On fractional logarithmic Schrödinger equations

: We study the following fractional logarithmic Schrödinger equation: where N 1 ≥ , Δ s ( ) − denotes the fractional Laplace operator, s 0 < and V x N (cid:2) (cid:2) ( ) ( ) ∈ . Under di ﬀ erent assumptions on the potential V x ( ) , we prove the existence of positive ground state solution and least energy sign - changing solution for the equation. It is known that the corresponding variational functional is not well de ﬁ ned in H s N (cid:2) ( ) , and inspired by Cazenave ( Stable solutions of the logarithmic Schrödinger equation , Nonlinear Anal. 7 ( 1983 ) , 1127 – 1140 ) , we ﬁ rst prove that the variational functional is well de ﬁ ned in a subspace of H s N (cid:2) ( ) . Then, by using minimization method and Lions ’ concentration - compactness principle, we prove that the existence results. 35R11

. Under different assumptions on the potential V x ( ), we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H s N ( ), and inspired by Cazenave (Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127-1140), we first prove that the variational functional is well defined in a subspace of H s N ( ). Then, by using minimization method and Lions' concentration-compactness principle, we prove that the existence results.

Introduction
In this article, we study the following fractional logarithmic Schrödinger equation: where P.V. stands for the principle value and C N s , is a normalization constant, see for instance [1] and references therein.
There are few results on fractional logarithmic Schrödinger equation. For example, Ardila [9] recently studied the existence and stability of standing waves for nonlinear fractional Schrödinger equation (1.2). In [10], d'Avenia et al. employed non-smooth critical point theory and obtained infinitely many standing wave solutions to equation (1.2).
Equation (  It is easy to see that u u log 2 2 N ∫ < +∞, but the functional fails to be finite since the logarithm is singular at origin. Indeed, let u be a smooth function that satisfies One can verify directly that u H s N ( ) ∈ but u u x log d 2 2 N ∫ = −∞. Thus, I fails to be 1 on H s N ( ). Due to loss of smoothness, the classical critical point theory cannot be applied for I . The same difficulty also occurs for s 1 = . In order to investigate the following logarithmic Schrödinger equation: several approaches developed so far in the literature. Cazenave [5] worked in a suitable Banach space endowed with a Luxemburg-type norm, in which the corresponding energy functional is well defined. Squassina and Szulkin [12] applied non-smooth critical point theory for lower semi-continuous functionals, see also [13][14][15]. Tanaka and Zhang [16] used penalization technique. Recently, via direction derivative and constrained minimization method, Shuai [17] proved the existence of ground state solutions and signchanging solutions for equation (1.6). Recently, Wang and Zhang [18] proved that the positive ground state solution of the power-law equations up to translations, which is the unique positive solution of the logarithmic equation They also proved that the same result holds for bound state solutions.
Inspired by [5], we first find a suitable Banach space, in which the energy functional I is well defined. Then, we study the existence of positive ground state solution and least energy sign-changing solution for equation (1.1).
Throughout this article, we assume V x and bounded from below. The following different types of potential are considered: , and the inequality is strict in a subset of positive Lebesgue measure.
which is equipped with the Luxemburg norm Now, we give the definition of weak solutions for equation (1.1).
Before stating our main results, we give some notations. Denote Our first result can be stated as follows. If u is a sign-changing solution to equation (1.1), then u ∈ , and we can easily check that Hence, by (1.8)-(1.10), one obtains which is totally different from the case s 1 = , because equation (1.1) is a nonlocal problem. And we cannot directly infer that I u c 2 ( ) > . This property is called energy doubling by Weth [19], which is crucial in overcoming the difficulty of lack of compactness. Now, we focus on whether c or m is achieved.
Theorem 1.2. If one of the following four conditions hold: then c is achieved. If one of the following three conditions hold: V 1 ( ); V 2 ( ) and N 2 ≥ ; V 4 ( ) and V 5 ( ), then m is achieved. In particular, if m is not achieved provided V 3 ( ) holds.
The proof of Theorem 1.2 is based on the concentration-compactness principle [20]. However, as we will see, the nonlocal operator Δ s ( ) − and the logarithmic nonlinearity cause some obstacles, which need some new technique and subtle analysis.
where p 2 2 s < < * . Even for equation (1.11), the results on the existence of sign-changing solutions are new.
In the rest of the article, we shall first prove some preliminary results and prove Theorem 1.1 in Section 2, and then we prove Theorem 1.2 in Section 3. We will use C to denote different positive constants from line to line.

Proof of Theorem 1.1
In this section, we first prove some technical lemmas, which is crucial for proving our main results.
is the unique pair of positive numbers such that α u β u u u . By direct computation, we have Without loss of generality, we assume that Combining (2.3) and (2.7), we deduce that On the other hand, by using a similar argument to (2.4) and (2.6), we obtain that . If there exists another pair Hence, by the above analysis, we conclude that Proof. Without loss of generality, we assume that Combining (2.10) with (2.11), we deduce that Proof. By the proof of Lemma 2.1, we deduce that α β , So it is sufficient to check that the maximum is not achieved on the boundary of × + + . Without loss of generality, we may suppose by contradiction that Proof. The proof is inspired by Proposition 2.2 in [22]. Let us define, for β 1 ≥ and T 0 > large enough, Obviously, φ is convex and differential function.
. Observe that by the Sobolev embedding theorem, we have the following inequality: where S N s , ( ) is the best fractional Sobolev constant defined by We point out that since φ u ( ) grows linearly, both sides of (2.16) are finite.
To prove this, we take R large to be determined later. Then, Hölder's inequality gives By the Monotone convergence theorem, we may take R so that In this way, the second term above is absorbed by the left-hand side of (2.16) to obtain in the right-hand side of (2.17) and letting T → +∞ in the left-hand side, since β 2 2 s 1 = * , we obtain can be written as be a non-negative solution of (1.1), we prove the conclusion by two cases.
be the solution of the equation . Now we look at the equation . Similar to the proof of Case 1, we can deduce that u s σ N loc On the other hand, it holds . Consequently, u is a positive ground state solution to equation (1.1).
(ii) Now, suppose that u ∈ and I u m ( ) = , then u + , u 0 ≠ ≤ . It is easy to check that  Now, we prove that  On the other hand, one has and u x u y u x u y x y x y d d 0.

Existence and nonexistence of minimizer
In this section, we study whether c or m is achieved under different types of potential. The following Brézis-Lieb-type lemma for u u log 2 2 is crucial.
Therefore, we can derive Taking α 0 > is small enough in ( Similarly, one can prove Thus, Lemma 2.1 yields a unique α β ,    , v n { } is still a bounded minimizing sequence of c, we may assume that, up to a subsequence, there exists v H\ 0

Periodic potential
a.e. on .
Thus, by a similar argument above, we also have which is a contradiction. As a result, we have J v 0 ( ) = . Consequently, we have Therefore, for any ε 0 > , we can choose a nonnegative function u ∈ such that I u c ε. ( ) < + Without loss of generality, we may assume that Combining this with u ∈ , direct computation yields that Hence, α β 1 = > . On the other hand, it follows from Let k → ∞, by (3.8) and (3.9), we have where o 1 0 k ( ) → as k → +∞. Therefore, α 1 → and β 1 → as k → ∞. It follows from Lemma 2.3 that Since ε is arbitrary, we deduce that m c 2 , ≤ which is contradiction to Theorem 1.1. Thus, we complete the proof. □

V α ( ) potential
In this subsection, we prove c and m are achieved under conditions V 4 ( ) and V 5 ( ). In order to overcome the difficulties caused by the loss of compactness, we first study the following problem in bounded domain, the solutions will be used as minimizing sequences.
Obviously, I is well defined on X B s R 0 ( ) and I X B , . Therefore, one can easily verify the following result.
where o 1 Moreover, it also holds that Therefore, . □ Next, we need to study the following limiting functional: It follows from Lemma 3.3 that c 0 > ∞ is achieved.
where α w is the constant such that which leads to a contradiction. Therefore, x n { } is bounded and thus u u n → in L p N ( ) for p 2 2 s < < * . Proceeding as the arguments in the proof of Lemma 3.2, we can prove that c is achieved. □ Lemma 3.9. Assume V 4 ( ) holds, and let u x ( ) be a positive ground state solution of equation ( Note that the function w is continuous away from the origin and w x 0 ( ) > on B R holds. We claim that w x 0 ( ) > on B R c as well. Suppose on the contrary that w is strictly negative somewhere in B R c . Since w x 0 ( ) → as x | | → +∞ and w x 0 ( ) > on B R , this implies that w attains a strict global minimum at some point x B R c 0 ∈ with w x 0 0 ( ) < . By using the singular integral expression for Δ s ( ) − , it is easy to see that This is a contradiction, and we conclude that w x 0 ( ) ≥ on N . Thus, we conclude Therefore, the result is directly from (3.29) and (3.30). □ From Lemma 3.8, there exists u ∈ , w ∈ ∞ such that I u c ( ) = and I w c ( ) = ∞ ∞ . Furthermore, with the same argument as that in proving Theorem 1.1, we may assume u 0 > and w 0 > . Similar to Lemma 3.9, one obtains We have the following estimate.