ENTIRE SOLUTIONS FOR CRITICAL p-FRACTIONAL HARDY SCHRÖDINGER KIRCHHOFF EQUATIONS

Existence theorems of nonnegative entire solutions of stationary critical p-fractional Hardy Schrödinger Kirchhoff equations are presented in this paper. The equations we treat deal with Hardy terms and critical nonlinearities and the main theorems extend several recent results on the topic. The paper contains also some open problems. 2010 Mathematics Subject Classification: Primary: 35R11, 35J60; Secondary: 35J20, 35B09.


Introduction
Nonlinear problems involving fractional diffusions appear in several areas of applied mathematics, as described by Caffarelli in [8,9] and by Vázquez in [42,43].Indeed, many recent papers are devoted to problems involving fractional and nonlocal operators, and concerning models in optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, and game theory.Several contributions have also been given for nonlinear fractional Schrödinger equations, fractional porous medium equations, and general nonlinear problems of any type.For a recent survey on some up-to-date developments we refer to the recent survey [39].
In particular, the paper deals with stationary fractional Kirchhoff p-Laplacian equations, involving critical nonlinearities, a topic of great appeal after the publication of the paper [22] due to Fiscella and Valdinoci.We refer e.g. to [2,10,16,29,30,32,36,37] and the references therein for details.But the equations treated here contain also Hardy terms, which make the analysis more delicate and quite interesting.For related problems we just mention [5,10,11,21] and the references cited in there.
Since we are interested in nonnegative entire solutions for applications in geometry and physics, the first equation we treat is of critical p-fractional Hardy Schrödinger Kirchhoff type, that is where γ and λ are real parameters, 0 < s < 1 < p < ∞, sp < N , and u + = max{u, 0}.The operator (−∆) s p is the fractional p-Laplacian, which for every function ϕ ∈ C ∞ 0 (R N ) may be defined, up to normalization factors, as Throughout the paper the weights K and V satisfy (K) K ≥ 0 a.e. in R N and K ∈ L ∞ (R N ), (V ) V ∈ C(R N ) and V (x) ≥ V 0 > 0 for all x ∈ R N , where V 0 is a positive constant, while the main Kirchhoff function M verifies the condition (M) M : R + 0 → R + 0 is a nonnegative continuous function and there exists θ ∈ [1, N/(N − ps)) such that tM (t) ≤ θM (t) for any t ∈ R + 0 , where M (t) = t 0 M (τ ) dτ .Note that • p,V is a uniformly convex norm on the weighted Lebesgue Banach space L p (R N , V ) by (V ).
The main framework for (1.1) is the space E, defined as the completion of C ∞ 0 (R N ) with respect to the norm • , introduced in (1.2).Denote by D s,p (R N ) the p-fractional Beppo-Levi space, that is the completion of C ∞ 0 (R N ) with respect to Gagliardo semi-norm [ • ] s,p .Theorems 1 and 2 of [31] give for all u ∈ D s,p (R N ), where C N,p is a positive constant depending only on N and p.Thus, the fractional Sobolev embedding D s,p (R N ) → L p * s (R N ) and the fractional Hardy embedding D s,p (R N ) → L p (R N , |x| −ps ) are continuous, but not compact.It is also evident that E → D s,p (R N ).Let us introduce the best fractional critical Sobolev and Hardy constants S = S(N, p, s) and H = H(N, p, s) given by Of course, the numbers S and H are strictly positive.We refer to Theorem 1.1 of [23] for the sharp Hardy constant H. Throughout the paper we require the following structural assumptions on f and g.
(F) f : R N × R → R is a Carathéodory function and there exists an exponent q ∈ (θp, p * s ) such that either (f 1 ) f (x, t) = w(x)(t + ) q−1 for a.a.x ∈ R N and all t ∈ R, where w > 0 a.e. in R N and w ∈ L ℘ (R N ), with ℘ = p * s /(p * s − q), or (f 2 ) f verifies both assumptions (a) there exists a positive function (G) g : R N × R → R is a Carathéodory function and there exist exponents r and µ in (θp, p * s ) such that for all ε > 0 there exists for a.a.x ∈ R N and all t ∈ R, and either (i) θp < µ < q and µG(x, t) ≤ t g(x, t) for a.a.x ∈ R N and all t ∈ R, where G(x, t) = t 0 g(x, τ ) dτ , or (ii) q ≤ µ < p * s and 0 ≤ µG(x, t) ≤ t g(x, t) for a.a.x ∈ R N and all t ∈ R. For examples of subcritical nonlinear terms which satisfy conditions (F) and (G) we refer to [10].The condition, assumed in [36], namely inf{G(x, t) : x ∈ R N , |t| = 1} > 0, is no longer required here and in [10] thanks to the possible presence of the nontrivial nonlinearity f .
For the next main existence result for (1.1), because of the possible presence of g, we assume also (V) There exists R > 0 such that for any c > 0 Condition (V) is weaker than the property V (x) → ∞ as |x| → ∞ usually required in Schrödinger problems.Assumption (V) was originally introduced by Bartsch and Wang in [6] to overcome the lack of compactness in problems defined in the entire space R N .In harmony with [10], we define κ = κ(q, µ, M ) by Clearly κ ∈ (0, a], being θ ≥ 1 and p ≤ θp < τ by assumptions (F) and (G).There are cases, besides the obvious one M ≡ a, in which κ = a, that is θ = 1 in (M), as shown in Section 2 of [10].
Thanks to the variational nature of (1.1), under the above structural assumptions, (weak) solutions of (1.1) are exactly the critical points of the underlying functional J γ,λ , which satisfies the geometry of the mountain pass lemma.The solutions constructed for problem (1.1) are given in terms of critical points u γ,λ of J γ,λ determined at special mountain pass levels.These solutions are briefly called mountain pass solutions.
Finally, if f , g : R N × R → R are Carathéodory functions, with the property that for a.a.
and if h is a nonnegative perturbation term of class L ν (R N ), where ν is the conjugate exponent of some fixed ν ∈ [p, p * s ], then the nonhomogeneous equation associated with (1.1), that is admits only nonnegative solutions in R N , provided that λ ≥ 0 and γ < a H.
The last part of Theorem 1.1, that is when g ≡ 0 in (1.1), takes somehow inspiration from the paper [41] and covers also the interesting case in which V is a positive constant.See also Theorem 1.1 of [10].Theorem 1.1 completes the picture given in Theorem 1.1 of [20].
As in [10], we can study equation (1.1), requiring only (V ) on the potential V , but including the term g, provided that K, V , f , and g are radial functions in x.
The second problem we consider comes from the equation where the parameters satisfy the previous assumptions, and f (x, t) = w(x)(t + ) q−1 is as in (F)-(f 1 ), that is q ∈ (θp, p * s ), the weight w > 0 a.e. in R N and of class L ℘ (R N ), with ℘ = p * s /(p * s − q).The function h can be viewed again as a nonnegative perturbation term and h is assumed in the second part of the paper to be nontrivial, nonnegative and of class L ν (R N ), where ν is the conjugate exponent of some fixed ν ∈ [p, p * s ].For problem (1.9) we also assume for simplicity that K > 0 a.e. in R N and crucially that M is in standard form, that is there exists θ ∈ [1, N/(N − ps)) such that for all t ∈ R + 0 .Clearly, condition (1.10) is stronger than the previous assumption (M).
A very natural appealing open problem is to prove existence of nontrivial solutions for problems (1.1), (1.8), and (1.9) in the degenerate case, that is when M (0) = 0 and M (t) > 0 for all t > 0.
The paper is organized as follows.In Section 2 we prove the existence Theorem 1.1 for the Hardy Schrödinger Kirchhoff problem (1.1) and the asymptotic behavior (1.6).Section 3 deals with the proof of Theorem 1.3.Finally, in Section 4 we extend Theorems 1.1 and 1.3 to settings having wider applications, replacing the fractional p-Laplacian operator by a general nonlocal integro-differential operator, generated by a singular kernel K, satisfying the natural assumptions described by Caffarelli, e.g., in [8].See also [39].

The non-degenerate Hardy-Schrödinger-Kirchhoff equation (1.1)
Here we prove the existence result for problem (1.1) and we recall that throughout this section (M), (K), (V ), and (1.3) hold.First, by Theorem 6.7 and Corollary 7.2 of [15] we have the following embedding result for the uniformly convex Banach space E defined in the introduction.The fact that E is a uniformly convex Banach space can be easily derived following the main arguments of Proposition A.9 of [3], or Lemma 10 of [36], or Lemma A.6 of [37].
From the proof of Lemma 2.2 it is apparent that if e is the nontrivial nonnegative radial function determined for some γ ∈ (−∞, aH) and λ 0 > 0, then e is such that J γ,λ (e) < 0 for all λ ≥ λ 0 and e ≥ 2 > ρ = ρ(γ + , λ), being ρ ∈ (0, 1]. We recall in passing that, if X is a real Banach space, a C 1 (X) functional J satisfies the Palais-Smale condition at level c ∈ R if any Palais-Smale sequence (u n ) n at level c, that is any sequence (u n ) n , with the property that (2.7) admits a strongly convergent subsequence in X. Fix γ ∈ (−∞, aH), λ > 0 and put Obviously, c γ,λ > 0 thanks to Lemma 2.2.
We are going to prove that J γ,λ satisfies the Palais-Smale condition at level c γ,λ in E. To this aim, we show an asymptotic property of the levels c γ,λ .This crucial idea is strongly related to Lemma 2.3 of [10] (see also Lemma 4.3 of [21] and Lemma 6 of [22] for a somehow similar fractional non-degenerate Kirchhoff Dirichlet problem in bounded regular domains).The next lemma indeed is useful to obtain (1.6) and, most importantly, to defeat the lack of compactness due to the presence of a Hardy term and a critical nonlinearity.In order to get the Palais-Smale condition at level c γ,λ in E, the presence of g forces to also assume (V) on the potential V , as we shall see in the proof of the main Lemma 2.4.

Fix a sequence (λ
Hence, there exists a subsequence of (λ k ) k , still relabeled (λ k ) k , and a number ≥ 0 such that t γ,λ k → as k → ∞.Assume by contradiction that > 0.Then, by (F), (G), and the Lebesgue dominated convergence theorem, we obtain since e ≥ 0 in R N and e ≥ 2. Hence, (2.8) and the argument above gives which is the desired contradiction.In conclusion, = 0, being the sequence (λ k ) k , with λ k → ∞, arbitrary.Consider now the path ξ(t) = te, t ∈ [0, 1], belonging to Γ.By Lemma 2.2, (F), and (2.6) Moreover, M (t p γ,λ e p ) → 0 as λ → ∞, by the continuity of M and the fact that t γ,λ = o(1) as λ → ∞.This completes the proof of the lemma, since e is independent of λ ≥ λ 0 .Now, following the key idea of the proof of Lemmas 2.4 and 4.5 in [10], we prove the validity of the Palais-Smale condition for J γ,λ at level c γ,λ in E. The crucial argument also appears in the proof of Lemma 4.5 in [21], given for Dirichlet problems in bounded domains, when M ≡ 1 and p = 2. Let us recall that κ is the constant given in (1.5).
As already noted in the introduction, besides the obvious case M ≡ a, in which κ = a, there are several non monotone Kirchhoff functions M for which κ = a, that is θ = 1.We refer to [10] for specific simple examples.We also point out that in the proof of the main Lemma 2.4 we use (V) only to get (2.20) and (2.21).Therefore, if g ≡ 0, the assertion of Lemma 2.4 continues to hold under the sole assumption (V ) on the potential V .
Finally, assume that f , g are Carathéodory functions, satisfying (1.7), and that h ∈ L ν (R N ) is nonnegative in R N , with ν = ν/(ν − 1) and ν ∈ [p, p * s ].Let γ < a H and λ ≥ 0 be fixed.Let u be a solution of (1.8) in E. Put u = u + − u − .It is not hard to show that for a.a.
and, as noted in the proof of Lemma 2.4, both u + and u − are still in E. Combining these facts and recalling that |u − (x) − u − (y)| ≤ |u(x) − u(y)| for a.a.x, y ∈ R N , we get at once that In conclusion, u, u − ≤ − u − p .Thus, by the definition of solution for (1.8), taking as test function ϕ = u − ∈ E, we have 3), (1.4), (1.7), and the fact that h ≥ 0 a.e. in R N .Hence, u − = 0, since γ < a H.In conclusion, u − = 0 a.e. in R N and so u is nonnegative in R N , as required.
From the proofs of Lemmas 2.2-2.4 it is evident that in condition (F)-(f 1 ) the function (x, t) → w(x)(t + ) q−1 can be replaced by (x, t) → w(x)|t| q−2 t, and similarly (x, t) → K(x)(t + ) p * s −1 by (x, t) → K(x)|t| p * s −2 t.Thus existence of a nontrivial mountain pass solution of in R N , as well as the validity of (1.6), can be obtained in a similar way.
Of course, we cannot conclude any longer about its sign.
In this section we prove the main existence result for (1.1) in the radial case.To apply the mountain pass theorem and the Ekeland variational principle, we need the following embedding result obtained combining Theorem II.1 of [28] with Lemma 2.1.
Lemma 2.5.Let N ≥ 2. For any p < ν < p * s , the embedding E rad → L ν (R N ) is compact, where E rad = {u ∈ E : u is radially symmetric with respect to 0}.
In order to avoid condition (V) we pass into the radial setting.In order to get the compactness of the embedding E → L ν (R N ), p < ν < p * s , we need to restrict the study searching solutions of (1.1) in E rad , where E rad = {u ∈ E : u is radially symmetric with respect to 0}.
Thus, until the end of the section K, V , f , and g are assumed to be radially symmetric functions in x, and that (M), (V ), (F), and (G) hold, without further mentioning.Again, the geometry stated in Lemma 2.2 continues to hold.Therefore, for any γ < a H and λ > 0 we now put where e ∈ E rad is the function constructed in Lemma 2.2.Then Lemma 2.3 continues to hold without significant adjustments.The notable changes now occur in the proof of Lemma 2.4.
Up to this moment, the function u γ,λ is a solution of (1.1)only in the E rad sense.Let us show that u γ,λ is a solution of (1.1) in the whole space E, that is in sense of definition (2.2).
To this aim we use a version of the well known principle of symmetric criticality, due to Palais in [34], in the form of Proposition 3.1 of [14], which holds in reflexive strictly convex Banach spaces as proved in Lemma 5.4 of [10].
Let SO(N ) denote the special orthogonal group, that is Next, consider the following subgroup of linear operators of E in itself , and since V is a radial function.Furthermore, E rad = {u ∈ E : au = u for all a ∈ G}.
To apply Lemma 5.4 of [10] to the functional J γ,λ , we need to show that J γ,λ • a = J γ,λ for all a ∈ G. Fixed u ∈ E, for all a ∈ G we have since K, V , f , and g are radial functions in x.Hence, J γ,λ satisfies Lemma 5.4 of [10].Now, u γ,λ is critical point of J γ,λ | E rad , that is J γ,λ (u γ,λ ), ϕ (E rad ) ,E rad = 0 for any ϕ ∈ E rad .
Then, Lemma 5.4 of [10] implies that u γ,λ is a critical point of J γ,λ in the whole space E. Thus, u γ,λ is a solution of (1.1) in the sense of definition (2.2).
3. The Schrödinger-Kirchhoff equation (1.9) Throughout the section we assume that (M) and (V ) hold, that q ∈ (θp, p * s ), that the weight w > 0 a.e. in R N and of class L ℘ (R N ), with ℘ = p * s /(p * s − q).We also treat, for simplicity, the only interesting case where K > 0 a.e. in R N .Problem (1.9) has a variational structure and the underlying functional is J γ,λ : E → R, given by Clearly, J γ,λ is well-defined and of class C 1 (E).We first prove that the geometry of the mountain pass lemma is still preserved for (1.9), provided that the nonnegative perturbation h is sufficiently small in the ν -norm, as shown first in [36] and then in [10] for more general equations.
Proof: Fix γ < aH and λ ≥ 0. Take a radial function v ∈ C ∞ 0 (R N ), with v ≥ 0 in R N and v = 1.By (2.3) we have for t → ∞ since λ ≥ 0, h, and v are nonnegative in R N , K > 0 a.e. in R N and of course p ≤ θp < p * s .Hence, taking e = τ 0 v, with τ 0 > 0 sufficiently large, we obtain at once that e ≥ 2 and J γ,λ (e) < 0. Clearly, e depends on neither λ nor h.
Using the notation above, we fix h ∈ L ν (R N ), with 0 < h ν ≤ δ and h ≥ 0 a.e. in R N .First, we claim that there exists a nonnegative function ψ ∈ C ∞ 0 (R N ) such that Since h ∈ L ν (R N ) \ {0}, and h ≥ 0 a.e. in R N , the function Then, there exists a sequence Hence, for n 0 ∈ N large enough we get Thus, by the Hölder inequality, we have , where ρ = ρ(γ + , λ) > 0 is the number given in the previous part of the lemma, by (3.3) we have for t ∈ (0, 1) small enough, since 1 < p ≤ θp < q < p * s .Thus, we obtain that m γ,λ = inf{J γ,λ (u) : u ∈ B ρ } < 0, where B ρ = {u ∈ E : u < ρ}.Then, by the Ekeland variational principle in B ρ and the first part of the lemma, there exists a sequence ( for all n ∈ N and for any v ∈ B ρ .Fixed n ∈ N, for all w ∈ S E , where S E = {u ∈ E : u = 1}, and for all σ > 0 so small that v n + σ w ∈ B ρ , we have Clearly, J γ,λ (u γ,λ ) ≥ m γ,λ , since u γ,λ ∈ B ρ by (3.1).Going if necessary to a subsequence, we may assume that as n → ∞ by Lemma 2.1 and the fact that (v − n ) n strongly converges to zero in E, as shown in the proof of Lemma 3.1.Moreover, by (1.10) and (3.1) we have as n → ∞ as well as u γ,λ H ≤ v n H and u γ,λ p * s ,K ≤ v n p * s ,K as n → ∞.Multiplying the expression in (3.6) by 1/θp and subtracting it below, by (3.5) and the fact that either θ = 1 or γ ≤ 0, we find as n → ∞ In conclusion, u γ,λ is a minimizer of J γ,λ in B ρ and J γ,λ (u γ,λ ) = m γ,λ < 0 < α ≤ J γ,λ (u) for all u ∈ ∂B ρ by Lemma 3.1.Thus in turn u γ,λ ∈ B ρ , so that J γ,λ (u γ,λ ) = 0 and this implies that u γ,λ is a nontrivial nonnegative solution of (1.9), as stated.
Obviously, the operator L K reduces to the fractional p-Laplacian (−∆) s p , when K(x) = |x| −N −ps and usually K 0 ≤ 1.Let us denote by D s,p K (R N ) the completion of C ∞ 0 (R N ) with respect to .
Then E K = (E K , • K ) is a separable uniformly convex Banach space, adapting the arguments of Proposition A.9 of [3].
It is clear that the embeddings E K → W s,p (R N ) → L p * s (R N ) are continuous by the above remarks and Lemma 2.1.A similar argument as in Lemma 2.1 of [10], combined with some ideas taken from Appendix B of [4], shows that the embedding E K → L q (R N , w) is compact.
A (weak) entire solution of It is worth pointing out, as in [2], that it is not restrictive to assume K to be even, since the odd part of K does not give any contribution in the integral above.Indeed, write K = K e + K o , where for all x ∈ R N \ {0} K e (x) = K(x) + K(−x) Actually, the entire solutions of problem (4.1) correspond to the critical points of the energy functional J γ,λ,K : E K → R, for all u ∈ E K defined by where u → H λ (u) is given exactly as in Section 2. Lemmas 2.2 and 2.3 continue to hold for all γ ∈ (−∞, a K 0 H) and all λ > 0 by (K 1 ), with obvious changes in their proof.Similarly, Lemma 2.4 can be proved in almost the same way as before, provided that γ ∈ (−∞, κ K 0 H) and λ > 0, where κ is given in (1.5) as usually.Thus we have proved Theorem 4.1.Suppose that (4.1) is non-degenerate, i.e., that (1.3) holds, and that (M), (V ), (V), (F), and (G) are satisfied.Then for every γ ∈ (−∞, κ K 0 H) problem (4.1) admits a nontrivial mountain pass solution u γ,λ for any λ > 0, whenever K ∞ = 0, and u γ,λ satisfies the asymptotic behavior (1.6).While, if K ∞ > 0, then there exists a threshold λ * = λ * (γ) > 0 such that for any λ ≥ λ * problem (4.1) admits a nontrivial mountain pass solution u γ,λ , satisfying again (1.6).Moreover, if g ≡ 0, then the assertion above continues to hold assuming only condition (V ) on the potential V .
Finally, if f , g : R N × R → R are Carathéodory functions, satisfying (1.7), and if h is a nonnegative perturbation term of class L ν (R N ), where ν is the conjugate exponent of some fixed ν ∈ [p, p * s ], then the non-homogeneous equation associated with (4.1), that is = λf (x, u) + g(x, u) + K(x)(u + ) p * s −1 + h(x) in R N , admits only nonnegative solutions in R N , provided that λ ≥ 0 and γ < a K 0 H.
[u] s,p,K = R 2N |u(x) − u(y)| p K(x − y) dx dy 1/p, which is well-defined by (K 1 ).Clearly, the embedding D s,pK (R N ) → D s,p (R N ) is continuous, being [u] s,p ≤ K −1/p 0 [u] s,p,K for all u ∈ D s,p K (R N ), by (K 2 ).Similarly, E K denotes the completion of C ∞ 0 (R N ) with respect to the norm u K = [u] p s,p,K + u p