Nonlinear fractional equations with supercritical growth

Recently, as observed in [16], a great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for the pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts such as, among the others, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, and water waves.


Introduction and main result
Recently, as observed in [16], a great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for the pure mathematical research and in view of concrete real-world applications.This type of operators arises in a quite natural way in many different contexts such as, among the others, the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, and water waves. 1 The author is supported by Research Fund of National Natural Science Foundation of China (No. 11601046), Chongqing Science and Technology Commission (No. cstc2016jcyjA0310), Chongqing Municipal Education Commission (No. KJ1600603), and Program for University Innovation Team of Chongqing (No. CXTDX201601026). 2 Corresponding author.
c Vilnius University, 2017 We consider the non-local fractional Laplacian equation (N > 1) where Ω be a bounded domain in R N with smooth boundary ∂Ω and 1 < q < 2.Here L K u is the non-local fractional Laplacian operator.The nonlocal operator L K is defined as follows: where K : R N \ {0} → (0, +∞) is a measurable function with the following property: where γ(x) = min{|x| 2 , 1}; (1) A typical example for K is given by singular kernel K(x) = |x| −(N +2s) .In this case, problem (P) becomes where (−∆) s u is the fractional Laplacian operator with (up to normalization factors) may be defined as [1, 2, 5-8, 12-30, 32-34] and the references therein for further details on the factional Laplacian operator.The weight function b will be possibly sign-changing and the assumption for b is as follows: (A1) b(x) ∈ C( Ω), and there is a nonempty open subset Ω of Ω such that b(x) > 0 in Ω .
A special case of our main result is the following theorem.
https://www.mii.vu.lt/NARemark 1.This type of equations have been studied extensively [1,2,[6][7][8] in the subcritical and critical case.But these equations have not been well studied in the supercritical case, that is r > 2N/(N − 2s).Applying Theorem 3 to (3), our theorem includes results in the supercritical one.The exponent r in Theorem 1 can be critical or supercritical in the sense of Sobolev embedding because the solutions (u n ) we obtained are small solutions with u n L ∞ (Ω) → 0 and we only give the assumptions for f near zero.We use a suitable cut-off technique to overcome the exponent r is supercritical.This idea is from [30].Now, we give the assumptions on f : The main result is as follows.
Theorem 2. Let 1 < q < 2 and assume (A1)-(A3) are satisfied.Then (P) has a sequence of solutions Following the same idea, we can also consider the so-called fractional p-Laplacian equation where Ω ⊂ R N is a bounded domain, L p K u is the fractional p-Laplacian operator where K : R N \ {0} → (0, +∞) is a measurable function with the following property: where γ(x) = min{|x| p , 1}; Moreover, 1 < p < ∞ and 1 < q < p.We need the following assumption for nonlinearity f : Theorem 3. Let 1 < q < p and assume (A1), (A3) and (A4) are satisfied.Then (P ) has a sequence of solutions Remark 2. For results on existence of multiple solutions for fractional Laplacian or p-Laplacian equations by using Nehari manifold, see, for example, [2,9,10].
L. Li et al.

Preliminarily
In this section, we first give some basic results and the functional space that will be used in the next section, which was introduced in [23].Let 0 < s < 1 be a real number and the fractional critical exponent 2 * s be defined as In the following, we denote and C = R N \ Ω. W is a linear space of Lebesgue measurable function from R N to R such that the restriction to Ω of any function u in W belongs to L 2 (Ω) and The space W is equipped with the norm We shall work in the closed linear subspace According to the conditions of K, we have that C ∞ 0 (Ω) ⊂ W 0 , and so W and W 0 are nonempty.The space W 0 is endowed with the norm defined by Since u ∈ W 0 , then the integral in ( 6) can be extended to all R 2N .Moreover, the norm on W 0 given in ( 6) is equivalent to the usual one defined in (4), by Lemma 6 in [23].For the framework of fractional Sobolev space, we refer the reader to the survey of Di Nezza, Palatucci and Valdinoci [4].
In the following, we denote by W s,2 (Ω) the usual fractional Sobolev space endowed with the norm (the so-called Gagliardo norm) Taking into account Lemma 5 in [23], we have the following result.
We will use the following theorem, which is a variant of a result due to Clark [3], to prove our main result.
, where X is a Banach space.Assume Φ satisfies the Palais-Smale (PS) condition, is even and bounded from below, and Φ(0) = 0.If for any k ∈ N, there exists a k-dimensional subspace X k and ρ k > 0 such that sup Last, we show that the weak solutions of (P) are bounded in L ∞ (Ω).This result was established in [31,Thm. 3.1] and proved by using the De Giorgi-Stampacchia iteration method.
Proposition 1.Let u ∈ W 0 be a weak solution of problem (P) and the nonlinearity is subcritical growth.Then u ∈ L ∞ (Ω), and there exists C > 0 possibly depending on N , s, Ω such that

Proof of Theorems 2 and 3
The proof is motivated by the arguments in [11,30].We shall only give the proof of Theorem 2 since the proof of Theorem 3 is similar.Denote by λ 1 the first eigenvalue of −L K with Dirichlet boundary condition on Ω.As in [30], we first modify f so that the nonlinearity is defined for all (x, u) ∈ Ω × R.
Choose β ∈ (0, θ/16) and F ∞ (u) = β|u| 2 .Using ρ and F ∞ , we define Then, for |u| 2α, we have It is easy to see that, for all (x, u) ∈ Ω × R, Therefore, α and f defined above satisfy all the properties stated in the lemma.
We now consider the modified problem https://www.mii.vu.lt/NA whose solutions correspond to critical points of the functional The construction of Ĩ together with (8) shows that Ĩ is C 1 , even, bounded from below, and coercive, and therefore satisfies the (PS) condition.
We are ready to prove Theorem 2.
Proof of Theorem 2. In order to apply Theorem 4 to Ĩ, we only need to find for any k ∈ N a subspace X k and ρ k > 0 such that sup X k ∩Sρ k Ĩ < 0. For any k ∈ N, we find k linearly independent functions e 1 , . . ., e k in C ∞ 0 (Ω ).We define X k := span{e 1 , . . ., e k }.By (A1), we may assume b(x) > b 0 > 0 in k i=1 supp e i for some constant b 0 .For u ∈ X k , using (8) in Lemma 2, we have which implies the existence of ρ k > 0 such that sup X k ∩Sρ k Ĩ < 0 since the dimension of In view of ( 7) and ( 9), we see that u k with k large are solutions of (P).The proof is complete.
Proof of Theorem 1.If r ∈ (p, ∞), then the result is a consequence of Theorem 3. If r ∈ (q, p), then we just apply Theorem 4 to the functional to obtain the result.

X
k is finite.According to Theorem 4, there exists a sequence of negative critical values c k Nonlinear Anal.Model.Control, 22(4):521-530 of Ĩ satisfying c k → 0 as k → ∞.For any k, let u k be a critical point of Ĩ associated with c k .Then u k are solutions of (9) and they form a (PS) sequence.Without loss of generality, we may assume that u k → u in W 0 as k → ∞.Then u satisfies Ĩ(u) = 0 = Ĩ (u), u .Therefore, u = 0 according to Theorem 4, and u k → 0 in W 0 as k → ∞.Proposition 1 shows that u k → 0 in L ∞ (Ω) as k → ∞.