Nonlinear equations involving the square root of the Laplacian

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u)&\mbox{ in } \Omega\\ u=0&\mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least two non-trivial $L^{\infty}$-bounded weak solutions is established for large value of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.


Introduction
This paper is concerned with the existence of solutions to nonlinear problems involving a non-local positive operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. More precisely, from the variational viewpoint, we study the existence and non-existence of weak solutions to the following fractional problem where Ω is an open bounded subset of R n (n ≥ 2) with Lipschitz boundary ∂Ω, λ is a positive real parameter, and β : Ω → R is a function belonging to L ∞ (Ω) and satisfying (2) essinf x∈Ω β(x) > 0.
Moreover, the fractional non-local operator A 1/2 that appears in (1) is defined by using the approach developed in the pioneering works of Caffarelli & Silvestre [12], Caffarelli & Vasseur [13], and Cabré & Tan [11], to which we refer in Section 2 for the precise mathematical description and properties. We also notice that A 1/2 which we consider, should not be confused with the integro-differential operator defined, up to a constant, as In fact, Servadei & Valdinoci in [39] showed that these two operators, although often denoted in the same way, are really different, with eigenvalues and eigenfunctions behaving differently (see also Musina & Nazarov [36]). As pointed out in [11], the fractions of the Laplacian, such as the previous square root of the Laplacian A 1/2 , are the infinitesimal generators of Lévy stable diffusion processes and appear in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids, population dynamics, geophysical fluid dynamics, and American options in finance. Moreover, a lot of interest has been devoted to elliptic equations involving the fractions of the Laplacian, (see, among others, the papers [1,2,3,5,8,14,24,28,35,40] as well as [7,25,27,30,31,32,34] and the references therein). See also the papers [4,37] for related topics.
In our context, regarding the nonlinear term, we assume that f : R → R is continuous, superlinear at zero, i.e.  (3) and (4) are quite standard in the presence of subcritical terms. Moreover, together with (5), they guarantee that the number (6) c f := max |t|>0 |f (t)| |t| is well-defined and strictly positive. Furthermore, property (3) is a sublinear growth condition at infinity on the nonlinearity f which complements the classical Ambrosetti and Rabinowitz assumption.
Here, and in the sequel, we denote by λ 1 the first eigenvalue of the operator −∆ in Ω with homogeneous Dirichlet boundary data, namely the first (simple and positive) eigenvalue of the linear problem The main result of the present paper is an existence theorem for equations driven by the square root of the Laplacian, as stated below.
Theorem 1.1. Let Ω be an open bounded set of R n (n ≥ 2) with Lipschitz boundary ∂Ω, β : Ω → R a function satisfying (2), and f : R → R a continuous function satisfying (3)- (5). Then the following assertions hold: (i) problem (1) admits only the trivial solution whenever (ii) there exists λ ⋆ > 0 such that (1) admits at least two distinct and non-trivial weak solutions u 1,λ , u 2,λ ∈ L ∞ (Ω) ∩ H Furthermore, in the sequel we will give additional information about the localization of the parameter λ ⋆ . More precisely, by using the notations clarified later on in the paper, we show that see Remark 1 for details. Theorem 1.1 will be proved by applying classical variational techniques to the fractional framework. More precisely, following [11], we transform problem (1) to a local problem in one more dimension by using the notion of harmonic extension and the Dirichlet to Neumann map on Ω (see Section 2). By studying this extended problem with the classical minimization techniques in addition to the Mountain Pass Theorem, we are able to prove the existence of at least two weak solutions whenever the parameter λ is sufficiently large (for instance when λ > λ 0 ). Finally, the boundedness of the solutions immediately follows from [11,Theorem 5.2].
We emphasize that Cabré & Tan in [11] and Tan in [41] studied the existence and non-existence of positive solutions for problem (1) with powertype nonlinearities, the regularity and an L ∞ -estimate of weak solutions, a symmetry result of the Gidas-Ni-Nirenberg type, and a priori estimates of the Gidas-Spruck type.
Along this direction, we look here at the existence of positive L ∞ -bounded weak solutions on Euclidean balls in presence of sublinear term at infinity. To this end, for every n ≥ 2 and r > 0, set With the above notations, a special case of Theorem 1.1 reads as follows.
Theorem 1.2. Let r > 0 and denote where ∂R n+1 + := R n × (0, +∞) and n ≥ 2. Moreover, let f : [0, +∞) → R be a continuous non-negative and non-identically zero function such that where S := {t > 0 : F (t) > 0}. Then the following nonlocal problem . The structure of this paper is as follows. After presenting the functional space related to problem (1) together with its basic properties (Section 2), we show via direct computations that for a determined right neighborhood of λ, the zero solution is the unique one (Section 3). In Section 4 we prove the existence of two weak solutions for λ bigger than a certain λ ⋆ : the first one is obtained via direct minimization, the second one via the Mountain Pass Theorem. Specific bounds for λ ⋆ are obtained in Remark 1.
We refer to the recent book [29], as well as [15], for the abstract variational setting used in the present paper. See the recent very nice papers [22,23] of Kuusi, Mingione & Sire on nonlocal fractional problems.

Preliminaries
In this section we briefly recall the definitions of the functional space setting, first introduced in [11]. The reader familiar with this topic may skip this section and go directly to the next one.
The operator A 1/2 is well-defined on the Sobolev space endowed by the norm , and has the following form 2.2. The extension problem. Associated to the bounded domain Ω, let us consider the cylinder For a function u ∈ H 1/2 0 (Ω), define the harmonic extension E(u) to the cylinder C Ω as the solution of the problem where The extension function E(u) belongs to the Hilbert space with the standard norm and can be characterized as follows In our framework, a crucial role between the spaces X 1/2 0 (C Ω ) and H 1/2 0 (Ω) is played by trace operator Tr : X The trace operator is a continuous map (see [11,Lemma 2.6]), and gives a lot of information, which we recall in the sequel. We also notice that and that the extension operator E : H for every u ∈ H 1/2 0 (Ω). Here, H 1/2 (Ω) denotes the Sobolev space of order 1/2, defined as Next, we have the following trace inequality . Before concluding this subsection, we recall the embedding properties of Tr(X More precisely, the embedding j : Tr(X Thus, if ν ∈ [1, 2 ♯ ], then there exists a positive constant c ν (depending on ν, n and the Lebesgue measure of Ω, denoted by |Ω|) such that . From now on, for every q ∈ [1, ∞], · L q (Ω) denotes the usual norm of the Lebesgue space L q (Ω).
As already said, we will consider the square root of the Laplacian, defined according to the following procedure (see, for instance, the papers [5,8,11]). By using the extension E(u) ∈ X 1/2 0 (C Ω ) of the function u ∈ H 1/2 0 (Ω), we can define the fractional operator A 1/2 in Ω, acting on u, as follows: where ν is the unit outer normal to C Ω at Ω × {0}.

Weak solutions.
Assume that f : R → R is a subcritical function and λ > 0 is fixed. We say that a function u = Tr(w) ∈ H C for every ϕ ∈ X 1/2 0 (C Ω ). As direct computations prove, equation (15) represents the variational formulation of (14) and the energy functional for every w ∈ X 1/2 0 (C Ω ). Indeed, as it can be easily seen, under our assumptions on the nonlinear term, the functional J λ is well-defined and of class C 1 in X 1/2 0 (C Ω ). Moreover, its critical points are exactly the weak solutions of the problem (14).
Thus the traces of critical points of J λ are the weak solutions to problem (1). According to the above remarks, we will use critical point methods in order to prove Theorems 1.1 and 1.2.
3. The main theorem: Non-existence for small λ Let us prove assertion (i) of Theorem 1.1. Arguing by contradiction, suppose that there exists a weak solution w 0 ∈ X 1/2 Testing (17) with ϕ := w 0 , we have and it follows that In the last inequality we have used the following fact , and the trace inequality (12). By (18), (19) and the assumption on λ we get , clearly a contradiction. Moreover, the functional J λ is weakly lower semicontinuous on X 1/2 0 (C Ω ). Indeed, the application w → Ω β(x)F (Tr(w)(x))dx is continuous in the weak topology of X 1/2 0 (C Ω ). We prove this regularity result as follows. Let {w j } j∈N be a sequence in X 1/2 0 (C Ω ) such that w j ⇀ w ∞ weakly in X 1/2 0 (C Ω ). Then, by using Sobolev embedding results and [9, Theorem IV.9], up to a subsequence, {Tr(w j )} j∈N strongly converges to Tr(w ∞ ) in L ν (Ω) and almost everywhere (a.e.) in Ω as j → +∞, and it is dominated by some function κ ν ∈ L ν (Ω) i.e. (20) |Tr(w j )(x)| ≤ κ ν (x) a.e. x ∈ Ω for any j ∈ N for any ν ∈ [1, 2 ♯ ). Due to (4), there exists c > 0 such that It then follows by the continuity of F and (21) that e. x ∈ Ω as j → +∞ and a.e. x ∈ Ω and for any j ∈ N.
Since q > 2 and ε is arbitrary, the first limit of (22) turns out to be zero. Now, if r ∈ (1, 2), due to the continuity of f , there also exists a number M ε > 0 such that for all t ∈ [δ ε , δ −1 ε ], where ε and δ ε are the previously introduced numbers. The above inequality, together with (23), yields for each t ∈ R and hence , for each w ∈ X 1/2 0 (C Ω ).
Therefore, it follows that for every w ∈ X 1/2 .
The following technical lemma will be useful in the proof of our result via minimization procedure.
For this purpose, first of all, note that Moreover, by the construction of u, (35) and the fact that F (0) = 0, it follows that (36) Consequently, relations (36) and (37) and again the definition of u yield thanks to (33). Clearly, this completes the proof of Lemma 4.2.
Now, let us prove item (ii) of Theorem 1.1.

4.5.
Second solution via MPT. The non-local analysis that we perform in this paper in order to use the Mountain Pass Theorem is quite general and may be suitable for other goals, too. Our proof will check that the classical geometry of the Mountain Pass Theorem is respected by the nonlocal framework. Fix λ > λ ⋆ , λ ⋆ defined in (38), and apply (24) with ε := 1/(2λc 2 2 ). For each w ∈ X 1/2 . Setting due to what has been seen before one has namely the energy functional possesses the usual mountain pass geometry. Therefore, invoking also Lemma 4.1, we can apply the Mountain Pass Theorem to deduce the existence of w 2,λ ∈ X 0 so that J ′ λ (w 2,λ ) = 0 and J λ (w 2,λ ) = c 2,λ , where c 2,λ has the well-known characterization: we have 0 = w 2,λ = w 1,λ and the existence of two distinct non-trivial weak solutions to (14) is proved. In conclusion, Tr(w 2,λ ) and Tr(w 1,λ ) are two distinct non-trivial weak solutions to (1).
Furthermore, by [11,Theorem 5.2], since (21) holds in addition to β ∈ L ∞ (Ω), it follows that u i,λ := Tr(w i,λ ) ∈ L ∞ (Ω), with i ∈ {1, 2}. The proof is now complete. Remark 1. The proof of Theorem 1.1 gives an exact, but quite involved form of the parameter λ ⋆ . In particular, we notice that (39) λ ⋆ := inf Indeed, by (6), one clearly has Moreover, since , for every w ∈ X 1/2 0 (C Ω ). Hence, inequality (39) immediately holds. We point out that no information is available concerning the number of solutions of problem (1) if Since the expression of λ ⋆ is quite involved, we give in the sequel an upper estimate of it which can be easily calculated. This fact can be done in terms of the same analytical and geometrical constants. To this end we fix an element x 0 ∈ Ω and choose τ > 0 in such a way that (40) B(x 0 , τ ) := {x ∈ R n : |x − x 0 | < τ } ⊆ Ω. Now, let σ ∈ (0, 1), t ∈ R and define ω t σ : Ω → R as follows: It is easily seen that Thus inequalities (41) and (42) yield Moreover, arguing as in Lemma 4.2, we have that there exist t 0 ∈ R and σ 0 ∈ (0, 1) such that (44) Due to (38) one has . More precisely, inequalities (43) and (44) yield λ ⋆ ≤ λ 0 , where .
Thus the conclusions of Theorem 1.1 are valid for every λ > λ 0 .
Proof of Theorem 1.2. For any t ∈ R, set and define in a natural way J + λ : X It is easy to see that the functional Ψ + is well-defined and Fréchet differentiable at any u ∈ X  on ∂Γ 0 r , admits at least two distinct and nontrivial weak solutions u 1,λ , u 2,λ ∈ L ∞ (Γ 0 r )∩ H 1/2 0 (Γ 0 r ). Since condition (7) holds, inequality (47) is satisfied for λ = 1. Hence, problem (8) admits at least two distinct L ∞ -bounded weak solutions.
In conclusion, we present a direct application of our main result.

Example 1.
Let Ω be an open bounded set of R n (n ≥ 2) with Lipschitz boundary ∂Ω. As a model for f we can take the nonlinearity f (t) := log(1 + t 2 ), ∀ t ∈ R.