Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in R^N involving fractional Laplacian

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ \alpha\in(0,1) $, $ N\ge 2 $, $\lambda>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.


Introduction
A celebrated result of Du and Ma [10] asserts that the uniqueness positive solution of −∆u = λu − u p in R N for N ≥ 1, λ > 0 and p > 1, is u ≡ λ 1 p−1 .Moreover, in [10], the authors also consider the following logistic type equation: where p > 1, a and b are positive smooth function in R N satisfying a(x) → a ∞ > 0 and b(x) → b ∞ > 0 as |x| → ∞.
Then they proved that problem (1.1) has a unique positive solution for each λ > 0. A similar problem for quasi-linear operator has been studied by Du and Guo [9].
In the present work, we are interested in understanding whether similar results hold for equations involving a nonlocal diffusion operator, the simplest of which is perhaps the fractional Laplacian.For α ∈ (0, 1), we study the following fractional elliptic problem: where N ≥ 2. The fractional Laplacian is defined, up to a normalization constant, by Our first main result is for some β > 0 and ω = 1/(1 + |y| N +2α ) is a nonnegative solution of (1.2).Then u must be a constant if p verifies Remark 1.1 We notice that As in [10] and [9], our proof of this result based on a comparison principle for concave sublinear problems (see Lemma 2.1) and involves boundary blowup solutions.We use a rather intuitive squeezing method to proof Theorem 1.1 as follows.Denote B R as a ball centered at the origin with radius R. Then problem has a unique positive solution v R if R is large enough for any fixed λ > 0.
On the other hand, the following boundary blow-up propblem for some g ∈ L 1 (R N \ BR , ω) and λ > 0, has a positive solution w R for any R > 0. The comparison principle implies that any entire positive solution of (1.2) satisfies v R ≤ u ≤ w R in B R .Moreover, one can show (see Lemmas 2.2 and 2.3 in Section 2) that both v R and w R converge locally uniformly to λ Next, we make use of Theorem 1.1 to study logistic type fractional elliptic problems with variable coefficients that are asymptotically positive constants.More precisely, we study the following problem where a and b are positive smooth function in R N .Moreover, we suppose that We can prove that We prove Theorem 1.2 by a similar argument as in the proof of Theorem 1.2, we consider the Dirichlet problem and the boundary blow-up problem in a ball B R .When R is large, these problems have positive solutions v R and w R respectively.By comparison principle, as R → ∞, v R increase to a minimal positive solution of (1.5) and w R decrease to a maximal positive solution of (1.5).Therefore, when (1.5) has a unique positive solution, v R and w R approximate this unique solution from below and above, respectively.
We mentioned that, in [10] and [9], the existence and uniqueness results hold provided p > 1, but in our Theorems 1.1 and 1.2 we require p satisfying (1.3).This is because we will use Perron's method (we refer the reader to User's guide [6] for the presentation of Perron's method which extends to the case of nonlocal equations, see for example [3,4,11]) to construct solution of problem 1.4 by applying Proposition 2.2 and choosing (2.13)).This implies Moreover, in [5], the authors proved that τ 0 (α) has a simplicity formula, that is, τ 0 (α) = α − 1.Thus, we have This article is organized as follows.In Section 2 we present some preliminary lemmas to prove a comparison principle involving the fractional Laplacian, existence and asymptotic behaviors of boundary blow-up solutions.Section 3 is devoted to prove the existence and uniqueness results of problems (1.2) and (1.5), i.e., Theorems 1.1 and 1.2.

Preliminary lemmas
In this section, we introduce some lemmas which are useful in the proof of our main results.The first important ingredient is the comparison principle involving the fractional Laplacian which is useful in dealing with boundary blow-up problems.
Lemma 2.1 (Comparison principle) Suppose that Ω is a bounded domain in R N , a(x) and b(x) are continuous functions in Ω with a L ∞ (Ω) < ∞ and b(x) nonnegative and not identity zero.Suppose u 1 , u 2 ∈ C 2α+β (Ω) for some β > 0 are positive in Ω and satisfy in Ω (2.1) In order to prove Lemma 2.1, we need the following proposition.
Proposition 2.1 For u ≥ 0 and v > 0, we have where Moreover, the equality holds if and only if u = kv a.e. for some contant k.
We note that Proposition 2.1 is a special case (p = 2) of Lemma 4.6 in [13] and we omit the proof here.
Proof of Lemma 2.1.Let φ 1 and φ 2 be nonnegative functions in C ∞ 0 (Ω).By (2.1), we obtain that For ε > 0, we denote ε 1 = ε and ε 2 = ε/2 and let By our our assumption, v i is zero near ∂Ω and in ≤ C and thus it remains to verify that the Gagliardo norm of v 1 in R N is bounded by a constant.Using the symmetry of the integral in the Gagliardo norm with respect to x and y and the fact that v 1 = 0 in R N \ Ω, we can split as follows Next, we estimate both integrals in the right hand side of (2.3) is finite.We first notice that, for any y ∈ R N \ D 0 , Hence, the second term in the right hand side of (2.3) is finite by the above inequality.In order to show the first term in the right hand side of (2.3) is also finite, we need the following estimates and Combining (2.4) and (2.5), we have In the last inequality of above estimate, we have used the fact u 1 , u 2 ∈ C 2α+β (Ω).This implies On the other hand, by Theorem 6 in [12], we know that v i can be approximate arbitrarily closely in the X α 0 (D 0 ) norm by C ∞ 0 (D 0 ) functions.Hence, we see that (2.2) holds when φ i is replaced by We notice that the integrands in the right hand side of (2.2) (with φ i = v i ) vanishing outside D(ε).Next, we prove the left hand side of (2.2) in nonpositive.We first divide R 2N into four disjoint region as: Therefore, ( It follows that Hence, A similar argument implies that By Proposition 2.1, we know that L(u 1 , u 2 )(x, y) ≤ 0 in D(ε) × D(ε).Therefore, Summing up these estimates from A 1 to A 4 , we know that the left hand side of (2.2) is nonpositive.
On the other hand, as ε → 0, the first term in the right hand side of (2.2) converges to while the last term in the right side of (2.2) converges to 0. Next, we show that D(0) = ∅.Suppose to the contrary that D(0) = ∅.Since the left side of (2.2) is nonpositive by the estimates from A 1 to A 4 and right hand side of (2.2) tends to 0 as ε → 0, we easy deduce This imply that b ≡ 0 in D(0) and Hence, by Proposition 2.1, we know u 1 = ku 2 in D(0) for some constant k.Since b ≡ 0 in Ω, it follows from the above that D(0) = Ω.Thus, D(0 On the other hand, we have . Therefore, we must have D(0) = ∅ and thus u 1 ≥ u 2 in Ω.We complete the proof of Lemma 2.1.
By applying this comparison principle together with the Perron's method for the nonlocal equation, we can obtain the following two lemmas.

Lemma 2.2
Let Ω be a bounded domain in R N with smooth boundary and p > 1. Suppose a and b are smooth positive functions in Ω, and let µ 1 denote the first eigenvalue of (−∆) α u = µa(x)u in Ω with u = 0 in R N \ Ω.Then equation has a unique positive solution for every µ > µ 1 .Furthermore, the unique solution u µ satisfies u µ → [a(x)/b(x)] 1/p−1 uniformly in amy compact subset of Ω as µ → +∞.
Proof.(Existence) The existence follows from a simple sub-and supersolution argument.In fact, any constant great than or equal to M = maxΩ[a(x)/b(x)] 1/(p−1) is a super-solution.Let φ be a positive eigenfunction corresponding to µ 1 (for the existence of the first eigenvalue and corresponding eigenfunction has been obtained in [13] and [15]), then for each fixed µ > µ 1 and small positive ε, εφ < M and is a sub-solution.Therefore, by the sub-and super-solution method (see [14]), there exist at least one positive solution.
(Uniqueness) If u 1 and u 2 are two positive solutions, by Lemma 2.1, we have u 1 ≤ u 2 and u 2 ≤ u 1 both hold in Ω.Hence, u 1 = u 2 .This proves the uniqueness.
On the other hand, let φ be a positive eigenfunction corresponding to µ 1 .Then we can find a small neighborhood of ∂Ω in Ω, say U, such that φ is very small in U. Therefore, for all µ > µ 1 + 1, we have (2.7) By shrinking U further if necessary, we can assume that Ū ∩ K = ∅ and φ < v 0 − ε in U. Next, we choose smooth function w ε as where l is a positive function such that w ε is smooth in Ω and satisfying l ≤ v 0 − ε/2.Moreover, we let otherwise we choose φ = φ/C for some constant C > 0 large replace φ.Then we can see that, for x ∈ Ω \ U, for all large µ.For x ∈ U, by (2.7) and (3.7), we have (2.10) for µ > µ 1 + 1.Finally, combining (2.9) and (2.10), we know w ε is a subsolution of our problem for all large µ.Since w ε < v ε , we deduce that Lemma 2.3 Let Ω, a and b be as in Lemma 2.2.Suppose p verifies (1.3), then equation has at least one positive solution for each µ > 0 if the measurable function g µ satisfying where positive constant C is independent of µ.Furthermore, suppose u µ is a positive solution of (2.11), then u µ satisfies u µ → [a(x)/b(x)] 1/(p−1) uniformly in amny compact subset of Ω as µ → +∞.
We first recall the following result in [4].Assume that δ > 0 such that the distance function where τ is a parameter in (−1, 0) and the function l is positive such that V τ is C 2 in Ω.
for τ ∈ (−1, 0) and χ (0,1) is the characteristic function of the interval (0, 1), Next, we will the existence result in Lemma 2.3 by applying Perrod's method and thus we need to find ordered sub and super-solution of (2.11).
As in [4], we begin with a simple lemma that reduce the problem to find them only in A δ .

Lemma 2.4
Let Ω, a and b as in Lemma 2.2.Suppose U and W are order super and sub-solution of (2.11) in the sub-domain A δ .Then there exists λ large such that U λ = U + λη and W λ = W − λη are ordered super and sub-solution of (2.11), where Proof.The proof is similar as Lemma 4.1in [4] and we just need replace V in Lemma 4.1in [4] to η for our lemma.So we omit the proof here.Now we in position to prove Lemma 2.3.Proof of Lemma 2.3.(Existence) We define Moreover, we know that G µ is continuous (see Lemma 2.1 in [4]) and nonnegative in Ω.Therefore, if u is a solution of (2.11), then u − gµ is the solution of and vice versa, if u is a solution of (2.14), then u + gµ is a solution of (2.11).
(Asymptotic behaviour) Let K be an arbitrary compact subset of Ω, 1) in Ω and ε > 0 any small positive number satisfies where τ is a parameter in (−1, 0), λ and η defined as in Lemma 2.4 and the function l is positive such that w ε is C 2 in Ω.By a similar argument as Proposition 3.2 in [4], there exists δ 1 ∈ (0, δ) and constants c > 0 and C > 0 such that c(1 for all x ∈ A δ 1 and τ ∈ (−1, α − 1).Hence, for x ∈ A δ and δ > 0 small, if µ is large enough.Hence, wε is sub-solution in A δ .By applying Lemma 2.4, we know that w ε = wε − λη + gµ is a sub-solution of problem (2.11) for all large µ > 0.
On the other hand, we define choose a function where τ is a parameter in (−1, 0), λ and η defined as in Lemma 2.4 and the function l is positive such that v ε is C 2 in Ω.By a similar argument as Proposition 3.2 in [4], there exists δ 1 ∈ (0, δ) and constants c > 0 and C > 0 such that for all x ∈ A δ 1 and τ ∈ (−1, α − 1).Hence, for x ∈ A δ and δ > 0 small, (2.14).Hence, ṽε is sub-solution in A δ .By applying Lemma 2.4, we know that v ε = ṽε + λη + gµ is a super-solution of problem (2.11) for all large µ > 0.
As w ε < v ε in Ω, we must have w ε ≤ u µ ≤ v ε in Ω.This implies that u µ → v 0 in K as µ → ∞, as required.We complete the proof of this Lemma.
uniformly in B r and thus we may also assume that b n ≥ b ∞ /2 in B r for all n.
It is easily to check that εφ n is a subsolution of (3.6) for every n if we choose ε small enough.Furthermore, (2a ∞ /b ∞ ) 1/(p−1) is a supersolution of (3.6) for all n.Then (3.6) has a positive solution w n satisfies εφ n ≤ w n ≤ (2a ∞ /b ∞ ) 1/(p−1) .Then, using the regularity results again, we know w n converges in C 2α+β loc (B r ) to some function w satisfying εφ Applying Lemma 2.2, we know the above problem has a unique positive solution.Therefore, w = w r is uniquely determined and the whole sequence w n converges to w r .By the comparison principle (see Lemma 2.1), we know that Next, we show u n has a uniformly bounded in R N for all n large enough, that is, there exists a positive constant C independent of n such that u n (x 0 ) ≤ C for any x 0 ∈ R N .We define, for any t > 0, u t,n (x) = u n [x 0 + t(x − x 0 )].
Then u t,n satisfying (−∆) α u = t 2α (ã n u − bn u p ) in R N , where ãn (x) = a n (x 0 + t(x − x 0 )) and bn (x) = b n (x 0 + t(x − x 0 )).On the other hand, since ãn → a ∞ and bn → b ∞ uniformly in B where B denote the unit ball with center x 0 , we may assume ãn ≤ 2a ∞ and bn ≥ b ∞ /2 in B for all n.
We consider the following problem (3.9) As a argument before, we know u t,n ∈ L 1 (R N , ω) for t and n large enough.Thus, by applying Lemma 2.3, we know this problem has at least one positive solution.Let v t is a solution of (3.9), then v t → (4a ∞ /b ∞ ) 1/(p−1) as t → ∞ at x = x 0 ∈ B. Then the comparison principle deduce that u t,n ≤ v t in B and thus u n (x 0 ) = u t,n (x 0 ) ≤ v t (x 0 ).
Letting t → ∞ in the above inequality we conclude that u n (x 0 ) ≤ (4a ∞ /b ∞ ) 1/(p−1) as we required.Hence, u n L ∞ (R N ) ≤ C for all n large enough, where constant C > 0 independent of n.On the other hand, u n ∈ C 2α+β loc (R N ) implies that u n converges uniformly to some function u ∞ and (−∆) α u n → (−∆) α u ∞ in B r is strongly as n → +∞.Hence, u ∞ is nonnegative and satisfies Furthermore, u ∞ ≥ w r > 0. Thus u ∞ is a positive solution and |u ∞ (0) − (a ∞ /b ∞ ) 1/(p−1) | ≥ ε 0 due to the choice of x n .
Choose a sequence r = r 1 ≤ r 2 ≤ • • • ≤ r m → ∞ as m → ∞.We can apply the above argument to each r m and then use a diagonal process to obtain a positive solution U of where v(y) = u(x) = u(θy).In fact, we can choose θ = (b ∞ ) −1/(2α) .Then applying Theorem 1.1 to the above equation, we have v ≡ (a ∞ /b ∞ ) 1/(p−1) .

Theorem 1 . 2
Let λ > 0. Suppose a and b are positive smooth function in R N and satisfying (1.6).Then equation (1.5) has a unique positive solution if p verifies (1.3).