Remarks on a nonlinear nonlocal operator in Orlicz spaces

We study integral operators $\mathcal{L}u(x)=\int_{\mathbb{R^N}}\psi(u(x)-u(y))J(x-y)\,dy$ of the type of the fractional $p$-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincar\'e inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel $J$ considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$ in a bounded domain $\Omega$, and boundary condition $u\equiv0$ on $\Omega^c$; both cases $f=f(x)$ and $f=f(u)$ are considred, including the generalized eigenvalue problem $f(u)=\lambda\psi(u)$.


Introduction
The aim of this paper is to study the properties of the nonlinear nonlocal operator Br denotes the ball Br = {z ∈ R N : |z| < r}. This set of hypotheses is assumed throughout the paper without further mention. We also denote q * = inf{q > : (H ) holds }, which measures in some sense the di erential character of the operator. The power case ψ(s) = k|s| p− s for some p > , J(z) = c|z| −N−σp/ for some < σ < , is known as the σfractional p-Laplacian operator. We want to consider here general functions ψ and J more than just powers, so we are led to study some Orlicz and Sobolev-Orlicz spaces, see below, which makes the study nontrivial. On the other hand, we are also interested in the limit case of integrability, which in our context means q * = , that is, the singularity of the kernel can be weaker than that of any fractional Laplacian or p-Laplacian. Some of the results also hold for more general kernels, J = J(x, y), satisfying only a lower estimate J(x, y) ≥ J (x − y), with J in the above hypotheses, but we prefer to keep the proofs in a simpler way.
For problems including operators like (1.1), in particular the fractional p-Laplacian, and the motivations for their study we refer to [9].

. The associated Orlicz spaces
Formula (1.1) makes sense pointwise for regular functions with some extra restriction on the nonlinearity ψ and the kernel J, see Section 2. In order to de ne the operator L in weak sense we consider the nonlocal nonlinear interaction energy (linear in the second variable) and we put Lu, φ = E(u; φ).
Clearly, by the symmetry properties of ψ and J we have E(u; φ) = R N Luφ for regular functions. But the above allows to de ne L also for functions in a Sobolev type space. To this end we de ne the functionals 3) with Ψ = ψ. The properties of ψ imply that Ψ is an strict Young function, so we can consider the Orlicz spaces Observe that in general E(u; u) ≠ cE(u) for any constant c > , the equality being true only in the power case ψ(u) = k|u| p− u, and then c = p. What we have is that E is the Euler-Lagrange operator associated to the functional E, that is, for every u, φ ∈ W J,Ψ (R N ). The above spaces do not have good properties unless we impose some conditions on the nonlinearity Ψ. The simplest case is when c s p− ≤ Ψ (s) ≤ c s p− , s > , p > , (1.7) so that the space L Ψ (R N ) coincides with L p (R N ), and the Sobolev space W J,Ψ (R N ) is denoted by W J,p (R N ). But we are interested in more general functions. Thus we consider the set, for some p ≥ q > , Γp,q = Ψ : R → R + , convex, symmetric, satisfying Ψ( ) = , Ψ( ) = , q ≤ sΨ (s) Ψ(s) ≤ p ∀s ≠ . (1.8) The condition Ψ( ) = is for normalization purposes and simpli es some expression. We thus deal with functions that lie between two powers, for instance a sum of powers, but we also allow for perturbation of powers like Ψ(s) = c|s| p | log( + s)| r , min{p, p + r} > . The rst property deduced from (1.8) is the relation between the interaction energy E and the functional E, Our main interest lies in studying the properties of the spaces (1.3) and (1.4) for nonlinearities Ψ in the class Γp,q. In particular we have that L Ψ (R N ) and W J,Ψ (R N ) are re exive Banach spaces, with norms de ned, for instance, in (2.1) and (2.3). On the other hand, if q > q * , see (H ), then the functional E(u) is well de ned and nite for functions satisfying F(∇u) < ∞, see Proposition 2.3. This means the inclusion W ,Ψ (R N ) ⊂ W J,Ψ (R N ), the former being the standard Orlicz-Sobolev space of functions in L Ψ (R N ) with gradient also in L Ψ (R N ).
When dealing with problems de ned in bounded domains Ω ⊂ R N , since the Dirichlet conditions must be prescribed in the complement Ω c ≡ R N \ Ω, instead of just on the boundary, precisely by the nonlocal character of the operator, it is convenient to consider the space Without imposing any singularity condition on the kernel J at the origin, besides of course being not integrable, that is q * may be zero, we show that a Poincaré inequality E(u) ≥ cF(u) holds, so that we have the embedding Observe that if J were integrable then W J,Ψ (Ω) ≡ L Ψ (Ω). Assuming now q * > (and some monotonicity near the origin, (3.5)) we obtain a better result, namely a Sobolev embedding which is compact. The borderline r = r * when q * < N produces also an embedding W J,Ψ (Ω) ⊂ L Ψ r * (Ω), provided J(z) ≥ c|z| −N−q * near the origin, but without compactness. In the limit case q * = , which would give r * = in (1.11), we do obtain compactness of the inclusion (1.10) by assuming a minimum of singularity on the kernel, the extra condition lim

. Elliptic problems
With this machinery we next study the problem in Ω c . (1.12) This problem must be considered in weak sense with the aid of the interaction energy E, that is, any solution u satis es We study rst the case f = f (x) in an appropriate space. We obtain existence and uniqueness of a solution, see Theorem 4.1. We also show some integrability properties of the solution in terms of the data f when ψ is restricted to the power-like case (1.7), see Theorems 4.3 and 4.4. In particular the solution is bounded provided f ∈ L m (Ω) with m > N/q * if q * > , see Theorem 4.5. For the corresponding results in the case of the fractional p-Laplacian see [4]. We do not address regularity issues in this paper. For Hölder regularity results in the case of the fractional p-Laplacian we refer to [12], [13], [22], [24]. We then pass to study the case f = f (u) in problem (1.12). In Theorem 5.1 we prove existence of a nonnegative nontrivial solution in the lower range, which roughly speaking in the power-like case ψ(s) ∼ s p− , f (t) ∼ ct m− , means m < p. The intermediate range p < m < m * = Np N−q * , below the Sobolev exponent, is studied in Theorem 5.2 using the Mountain Pass Theorem. We also use a Pohozaev inequality in order to show nonexistence, in the exact power case, for supercritical powers m > m ** = Np N−δ , δ > being a constant depending on the kernel J, see Corollary 5.4. We must remark that all the conditions on the reaction f are very involved in terms of ψ, and are not as clean as suggested by the above, see the precise conditions (5.2) and (5.4). We refer to [20] for the study of nonlinear problems like the above, even with more general reactions, for the fractional p-Laplacian case.
We nally are interested in the limit case m = p, which corresponds to the so called generalized eigenvalue problem Lu = λψ(u), in Ω, We prove that there exists a rst positive eigenvalue and a rst positive eigenfunction, which is bounded provided q * > , Theorem 6.1. The fractional p-eigenvalues have been studied in [22] and [17].

. Organization of the paper
We begin with a preliminary Section 2 where we study the properties of the Orlicz spaces L Ψ (R N ) and W J,Ψ (R N ) by means of some inequalities satis ed by the nonlinearity Ψ and the functionals F and E. Section 3 shows the Sobolev inclusions of the spaces W J,Ψ (Ω). Finally Sections 4-6 are devoted to the study of problem (1.12) for the di erent reactions commented upon before. In what follows the letters c or c i will denote some constants that do not depend on the relevant quantities, and may change from line to line.
After the writing of this paper was completed, we learned that, at the very same time and independently of us, Fernández-Bonder and Salort proved in [15] some properties of fractional Sobolev-Orlicz spaces de ned in a way related to ours.

Preliminaries
In this section we study in detail the properties of the Orlicz spaces L Ψ (R N ) and W J,Ψ (R N ) de ned in (1.5) and (1.6), and the corresponding spaces in a bounded domain Ω. We refer to [25] for instance for the general theory of Orlicz spaces. We begin by studying the Young functions in the set Γp,q. First observe that Ψ ∈ Γp,q, p ≥ q ≥ , implies min{|s| p , |s| q } ≤ Ψ(s) ≤ max{|s| p , |s| q }.
Associated to any given positive function g : R + → R + we consider its characteristic functions, for s > , These are nondecreasing functions that satisfy Lemma 2.1. For any Ψ ∈ Γp,q, p ≥ q > , Proof. If s > we have that These two functions satisfy the Young inequality and equality holds only if b = Ψ (|a|)sign a. From this point on we always assume q > . Let Ψ ∈ Γp,q be xed and consider the corresponding Orlicz space L Ψ (R N ). It is a linear space that satis es and in the case of bounded domains Also it is a Banach space with norm, called Luxemburg norm, We recall that other equivalent norms are also used in the literature. The following result allows us to use F(u) instead of u L Ψ in most calculations.
Proof. Let a = u L Ψ . We clearly have F u/a ≤ . Then On the other hand, for every ε > we have F u/(a + ε) > , so that where Φ is the complementary function, and thus they are both re exive Banach spaces. We also consider the Sobolev type space W J,Ψ (R N ). In the same way as before it is a Banach space with norm The second term is a kind of Gagliardo seminorm in the context of Young functions. For this seminorm an analogous property as that of Lemma 2.2 also holds, In order to show that this space is re exive as well we consider the weighted space ) is a closed subspace of M. It follows that T(W J,Ψ (R N )) is re exive (see [7,Proposition 3.20]), and consequently W J,Ψ (R N ) is also re exive. We now take a look at the properties of the space W J,Ψ (R N ) in terms of the properties of the kernel J, in particular its singularity at the origin, which is re ected in the exponent q * , see (H ).
Proof. We decompose the integral The far away integral is easy to estimate As to the inner integral, we have since γ + (|z|) = γ + Ψ (|z|) ≤ |z| q in the set {|z| < }, and using hypothesis (H ).
If the kernel J behaves like that of the fractional Laplacian then we also have the following interpolation estimate Proposition 2.4. If J satis es (2.6) for some α > , and Ψ ∈ Γp,q with p ≥ q > α then Proof. We apply inequality (2.5) to the rescaled function u λ (x) = u(λx). We rst observe that by (2.6), .
From this we easily deduce (2.7).
In the power-like case we obtain from the above the well-known interpolation result.
Corollary 2.5. If J satis es (2.6) for some α > , and ψ satis es (1.7) with p > α then or which is the same We now turn our attention to the operator L. The pointwise expression (1.1) does not always have a meaning. Let us look at some easy situations where Lu is well de ned. We may take, for instance, Ψ nondecreasing and u ∈ C (R N ). Another less trivial example is q > q * + and We now show some useful inequalities. The rst one is a Kato type inequality, that is, the result of applying the operator L to a convex function of u. We refer to [21] and [10], respectively, for the well-known inequalities Proposition 2.6. If A is a positive convex function and Lu is well de ned, then L(A(u)) is also well de ned and Proof. We just observe that since A is convex and ψ is nondecreasing, we have Now integrate with respect to J(x − y) dy to get the result.
As a Corollary we obtain an integral version of the Kato inequality, useful in the applications.
Of later use are also the following two inequalities whose proof is immediate just looking at the signs of the corresponding functions. Related to those inequalities is the well known Stroock-Varopoulos inequality, see [29] for the linear case Ψ(s) = |s| and J(z) = |z| −N−σ for some < σ < , and [5] for general Lévy kernels J. It is of the type of the integral Kato inequality, but the functions for which it holds is di erent. In the case of powers they coincide but for the coe cient, which is always better in the Stroock-Varopoulos inequality. We show here a generalized Stroock-Varopoulos inequality.
Proof. The proof follows from a calculus estimate. For any d > c we have that The same inequality is obtained for d ≤ c. We now deduce (2.9) by choosing d = u(x), c = u(y) and integrate with respect to J(x − y) dxdy.
For instance in the case of a sum of powers, All the above inequalities hold also, with di erent constants, for nonlinearities that behave like a power, i.e., when they satisfy (1.7) instead of (1.8). In particular in that case the integral Kato inequality and the Stroock-Varopoulos inequality coincide, but for the coe cient, both giving (2.10) We also obtain some calculus inequalities needed in proving uniqueness results in the last sections. We borrow ideas from [23] and [18] that deal with the exact power case. Lemma 2.9. Let ψ be a nonnegative, nondecreasing, continuous odd function and let ψ = Ψ .
iii) If ψ satis es c |s| p− ≤ ψ (s) ≤ c |s| p− for some < p < and every s ≠ , (2.14) then Proof. i) We begin by proving a Clarkson inequality. Condition (2.11) implies that the function g(s) = Ψ( |s|) is convex. Therefore and also This gives and reversing the roles of a and b, Adding these two inequalities we get (2.12).
ii) Developing the function Ψ around the point s = a we get We have used that |a + s(b − a)| ≤ |a| + |b| and Ψ is nonincreasing in ( , ∞). Observe that though Ψ is singular at zero, the integral is convergent. We conclude as before.
iii) As ii), using (2.14) in the last step.
To end this section devoted to the preliminary properties of E and F, we point out a result on symmetrization that says that the energy E(u) decreases when we replace u by its symmetric rearrangement (the radially deacreasing function with the same distribution function as u).
Theorem 2.10. If u ∈ W J,Ψ (R N ) and u * is its decreasing rearrangement, then This property is well known for the norm in W σ/ ,p (Ω), < σ ≤ , p > , see [1], and is proved in [11] for general kernels when p = . The same proof can be used to get the result in our situation, so we omit the details. See also [3].

Sobolev inclusions
In this section we consider a nonlinearity Ψ ∈ Γp,q, p ≥ q > max{q * , } xed. As to the kernel J, besides condition (H ) we also consider, for some results, the singularity condition at the origin Clearly it must be α ≤ q * . In fact in the fractional p-Laplacian case it is α = σp/ . Other kernels could also be considered, for instance J(z) = |z| −N−µ log(|z|/ ) β , for < |z| < , with µ ≥ , (and β ≥ − if µ = ). In that case it is q * = µ. If µ > then J satis es (3.1) with α = µ if β ≥ , but if β < it satis es (3.1) only with < α < µ. A more intricate example can be constructed by the following piecewise de nition of J, Here we have q * = µ while condition (3.1) does not hold for any α > . Assume now that u has support contained in Ω. Then This gives the Poincaré inequality E(u) ≥ AF(u), (3.2) and the inclusion W J,Ψ (Ω) ⊂ L Ψ (Ω). We remark that in the case of integrable kernel J we immediately would get E(u) ≤ c J F(u), and thus W J,Ψ (Ω) ≡ L Ψ (Ω). In order to obtain better energy estimates in the case q * > , which would yield better space embeddings, we need a better estimate of the function Λ(Ω; ·) in terms of the kernel J. The following result is essentially contained in [28, Lemma A.1]. This estimate allows us to prove, assuming condition (3.1), the Sobolev embedding W J,Ψ (Ω) ⊂ L Ψ r (Ω) for every ≤ r ≤ r * ≡ N N−α , if α < N, for every ≤ r < ∞ if α ≥ N. The proof uses ideas of [27] and [14]. If α ≥ N we obtain the result substituting α by any number below N and close to N, since (3.1) still holds for that exponent.

Ψ(u) r ≤ CE(u). (3.3)
Proof. We prove the inequality for r = r * , and then the result for r < r * follows by Hölder inequality. We can assume, without loss of generality, that u is radially deacreasing and Ω = B R * , since substituting u by its symmetric decreasing rearrangement u * , we have by Theorem 2.10, We may also consider the case of u bounded, since if not, taking the sequence u T = min{u, T}, and thanks to the Dominated Convergence Theorem, we would get the result in the limit T → ∞. We now de ne We have A k = B R k , with R k+ ≤ R k ≤ R * . Also a k = d k = for all large k, say for k > M. Now we compute, The second term can be estimated as We deduce the estimate We conclude, using [27,Lemma 5], since r = − α N , We also prove that the above embedding is compact provided r < r * . To this end we rst show the compactness of the inclusion for r = and then interpolate with the continuity for r = r * . It is important to remark that the inclusion W J,Ψ (Ω) → L Ψ (Ω) is compact even when q * = , which implies r * = , provided the following conditions on the kernel at the origin hold for every < |z | ≤ |z | ≤ , and some c > . (3.5) This implies some kind of minimal singularity and some monotonicity near the origin. In particular this allows to consider for instance a kernel of the form J(z) = |z| −N |log |z|| β , β > , for |z| ∼ . See [11] for the case Ψ(s) = |s| . Proof. The idea of the proof goes back to the Riesz-Fréchet-Kolmogorov work. We follow the adaptation to the fractional Laplacian framework performed in [14]. Let A ⊂ W J,Ψ (Ω) be a bounded set. We show that A is totally bounded in L Ψ (Ω), i.e., for any ϵ ∈ ( , ) there exist β , ..., β M ∈ L Ψ (B ) such that for any u ∈ A there exists j ∈ { , ..., M} such that We take a collection of disjoints cubes Q , ...Q M of side ρ < such that Ω ⊂ M j= Q j . For any x ∈ Ω we de ne j(x) as the unique integer in { , ..., M } for which x ∈ Q j(x) . Also, for any u ∈ A, let and that Q(u) is constant, say equal to q j (u), in any Q j , for j ∈ { , ..., M }. Therefore, we can de ne We observe that and also, by Jensen inequality and (3.  In the same way, for every constant a. In particular from (3.8) we obtain that the set S(A) is bounded in R M and so, since it is nite dimensional, it is totally bounded. Therefore, there exist b , ..., b K ∈ R M such that where Bη(b i ) are the -balls of radius η centered at b i . For any i ∈ { , ..., K}, we write the coordinates of b i and so q j (β i ) = Ψ − (ρ −N b i,j ); thus S(β i ) = b i . Furthermore, again by Jensen inequality, where is some function satisfying (z) ≤ |z| N J(z). Using (3.4), (3.5) we can take radial satisfying ( + ) = ∞. Consequently, for any j ∈ { , ..., K}, recalling (3.7) and (3.10) ) . Now recalling (3.9) we take j ∈ { , ..., K} such that S(u) ∈ Bη(b j ), that is We conclude by choosing ρ and η small so as to have c (ρ) + η < ϵ.
As a corollary we obtain the full compactness result in the fractional case.

Theorem 3.4. Assume J satis es (3.1) and (3.5). Then the embedding W J,Ψ (Ω) → L Ψ r (Ω) is compact for every
Proof. As before if α ≥ N we obtain the result substituting α by any number below N. By classical interpolation where r = λ + −λ r * . Therefore we can obtain, instead of (3.6), the estimate and we are done.

The problem with reaction f = f (x)
We start with this section the study of some elliptic type problems associated to our nonlinear nonlocal operator L.
Here we consider the problem in Ω c . (4.1) Given any f ∈ W J,Ψ (Ω) , the dual space, we say that u ∈ W J,Ψ (Ω) is a weak solution to (4.1) if (1.13) holds.
We next show that problem (4.1) has a weak solution. We do not know if this solution is a strong solution, that is if Lu is de ned pointwise and the equality in (4.1) holds almost everywhere. On the other hand, we are able to show uniqueness assuming some extra conditions on the function Ψ. In the exact power case Ψ(s) = |s| p these extra conditions cover the full range p > .
Clearly it is well de ned, lower semicontinuous and Fréchet di erentiable with for every v, φ ∈ W J,Ψ (Ω). To see that it is coercive we rst observe that v W J,Ψ → ∞ implies E(v) → ∞. Actually, by (2.2), (2.4) and Poincaré inequality,

Now use Hölder inequality in Orlicz spaces
, The last quantity is known as the Orlicz norm of f in W J,Ψ (Ω) , an is equivalent to the Luxemburg norm, see [25]. We thus get as v W J,Ψ → ∞. Therefore there exists a minimum of I, attained by compactness for some function u ∈ W J,Ψ (Ω), which is a weak solution to our problem. We now show uniqueness. Suppose by contradiction that there exist two functions u , u ∈ W J,Ψ (Ω) such that Assume rst that (2.11) holds. We have, denoting a = u (x) − u (y), b = u (x) − u (y), and using (2.12), by (4.2). This implies u ≡ u . Assume now condition (2.14). We calculate, using Hölder inequality and (2.15), A maximum principle is easy to obtain.
Proof. Since u − ≥ and u − ∈ W J,Ψ (R N ), we have, by (2.8), Hence u − ≡ . We now study the integrability properties of the solution in terms of the integrability of the datum in the power-like case (1.7). In the exact power case of the fractional p-Laplacian these integrability properties have been obtained in [4]. Our proofs in the more general case treated in this paper di er from theirs in that we are using Stroock-Varopoulos inequality instead of Kato inequality, and that we allow for the limit case q * = , which does not make sense in the fractional p-Laplacian. All the proofs are based on the well known Moser iteration technique for the standard Laplacian case, see for example the book [19].
The rst result uses no singularity condition on the kernel J, besides being nonintegrable. Of course this result is not trivial only if m > p p− , since u being a weak solution it belongs to W J,Ψ (Ω) ⊂ L p (Ω). Proof. Without loss of generality we may assume u ≥ , and this simpli es notation; the general case is obtained in a similar way. We de ne for β ≥ and K > the function We choose as test function φ = G(u) = u Ψ(H (s)) ds. It is easy to check that φ ∈ W J,Ψ (Ω). In fact We obtain on one hand, using the Stroock-Varopoulos inequality (2.9) and the Poincaré inequality (3.2), and on the other hand, using Hölder inequality, The same proof allows to gain more integrability when condition (3.1) holds. Even more, assuming a better integrability condition on f we get that the solution is bounded. This is a well known result for the standard Laplacian or the fractional Laplacian. Proof. We change here slightly the test function used in the previous two proofs. We de ne for β ≥ and K ≥ k (k to be chosen later) a C ([k, ∞)) function H, as follows: Let us also de ne v = u + k, and choose as test function φ = G(v) = v k Ψ(H (s)) ds. We obtain on one hand, using the Stroock-Varopoulos inequality (2.9) and the Sobolev inequality (3.3), and on the other hand, using Hölder inequality, since v ≥ k. Inequality (4.6) together with (4.7), and the properties of Ψ, lead to We choose k = (c f m) p− and let K → ∞ in the de nition of H, so that the inequality (4.8) becomes Hence for all β ≥ the inclusion u ∈ L pm β (Ω) implies the stronger inclusion u ∈ L r * pβ (Ω), since Observe that u being a weak solution it belongs to W J,Ψ (Ω), and thus u ∈ L Np N−α (Ω). The result follows now iterating the estimate starting with β = N(m− ) (N−α)m > , see for example [19,Theorem 8.15] for the details in the standard Laplacian case. This gives u ∈ L ∞ (Ω). In fact we get the estimate

The problem with reaction f = f (u)
We study in this section the nonlinear elliptic type problem in Ω c . (5.1) We rst show existence in the lower case, i.e., when f : [ , ∞) → R is a continuous function satisfying In the power-like case (1.7) with f (t) = t m− this means < m < p. See [8] for the classical sublinear problem for L = −∆ and [11] for general L with q * ≥ , both in the case Ψ(s) = |s| . Proof. We de ne the energy functional I : W J,Ψ (Ω) → R de ned by where G(u) = u f (s) ds. This functional is easily seen to be weakly lower semicontinuous, and is well de ned On the other hand, this same estimate also gives coercivity since µ < , and then

I(u).
This sequence is bounded in W J,Ψ (Ω), and therefore we can assume that there is a subsequence, still denoted {vn}, such that vn u in W J,Ψ (Ω). Therefore vn → u in L Ψ (Ω). We thus deduce by (5.3) This shows that I(u) = ν and u is a global minimum for I, hence a solution to (5.1). It is easy to see that we can replace u by |u| since I(|u|) ≤ I(u). In order to show that u is nontrivial let us check that I(u) < . In fact, given any v ∈ W J,Ψ (Ω) we have for small ε > , since q > + pµ. We deduce that ν < and u ̸ ≡ .
Unfortunately we are only able to prove uniqueness in the exact power case Ψ(s) = |s| p . In fact uniqueness follows in that case using a standard argument by means of a Picone inequality proved in [16], see [8] and [11]. Though a Picone inequality could be obtained also assuming that (1.7) is satis ed, it is not sharp enough to prove uniqueness. In the more general case of Ψ ∈ Γp,q such Picone type inequality is not even known to hold. We now assume condition (3.1) and consider nonlinear functions f in the intermediate range, that is above the power p − but subcritical in the sense of Sobolev, see Theorem 3.2. The precise conditions on f are where G = f . When f (t) = t m− these condition hold with ρ = m provided p < m < Nq N−α .
Proof. As before we consider the functional whose critical points are the solutions to our problem. This functional is well de ned in W J,Ψ (Ω) thanks to the Sobolev embedding and the second condition in (5.4). We therefore apply the standard variational technique based on the Mountain Pass Theorem [2]. We only have to prove that the functional satis es the Palais-Smale condition and has the appropriate geometry. We rst prove that any Palais-Smale sequence has a convergent subsequence. Let {vn} be a sequence satisfying The exponent r * in (5.4) is sharp in the fractional p-Laplacian case. In fact, in the fractional Laplacian case p = this has been proved in [26] by means of a Pohozaev identity when Ω is star-shaped. Their proof was adapted in [11] for more general kernels again with Ψ(s) = |s| , obtaining an exponent which depends on the kernel and is presumed not to be optimal. The proof of this last result works verbatim for general powers Ψ(s) = |s| p , but not for other functions, since homogeneity is crucial in the argument. Let, for λ > , and assume µ(λ) < ∞ for λ close to 1. We observe that this nonexistence result depends not only on the behaviour of the kernel at the origin, but on its global behaviour, see (5.5). In fact when the kernel is α < p, α > , we get σ = max{α , α }. It will be interesting to know if only the singularity of J at the origin determines by its own the existence or nonexistence of solution. If this is the case we would get, in the critical singularity exponent q * = in (H ), that there is no solution for any m > p. This, together with the existence result for m < p of Theorem 5.1, leaves only the case m = p to be studied. We dedicate next section to this task.

The generalized eigenvalue problem
In this last section we study the parametric problem in Ω c .
Though the problem is not homogeneous due to the presence of the non homogeneous function ψ, since it is precisely this same function ψ that de nes both, the operator and the reaction, it can also be called generalized eigenvalue problem, as is usual for the p-Laplacian or the fractional p-Laplacian, see [22], [23]. The rst (generalized) eigenvalue and eigenfunction are obtained minimizing In fact, if u is a minimum, the function g(t) = I(u + tφ), for any admisible function φ satis es g( ) = E(u) F(u) = λ , g ( ) = , that is, E (u), φ = λ F (u), φ , which is the associated Euler-Lagrange equation, the weak formulation (1.13). Therefore for every φ ∈ W J,Ψ (Ω). The fact that the eigenfunction is nonnegative or nonpositive follows by (2.8) which implies I(±|u|) ≤ I(u). The boundedness of u assuming (3.1) is easily proved again by the Moser iterative scheme as performed in [6]. The key point is the use of the Stroock-Varopoulos inequality (2.10) and condition (1.7), and nally apply Theorem 4.5. See also [17].