EXISTENCE AND MULTIPLICITY RESULTS FOR SUPERCRITICAL NONLOCAL KIRCHHOFF PROBLEMS

. We study the existence and multiplicity of solutions for the non-local perturbed Kirchhoﬀ problem

In particular, under assumptions (A1)-(A4), Faraci and Silva proved in [4] that problem (1.4) admits at least a nonzero solution if one of the following conditions holds Here, c 2 * is the best constant for the embedding L 2 * (Ω) → W 1,2 0 (Ω). Moreover, a second solution is proved to exist for λ large, under the following more strict condition on a, b (1.5) Problem (1.1) is associated with the stationary version of the well known equation proposed by Kirchhoff to describe the transversal oscillations of a stretched string. For more details, we refer the reader to [4] or [7] and references therein. To the best of our knowledge, the case in which problem (1.1) involves nonlinearities of arbitrary growth has been addressed in few papers. Among them, we can cite [1,2,3,6]. However, in these papers only existence results were established. We stress out that variational methods are not directly applicable when supercritical nonlinearities are involved. Usually, in this case, an auxiliary problem involving a suitable truncation of the supercritical nonlinearity is introduced. After that, one shows, by using L ∞ -norm estimates, that the solutions of the auxiliary problem are also solutions of the original problem. We will make use of this technique to prove our main results. Now we recall some basic concepts of variational methods. Let h : Ω × R → R be a Carathéodory function and let H : Ω × R → R be the primitive of h, defined by (1.6) Consider the set X h ⊆ W 1,2 0 (Ω) given by By Sobolev embeddings, the set X h is the whole W 1,2 0 (Ω) whenever ess sup (x,t)∈Ω×R |H(x, t)| 1 + |t| 2 * < +∞.
Throughout the paper, if h : Ω × R → R and X h are as above, we denote by I h : X h → R the energy functional associated with the problem which is defined by for all u ∈ X h . By a solution of problem (1.7) we mean any function u ∈ W 1,2 0 (Ω) satisfying, for each v ∈ W 1,2 0 (Ω), the following conditions: the function x ∈ Ω → h(x, u(x))v(x) belongs to L 1 (Ω), and When X h = W 1,2 0 (Ω) and I h is differentiable in W 1,2 0 (Ω), the solutions of (1.7) are exactly the critical points of I h . We denote byĨ λ the energy functional associated with problem (1.4), that is A key ingredient in the proofs of the results in [4] is the sequential weak lower semicontinuity of the functional when q = 2 * . It is well known that Φ is sequentially weakly lower semicontinuous for 0 < q < 2 * , but this is not true, in general, if q = 2 * . In [4] the condition a N −4 2 b ≥ C 1 (N ) assumes a key role since it just ensures the sequential weak lower semicontinuity of Φ in the case q = 2 * . Thus, if one assumes a N −4 2 b ≥ C 1 (N ) and the subcritical growth condition i) on g, one gets the sequential weak lower semicontinuity ofĨ λ . When q > 2 * , the set X h corresponding to h(x, t) = λg(x, t) + |t| q−2 t is strictly contained in W 1,2 0 (Ω) and, moreover, the functional Φ (and therefore also the functionalĨ λ ) is never sequentially weakly lower semicontinuous in X h . Thus, the arguments used in [4] cannot be applied when q > 2 * and, in general, when a nonlinearity of arbitrary growth is involved.
As said above, in the present paper, we address the question of the existence and multiplicity of solutions to problem (1.1) in the case g has an arbitrary growth. We will establish existence and multiplicity results by assuming only condition α g ) on g, and imposing (as in [4]) some constrains on a, b. However, differently to the problem (1.4) considered in [4], where the parameter λ is multiplied by the subcritical nonlinearity, in our case the parameter λ is multiplied by the nonlinearity of arbitrary growth. This allows to deduce that the solutions of the auxiliary truncated problem are also solutions of the original problem, for λ small enough.
Besides (A1), we assume on the nonlinearity f the following two additional conditions: (A5) lim sup ξ→0 Here, Under (A1), (A2), (A5), and (A6), we will be able to prove a multiplicity result for problem (1.1) for all a, b > 0, with b ≤ β(a), where β(a) is a suitable number depending on a), and for all λ small enough. We will also show that, if conditions (A5) and (A6) are replaced by 2 , uniformly for a.a. x ∈ Ω, an existence result can be proved, again for λ small, without imposing any constrain on a, b.

Notation and preliminary lemmas
Throughout this paper, we use of the following notation: u m u denotes the best constant for the Sobolev embedding L m (Ω) → W 1,2 0 (Ω). Note that λ 1 := c −2 2 . (5) for each λ ∈ R and C > 0, g C : Ω × R → R and h λ,C : Ω × R → R are the functions defined by (2.1) The next lemmas provide regularity estimates for the solutions of the problem In particular, by these estimates we will infer that, for certain values of C and λ, every solution of (2.2) is also a solution of (1.1).
Let ψ be the positive eigenfunction associated with λ 1 and normalized with respect to the norm · . Moreover, put θ = 2τ |Ω| µ . By the above inequality, for b < β(a) := µ 2 τ |Ω| , one gets Thus, if we consider the functional I f : W 1,2 0 (Ω) → R defined by we realize that inf W 1,2 0 (Ω) Now, by (A1) and (A5), we can also find two constants δ, η > 0 such that for each ξ ∈ R and a.a. x ∈ Ω. Consequently, for each u ∈ W 1,2 0 (Ω). Therefore, since p > 2, if we fix 0 < < ηp δc p p 1 p−2 and take (3.1) into account, we obtain, for all b ∈ (0, β(a)), Now, let λ ∈ R, and let C = C(a, b) > 0 be the constant defined in (2.9). Moreover, let h λ,C be the function defined in (2.1). Since p < 2 * < 4, by assumptions (1. Since, by standard results, I h λ,C is sequentially weakly lower semicontinuous in W 1,2 0 (Ω), we infer that I h λ,C is bounded below on W 1,2 0 (Ω) as well. Consequently, we can consider the functions ω, ω 1 : R → R defined by , for each λ ∈ R. Since λ ∈ R → I h λ,C (u) is an affine function for each u ∈ W 1,2 0 (Ω), we have that the functions ω, ω 1 are both the difference of two concave functions, and so they are continuous in R. By (3.2), we also have Thus, by the continuity of ω and ω 1 , we can find λ(a, b) ∈ (0, ρ g (C) −1 ) such that λ(a, b)]. By the above two inequalities and by the sequential weak lower semicontinuity of I h λ,C , one infers that • I h λ,C admits a local minimum point u λ ∈ W 1,2 0 (Ω), such that u λ < ; • I h λ,C admits a global minimum point v λ ∈ W 1,2 0 (Ω), with Of course, u λ , v λ are critical points of I h λ,C . Observe also that the inequality I h λ,C (v λ ) < inf u ≤ I h λ,C (u) implies v λ > . Hence, in particular, the functional I h λ,C turns out to have the mountain pass geometry. In addiction, we know, again by standard results, that: • the functional is differentiable in W 1,2 0 (Ω) with compact derivative. Therefore, taking (3.3) into account, we infer that I h λ,C satisfies the Palais-Smale condition (see, for instance, Example [9, 38.25]). By applying the classical Mountain Pass Theorem by Ambrosetti-Rabinowitz, we derive the existence of a third critical point w λ for I h λ,C , which is of mountain pass type. Finally, since λ ∈ (0, ρ g (C)) and C is as in (2.9), by Lemma 2.5 we conclude that u λ , v λ , w λ are three distinct solutions of (1.1).
Let ψ be the positive eigenfunction associated with λ 1 and normalized with respect to the norm · . Since p > 2, one has for θ > 0 small enough. Fix such a θ and let C = C(a, b) be the constant defined in (2.9). By the previous inequality, we can find λ(a, b) ∈ (0, ρ g (C)) such that, for λ ∈ [0, λ(a, b)] and h λ,C as in (2.1), one has This means that inf W 1,2 0 (Ω) I h λ,C < 0.
Since I h λ,C also satisfies the coercivity condition (3.3), then I h λ,C admits a nonzero global minimum point u λ , which is a solution of (1.1) in view of the condition λ < ρ g (C) −1 and Lemma 2.5.

Conclusion
In this paper, we have considered a supercritical non local problem of Kirchhoff type and we have proved, via variational methods and truncation arguments, both existence and multiplicity results. The main feature of these results is that the presence of the nonlocal term allows to obtain the multiplicity of solutions even in the supercritical case. We point out that we found very few results where the multiplicity of solutions is established for critical or supercritical problems. Among them, we have mentioned the interesting paper [4]. In [4], the right hand-side in the problem considered there is a sum of a subcritical nonlinearity multiplied by a parameter λ and a critical nonlinearity (of power-type). Therefore, the problem considered in [4] is different from problem (1.1) considered here, where, instead, the parameter λ multiplies the supercritical term. We think that an interesting question is to investigate, by the approach used in present paper, the possible extension to the supercritical case of the multiplicity result obtained in [4].