On the fractional p-Laplacian problems

This paper deals with nonlocal fractional p-Laplacian problems with difference. We get a theorem which shows existence of a sequence of weak solutions for a family of nonlocal fractional p-Laplacian problems with difference. We first show that there exists a sequence of weak solutions for these problems on the finite-dimensional subspace. We next show that there exists a limit sequence of a sequence of weak solutions for finite-dimensional problems, and this limit sequence is a sequence of the solutions of our problems. We get this result by the estimate of the energy functional and the compactness property of continuous embedding inclusions between some special spaces.


Introduction
The nonlocal fractional p-Laplacian problems with difference appear in the models of nonlinear fractional Laplace flows such as the parabolic boundary value problems with time derivative and the fractional p-Laplacian differential operators. The fractional Laplacian flows arise in applications of nonlinear elasticity theory, electro rheological fluids, non-Newtonian fluid theory in a porous medium (cf. [9,31,40]).
In this paper we consider a family of the fractional p-Laplacian problems of Rothe type with difference under boundary and initial conditions: (-) s g p u n + λV (x)|u n | p-2 u n + |u n | r-1 u n -|u n-1 | r-1 u n-1 h = 0 in , (1.1) where is a bounded domain of R N , N ≥ 3, with smooth boundary ∂ , s ∈ (0, 1), p is a real constant, 2 ≤ p ≤ N , r = p * s -1 = Np N-sp -1, g p is a continuous function defined by g p (t) = |t| p-2 t, t = 0, g p (0) = 0, λ > 0, V : → [0, ∞) is a continuous function, and u n is a measurable function defined on with valued into R, n = 1, 2, . . . , and (-) s g p is the fractional p-Laplacian operator defined as follows: for each x ∈ R N and any u ∈ C ∞ 0 ( ), (-) s g p u(x) = P.V. g p |u(x)u(y)| |x -y| s u(x)u(y) |u(x)u(y)| dy |x -y| N+s = P.V.
In the last years, for pure mathematical research and concrete real-world applications, the fractional p-Laplacian operator has been studied on the fractional Sobolev space W s L g p ( ) = u ∈ L g p ( ) : |u(x)u(y)| p |x -y| N+sp dx dy < ∞ , where L g p ( ) is the Banach space defined by L g p ( ) = u : → R is a measurable function : The fractional p-Laplacian operator and the fractional Sobolev space arise in many fields of science, for example, elastic mechanics (see [40]), electro-rheological fluid dynamics(see [31]), and image processing (see [6]) and the references therein. When 0 < s < 1, (-) s is the usual fractional Laplacian operator defined by: for each x ∈ R N and any u ∈ C ∞ 0 ( ), where P.V. denotes the Cauchy principle value. Since 0 < s < 1, (-) s is called the fractional Laplacian operator. For the fractional Laplacian operator, see [8,10,19] and the references therein. The fractional Laplacian problems arise from continuum mechanics, phase transition phenomena, population dynamics, minimum surfaces, and game theory. The body of literature on the fractional Laplacian operators and their applications is quite large. We refer the reader to [3,12,13,[24][25][26][27][28][29][33][34][35][36][37][38] and the references therein. For the basic properties of the fractional Sobolev spaces, we refer the readers to [10]. If s → 1 -, (-) 2 reduces to -. For s = 1, we identify (-) s with the classical Laplacian operator -. If 2 < s < ∞, (1.1) is called s-exponent problems of elliptic type. The s-exponent Laplacian problems of elliptic type appear in a lot of applications, for example, elastic mechanics, electro-rheological fluid dynamics, and image processing. We refer the readers to [2,9,11,22,23,31] and the references therein. In [5,6,16], there are some papers concerning related equations involving the fractional Laplacian operator, but results for fractional Sobolev spaces and the fractional Laplacian operator with exponent are few. In particular, the fractional Laplacian operator with variable exponent was suggested firstly by Lorenzo and Hartley [20]. The fractional Laplacian operator with variable exponent and the variable exponent fractional Sobolev space have appeared in a nonlinear diffusion process. Some diffusion processes reacting to temperature changes can be explained well by fractional derivatives in a nonlocal integro differential operator (see [21]). In [17,18,30], the authors consider the pseudodifferential equations on the fractional Sobolev spaces.
In recent years, the Kirchhoff equations involving fractional p-Laplacian have attracted interest and have been researched by some mathematicians. In particular, when s = 1 and p = 2, -is the classical Laplace operator. Ji, Fang, and Zhang [15] provided multiplicity results of solutions for asymptotically linear Kirchhoff equations by using a variant version of mountain pass theorem and the variational method. When 0 < s < 1 and p = 2, Fiscella [14] provided the existence of solution for a class of Kirchhoff-type problems involving fractional Laplacian operator singular term and a critical nonlinearity. When 0 < s < 1 and 1 < p < N/s, Xiang, Zhang, and Rǎdulescu [39] obtained multiplicity results for superlinear Schrödinger-Kirchhoff equations involving fractional N/s-Laplacian with critical exponential nonlinearity by using the concentration compactness principle in the fractional Sobolev and mountain pass theorem. When 0 < s < 1 and p = N/s, Mingqi, Rǎdulescu, and Zhang [25] provided existence and multiplicity of solutions for Kirchhoff equations involving fractional N/s-Laplacian with critical nonlinearity by the mountain pass geometry and Ekeland's variational principle. They [26] also obtained the existence and multiplicity results of solutions for Kirchhoff equations involving fractional N/s-Laplacian with singular exponential nonlinearity by using the same methods.
The weak solutions u n ∈ W s L g p ( ) of (1.1) are a measurable function defined on with valued into R, n = 1, 2, . . . , and satisfy the following in weak sense: for any w ∈ W s L g p ( ) and Our main result is as follows. The outline of the proof of Theorem 1.1 is as follows: We first prove the existence of a sequence of weak solutions for a family of the fractional p-Laplacian difference equations defined on the finite-dimensional subspace. We next show that there exists a limit sequence of the sequence of weak solutions for the finite-dimensional problem, and this limit sequence is the sequence of the solutions of our problem. We get this result by the estimate of the energy functional and the compactness property of the continuous embedding inclusions between some special spaces. In Sect. 2, we introduce the fractional Lebesgue space with exponent and the fractional Sobolev space and give some properties. In Sect. 3, we first prove that problem (1.1) defined on the finite-dimensional subspace has a sequence of weak solutions for each n = 1, 2, . . . . In Sect. 4, we show that there exists a limit sequence of the sequence of weak solutions for finite-dimensional problem, and this limit sequence is a sequence of solutions of our problem (1.1).

Preliminaries
For the variational setting for our problem, we introduce some definitions and theories on the fractional Lebesgue space with exponent and the fractional Sobolev space.
Let N ≥ 3 and be a bounded open domain in R N with smooth boundary ∂ . Let 2 ≤ p < ∞ and r = Np N-sp -1. The Lebesgue space with p-exponent is The Sobolev space with p-exponent is Then L p ( ) and W 1,p ( ) are Banach spaces. We also define the Sobolev space W 1,p 0 ( ) as the closure of C ∞ 0 ( ) in W 1,p ( ). The space is also a reflexive Banach space. If p is bounded, then norm · W 1,p ( ) is equivalent to the norm [·] W 1,p ( ) . If p = ∞, L ∞ ( ) is the Banach space of essentially bounded. If p is bounded and p is the conjugate exponent of p defined by p = p p-1 , then the dual space (L p ( )) can be identified with L p ( ). If 1 < p < ∞, then the Lebesgue space L p ( ) with p-exponent is separable and reflexive. In L p ( ), Let L g p ( ) be the space defined by Then (L g p ( ), u L gp ) is a Banach space whose norm is equivalent to the Luxemburg norm In L g p ( ), Hölder's inequality is valid: Now we introduce the fractional Sobolev space with p-exponent. Let 0 < s < 1 and 2 ≤ p < ∞. The fractional Sobolev space with p-exponent is defined by Let W s 0 L g p ( ) denote the closure of C ∞ 0 ( ) in the norm u s,g p . The following lemma shows that the norm [·] s,g p is a norm of W s L g p ( ) equivalent to · s,g p .

Lemma 2.2 ([32]
; Generalized Poincaré inequality) Let 0 < s < 1 and 2 ≤ p < ∞. Then there exists a positive constant C > 0 such that That is, the embedding is continuous and compact. Furthermore, [u] s,g p is a norm of W s 0 L g p ( ) equivalent to · s,g p .
, then there exists a constant C 1 = C 1 (N, p, q, s) > 0 such that Proof By Theorem 6.7 and Theorem 6.9 of [10], for N > sp and any fixed constant exponent q ∈ (1, Np N-sp ], W s 0 L g p ( ) is continuously embedded into L g q ( ). It follows that (2.2) holds. By combining inequalities (2.1) and (2.2), [u] s,g p is an equivalent norm of W s 0 L g p ( ). It follows that (2.3) holds.

Lemma 2.4
Let 0 < s 1 < s < s 2 < 1 and 2 ≤ p < ∞. Then the embeddings Moreover we have It follows from this inequality that we can easily verify that the embedding W s 2 0 L g p ( ) → W s 0 L g p ( ) is continuous. Similarly, for any u ∈ W s 0 L g p ( ), we have Thus we have It follows that the embedding W s 0 L g p ( ) → W s 1 0 L g p ( ) is continuous. Thus the proof of the lemma is complete. Furthermore, [u] s,g p is a norm of W s 0 L g p ( ). Moreover, there exists a constant C 2 = C 2 (N, p, s) > 0 such that Since 0 < s < 1 and N > sp, there exists a constant τ 1 > 0 such that Since is bounded, there exist a constant > 0 and l numbers of disjoint hypercubes on V i , i = 1, 2, . . . , l. By Lemma 2.3, Theorem 6.7, and Theorem 6.9 of [12], there exists a constant D = D(N, s, p) such that By Hölder's inequality, if q ∈ (1, p * s ], we have We note that Thus we have It follows from (2.4) that Thus the embedding W s L g p ( ) → L g q ( ) is continuous. Furthermore, we show that the embedding is compact. In fact, in the constant p * s on V i , for q ∈ (1, p * s ] on V i , the embedding W s L g p (V i ) → L g q (V i ) is compact. Thus the embedding W s L g p ( ) → L g q ( ) is compact. It follows that there exists a constant D = D(N, p, s) > 0 such that u g q ≤ D u s,g p . By Lemma 2.2, we have the following lemma.
We need the following inequality for the p-Laplacian operator.
Then there exist constants C 1 and C 2 depending on p and N such that, for any ξ , η ∈ R N , We recall a fundamental fact, which is a crucial role for our main result.

Lemma 2.8 ([4]) Assume that Q is a continuous vector field from R N to R N and satisfies
for some ρ > 0. Then there exists a point x ∈ B ρ (0) such that where B ρ (0) denotes a ball centered at the origin with radius ρ in R N .

Existence of approximating solutions
In this section we show that there exists a unique approximating solution for (1.1) on each finite-dimensional subspace. Let us choose a family of bases is dense in W s 0 L g p ( ), any element u n in W s 0 L g p ( ) and the initial data u 0 can be expanded as Let us define the finite subspace F k of W s 0 L g p ( ) by Let N be any positive integer which shall be sent to infinity and h be any small positive number. For any fixed integer k = 1, 2, . . . , let u n,k = k i=1 a i n,k φ i (x) be a family of the Galerkin approximating solutions for a family of fractional Laplace equations with p-exponent and difference defined on the finite-dimensional subspaces.
Let us set Let us define the functional J i n,k (ρ) by Let us define the functional J n,k = (J 1 n.k , . . . , J k n,k ) : R k → R k . Then J n,k is continuous on ρ and satisfies We claim that J n,k (ρ) · ρ ≥ 0. In fact, by Young's inequality and generalized Poincaré's inequality of Lemma 2.2, for any > 0, there exists a constant C > 0 such that
Proof (i) The sequence u n-1 ∈ L g r+1 ( ) is defined inductively and by Lemma 3.2, {u n,k } is bounded in W s 0 L g p ( ). Since the embedding W s 0 L g p ( ) → L g q ( ) is continuous and compact for any q with 1 ≤ q < Np N-p = r + 1, the embedding W s 0 L g p ( ) → L g r ( ) is continuous and compact. Thus the sequence {u n,k } has a subsequence, up to a subsequence, {u n,k } converging strongly to lim k→∞0 u n,k = u n in L g r ( ).
(ii) By Lemma 2.7 (i), there exist constants C > 0 and C > 0 such that |u n,k | r-1 u n,k -|u n | r-1 u n dx ≤ C |u n,k | r-1 + |u n | r-1 |u n,ku n | dx ≤ C |u n,k | r + |u n | r dx r-1 r |u n,ku n | r dx 1 r ≤ C u n,ku n L gr .
Since by (i) u n,k → u n strongly as k → ∞ in L g r ( ) and u n ∈ L g r ( ), it follows that |u n,k | r-1 u n,k -|u n | r-1 u n ∈ L 1 ( ).
Proof of Theorem 1.1 By Lemma 3.1, for each n = 1, 2, . . . , N and k = 1, 2, . . . , there exists a unique weak solution u n,k ∈ F k ⊂ W s 0 L g p ( ) of (3.1). By Lemma 4.1, there exists a subsequence, up to a subsequence, {u n,k } converging strongly to lim k→∞∞ u n,k = u n in L g r ( ).
We shall show that u n satisfies (1.1). That is, we shall show that, for any w ∈ W s 0 L g p ( ), (-) s g p u n · w dx + λ V (x)|u n | p-2 u n · w dx + |u n | r-1 u n -|u n-1 | r-1 u n-1 h · w dx = 0, i.e., |u n (x)u n (y)| p-2 |x -y| s(p-2) u n (x)u n (y) |x -y| s w(x)w(y) |x -y| N+s dx dy In fact, for any w ∈ W s 0 L g p ( ), let w k = k i=1 h n,i φ i (x) be the approximating sequence which converges to w in W s 0 L g p ( ). By Lemma 2.7 (ii), there exists a constant C 2 > 0 such that On the other hand, putting w = u n,k in (3.1), we have -(-) s g p u n,k · u n,k dx = λ V (x)|u n,k | p-2 u n,k · u n,k dx + |u n,k | r-1 u n,k -|u n-1 | r-1 u n-1 h · u n,k dx. Taking the test function as w ku n,k in (3.1), we have -(-) s g p u n,k · (w ku n,k ) dx = λ V (x)|u n,k | p-2 u n,k · (w ku n,k ) dx + |u n,k | r-1 u n,k -|u n-1 | r-1 u n-1 h · (w ku n,k ) dx. By adding (4.1) and (4.2), we have ((-) s g p w k · (w ku n,k ) + λ V (x)|u n,k | p-2 u n,k · (w ku n,k ) dx + |u n,k | r-1 u n,k -|u n-1 | r-1 u n-1 h · (w ku n,k ) dx ≥ 0. By the energy estimate theorem Lemma 3.2, there exists a constant C 1 > 0 such that (-) s g p u n,k · u n,k dx + λ V (x)|u n,k | p dx + r (r + 1)h |u n,k | r+1 dx = |u n,k (x)u n,k (y)| p |x -y| N+sp dx dy + λ V (x)|u n,k | p dx + r (r + 1)h |u n,k | r+1 dx it follows that the sequence {u n,k } is bounded in L g p ( ) and so, up to a subsequence, u n,k converges to u n weakly in L g p ( ).
Passing to the limit as k → ∞, we have that the first part and the second part of the lefthand side of (4.4) ((-) s g p w k · (w ku n,k ) + λ V (x)|u n,k | p-2 u n,k · (w ku n,k ) dx − → ((-) s g p w · (wu n ) + λ V (x)|u n | p-2 u n · (wu n ) dx. On the other hand, by (ii) of Lemma 4.1, u n,k → u n a.e., in and by Vitali's converging theorem, up to a subsequence, |u n,k | r-1 u n,k converges to |u n | r-1 u n weakly in L r+1 r ( ).