood–Sobolev Inequality).

Let 0<α<N and 1<p<q<∞ with 1q=1p-αN. Then there exists C=C⁢(p,α,N)>0 such that

Remark 2.5.

Since, in the previous result, we have q=N⁢pN-α⁢p, we can simply say that Iα*f∈LN⁢pN-α⁢p⁢(ℝN) and

Theorem 2.6 (Stein–Weiss Inequality).

Let

with 0<λ<N, 1<p<∞, α<N-Np, β<Nq, α+β≥0 and

If p≤q<∞, then ∥Tλ⁢f⁢(x)⁢|x|-β∥Lq⁢(RN)≤A⁢∥f⁢(x)⁢|x|α∥Lp⁢(RN), where A=A⁢(p,α,β,λ).

Proposition 2.7.

Let N>2⁢s and α∈(0,N). If V∈C⁢(RN), V≥V0>0, satisfies

then there are two well-defined mappings

which are continuous for p∈(1,N+αN-2⁢s). If, in addition,

the mappings above are of weak to strong type.

Proof.

The sign of the exponent in (2.7), that is, N+αp-N, gives us the asymptotic behavior of V⁢(x) when |x|→∞, so we study separately the cases N+αp-N<0 and N+αp-N≥0.

(I) The case p>N+αN.

Continuity and weak to strong property for φ. In this case, |x|N+αp-N→0 as |x|→∞, so (2.7) implies that we are in the case γ=0 of Proposition 2.2, that is, lim inf|x|→+∞⁡V⁢(x)>0. Take u∈HVs⁢(ℝN); then, by (2.1), we have HVs⁢(ℝN)↪Hs⁢(ℝN) and

By the fractional Sobolev embedding, we have Hs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN) and

provided that the exponent 2⁢N⁢pN+α satisfies the condition

This implies

which is indeed the case we are considering. Thus, taking u∈HVs⁢(ℝN), we have u∈L2⁢N⁢pN+α⁢(ℝN), and so |u|p∈L2⁢NN+α⁢(ℝN), being

By the Hardy–Littlewood–Sobolev inequality with 0<α2<N, p=2⁢NN+α and f=|u|p, being

since N+α>α and α<N, and

we get

This means that the Riesz integral operator, which maps

is a linear and bounded operator. In the end, we get that the first map we are considering, that is, (2.8) is well defined.

As for the continuity, by (2.1), we have HVs⁢(ℝN)↪Hs⁢(ℝN), and by the fractional Sobolev embedding, Hs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN). Moreover, the Nemytskii operator

is continuous; see [19]. Then, as we said before, the Riesz integral operator (2.15) is linear and bounded, so it is continuous. It follows that the composition of these maps (2.8) is continuous, so we get the first part of the claim.

If (2.10) holds, the embedding HVs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN) is compact for N-2⁢sN+α<1p<NN+α by Proposition 2.2. As a consequence, if we take (un)n in HVs⁢(ℝN) such that un⇀u in HVs⁢(ℝN), we have un→u in L2⁢N⁢pN+α⁢(ℝN). From the continuity of the previous maps, we get Iα2*|un|p→Iα2*|u|p in L2⁢(ℝN). This proves that (2.8) is of weak to strong type, as we claimed.

Continuity and weak to strong property of ψ. For the second map, starting with u∈HVs⁢(ℝN), we have, as before,

for p such that (2.13) holds. Then, from u∈L2⁢N⁢pN+α⁢(ℝN), it follows that |u|p∈L2⁢NN+α⁢(ℝN). Now, we can use again the Hardy–Littlewood–Sobolev inequality with the same p and f as before, but with α in place of α2, being

so we get

By u∈L2⁢N⁢pN+α⁢(ℝN), we also have |u|p-2⁢u∈L2⁢NN+α⁢pp-1⁢(ℝN) since

Now, we want to use the fact that Iα*|u|p∈L2⁢NN-α⁢(ℝN) and |u|p-2⁢u∈L2⁢NN+α⁢pp-1⁢(ℝN) to prove that

To do this, we will use the Hölder inequality. We have the exponents 2⁢NN-α and 2⁢NN+α⁢pp-1, so, from

we get

So, with these exponents, we can use the Hölder inequality to get

which means that (2.17) holds, as we claimed. Since 2⁢N⁢pN+α is the Hölder conjugate of 11-N+α2⁢N⁢p, we can identify L11-N+α2⁢N⁢p⁢(ℝN) with the dual of L2⁢N⁢pN+α⁢(ℝN), so we have

Now, since HVs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN), by duality, (L2⁢N⁢pN+α⁢(ℝN))′↪(HVs⁢(ℝN))′, so (Iα*|u|p)⁢|u|p-2⁢u∈(HVs⁢(ℝN))′. In the end, we obtain that the second map (2.9) is well defined.

As for the continuity again, we start with the continuous embeddings

As above, the map (2.16) is continuous, and with the Hardy–Littlewood–Sobolev inequality, we showed that the linear map

is bounded; hence it is continuous. With the same arguments, we also have that the map

is continuous, and by the Hölder inequality, the map

is continuous. Then, as before, we identify L11-N+α2⁢N⁢p⁢(ℝN) with the dual of L2⁢N⁢pN+α⁢(ℝN), that is,

Then, from the continuous embedding HVs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN), considering the dual spaces, the embedding (L2⁢N⁢pN+α⁢(ℝN))′↪(HVs⁢(ℝN))′ is continuous. So, composing the maps, we get (2.9) that is, a continuous map, as we claimed.

If, in addition, (2.10) holds, then the embedding HVs⁢(ℝN)↪L2⁢N⁢pN+α⁢(ℝN) is compact for N-2⁢sN+α<1p<NN+α by Proposition 2.2. Hence, if we take a sequence (un)n in HVs⁢(ℝN) such that un⇀u in HVs⁢(ℝN), then un→u in L2⁢N⁢pN+α⁢(ℝN). Again from the continuity of the maps, it follows that (Iα*|un|p)⁢|un|p-2⁢un→(Iα*|u|p)⁢|u|p-2⁢u in (HVs⁢(ℝN))′. So we proved that the map (2.9) is of weak to strong type, as we wanted.

(II) The case p≤N+αN(<2).

Continuity and weak to strong property of φ.

Again, we start with u∈HVs⁢(ℝN). Since N+αp-N≥0, by Proposition 2.2 with γ=N+αp-N, we have a continuous embedding

and

Moreover, the operator

is well defined since

Then, using the Stein–Weiss inequality with λ=N-α2, β=0, q=2, N+α-N⁢p2 in place of α and 2p in place of p, we claim that

that is,

Now, we show that the conditions on the exponents hold. Indeed, we have

since N>α. Then, from p≤N+αN, we get

Finally, the equation

is verified, and we can apply the Stein–Weiss inequality, as claimed.

As a consequence, the linear operator

is bounded, and so it is continuous. In the end, the composition map

is well defined.

As for the continuity, from the continuous embedding (2.18), we get that the map

is continuous. As before, the map (2.20) is continuous. Then, by (2.22), the map (2.23) is continuous. By composition, we obtain that (2.24) is continuous.

If we are in the case (2.10), then the weak to strong property follows from Proposition 2.2, which gives the compactness of the embedding (2.18). So, if we take a sequence (un)n in HVs⁢(ℝN) such that un⇀u in HVs⁢(ℝN), then un→u in L2⁢(ℝN;|x|N+αp-N⁢d⁢x). From the continuity of the maps, the strong convergence still holds, so we have Iα2*|un|p→Iα2*|u|p in L2⁢(ℝN). This means that the desired map (2.24) is of weak to strong type, so we get the first part of the claim.

Continuity and weak to strong property of ψ. For the second map, starting again with u∈HVs⁢(ℝN), by Proposition 2.2, we have the continuous embedding (2.18) with

so u∈L2⁢(ℝN;|x|N+αp-N⁢d⁢x). Then, as before, the operator (2.20) is well defined, being

Similarly, the operator

is well defined; in fact,

By the Stein–Weiss inequality, we claim that

that is,

This implies that the linear operator

is bounded, so it is continuous as well. To prove this, we use again the Stein–Weiss inequality with λ=N-α, f=|u|p, q=22-p, β=N+α-N⁢p2, 2p in place of p and N+α-p⁢N2 in place of α.

Now, using Iα*|u|p∈L22-p⁢(ℝN;|x|-N+α-p⁢N2-p⁢d⁢x) together with |u|p-2⁢u∈L2p-1⁢(ℝN;|x|N+αp-N⁢d⁢x), we want to prove that

Indeed, using the Hölder inequality with exponents 1p-1 and 12-p, being p-1+2-p=1, we have

From this, we have that

is well defined. Now, we claim that

As a matter of fact, starting with u∈L2⁢(ℝN;|x|N-N+αp⁢d⁢x), we can consider u¯:=|x|N2-N+α2⁢pu∈L2(ℝN) and define

so that T∈(L2⁢(ℝN;|x|N+αp-N⁢d⁢x))′.

On the other hand, starting with T∈(L2⁢(ℝN;|x|N+αp-N⁢d⁢x))′, by the Riesz representation theorem, there exists a unique u¯∈L2⁢(ℝN;|x|N+αp-N⁢d⁢x) such that

Now, we can define u:=u¯|x|N+αp-N so that u∈L2⁢(ℝN;|x|N-N+αp⁢d⁢x). So, for every f∈(L2⁢(ℝN;|x|N+αp-N⁢d⁢x))′, we proved that there exists a unique u∈L2⁢(ℝN;|x|N-N+αp⁢d⁢x), and thus we get (2.28), as claimed.

From this, we know that (Iα*|u|p)⁢|u|p-2⁢u∈(L2⁢(ℝN;|x|N+αp-N⁢d⁢x))′. Then, from Proposition 2.2, we have the continuous embedding HVs⁢(ℝN)↪L2⁢(|x|N+αp-N⁢d⁢x;ℝN), so, reasoning with the dual spaces, we get that the embedding

is continuous. Thus, composing the maps, we get

which is well defined.

For the continuity, we take u∈HVs⁢(ℝN), and from Proposition 2.2, we have the continuous map

As above, the map (2.20) is continuous, as well as the map (2.25), doing the same calculations with p-1 instead of p. From the Stein–Weiss inequality, we obtained that the linear map (2.26) is bounded, so it is continuous. Then, combining the last two maps with the Hölder inequality, we obtain that the map (2.27) is continuous. Then we identify L2⁢(ℝN;|x|N-N+αp⁢d⁢x) with the dual of L2⁢(ℝN;|x|N+αp-N⁢d⁢x) as before. As a consequence of Proposition 2.2, we have the continuous embedding (2.29), so the map

is continuous. Composing the maps, we obtain (2.30), which is a continuous map, as we stated.

If, in addition, (2.10) holds, as in the other cases, the weak to strong property again follows from the compactness of the embedding (2.18) given again by Proposition 2.2. So we take (un)n such that un⇀u in HVs⁢(ℝN), and then un→u in L2⁢(ℝN;|x|N+αp-N⁢d⁢x). From the continuity of the maps, we obtain (Iα*|un|p)⁢|un|p-2⁢un→(Iα*|u|p)⁢|u|p-2⁢u in (HVs⁢(ℝN))′. So the map (2.30) is of weak to strong type, and this concludes the proof. ∎