A-priori bounds for quasilinear problems in critical dimension

1 Introduction and main results

The study of a-priori bounds for solutions of elliptic boundary value problems, that is, establishing the existence of a positive constant C such that ∥u∥_L^∞(Ω) ≤ C for all solutions u, is an interesting and important issue. Indeed, on the one hand it is a key point to show existence of solutions by means of the degree theory. Moreover, a large number of different techniques have been developed to get such results for problems of the kind

where Ω ⊂ ℝ^N is typically a smooth bounded domain, Δ_mu := div(|∇u|^m–2∇u) and f : Ω × ℝ → ℝ⁺ has a subcritical growth in the second variable. Here subcriticality is meant in the sense of the Sobolev embeddings.

When m = 2, namely for second-order semilinear problems, uniform a-priori bounds were firstly obtained for a (non-optimal) class of subcritical nonlinearities by Brezis and Turner in [3] by means of Hardy-Sobolev inequalities. Some years later, Gidas and Spruck in [4] and de Figueiredo, Lions and Nussbaum in [5] improved this result, applying respectively a blow-up method and a moving-planes technique together with a Pohožaev identity. The main assumption therein was that the growth at ∞ of f is controlled by a suitable power with exponent 2^* – 1 = N+2N−2 , 2^* being the critical Sobolev exponent. We also point out the more recent work of Castro and Pardo [6] which enlarges the class of nonlinearities involved.

Regarding quasilinear equations, similar results have been achieved for 1 < m < N by Azizieh and Clément [7], Ruiz [8], Dall’Aglio et al. [9], Lorca and Ubilla [10] and Zou [11]. In these works, the nonlinearity may depend also on x and on the gradient, nonetheless its growth at infinity with respect to u should be less than a subcritical power. Originally, most of these results concerned the case 1 < m ≤ 2 due to some symmetry and monotonicity arguments for solutions to m-laplacian equations which were at that time available only for that range of m; however, those techniques have been later extended also to the case m > 2 in the papers [12, 13]. See also the recent work of Damascelli and Pardo [2], which further extends the class of nonlinearities for which the a-priori bound applies, in the spirit of [5, 6].

When, on the other hand, we consider the limiting case m = N, the Sobolev space W01,N (Ω) compactly embeds in all Lebesgue spaces and the well-known Trudinger-Moser inequality shows that the maximal growth for a function g such that ∫_Ωg(u)dx < + ∞ for any u ∈ W01,N (Ω) is g(t)=exp⁡{αNtNN−1}, the constant α_N > 0 being explicit. Therefore, we can consider problems of kind (1) with suitable exponential nonlinearities. Nevertheless, a-priori bounds may be found only up to the threshold t ↦ e^t, as the work [14] of Brezis and Merle shows for m = N = 2 (see also Section 5). In addition, the authors proved local a-priori bounds when the nonlinearity is of kind f(x, t) = V(x)e^t. This, together with the boundary analysis [5], yields the desired a-priori bound for convex domains. The results in [14] have been later on extended in the quasilinear setting by Aguilar Crespo and Peral Alonso in [15] and we refer also to [16] for concentration compactness issues in this direction.

The boundary value problem (1) with m = N and general subcritical nonlinearities (in the sense of the Trudinger-Moser inequality) has been studied by Lorca, Ruf and Ubilla in [1], where the authors considered superlinear growths either controlled by the map t ↦ e^tα for some α ∈ (0, 1), or which are comparable to t ↦ e^t. To prove a uniform a-priori bound in the first alternative, techniques involving Orlicz spaces are applied, while for the second the authors adapt the strategy in [14].

The goal of our work is to fill the gap between these growths. Indeed, Passalacqua improved the results of [1] in his Ph.D. Thesis [17] by means of a subtle modification of the argument therein, but the gap was still not completely filled and this seems to be out of reach with those techniques. Here, instead, we apply a blow-up method inspired by [18] to deduce the desired estimates in the interior of the domain. Moreover, our results complete the analysis in [2] for the case m = N and, further, to the best of our knowledge, they seem to be new even for the semilinear problem when N = 2.

We note that a similar approach has been also applied by Mancini and the author in [19] to study higher-order problems.

Throughout all the paper, Ω is a bounded and strictly convex domain in ℝ^N with C^1,α boundary and the nonlinearity f satisfies the following conditions:

1. f ≥ 0, f ∈ C¹(ℝ) and f′ ≥ 0 on [M, +∞) for some M ≥ 0;

2. there exists a positive constant d such that lim inft→+∞f(t)tN−1+d=+∞;

3. there exists limt→+∞f′(t)f(t):=β∈[0,+∞).

The second assumption is standard (cf. [1, 2]) and in the linear case N = 2 is equivalent to say that f is slightly more than superlinear. Assumption (A3) is a control on the growth at ∞ of f. Indeed, Gronwall Lemma implies that for any ε > 0 there are constants C_ε, D_ε > 0 such that

We refer to [19, Lemma 1.1] for a detailed proof.

In the blow-up analysis, it will be useful to distinguish between the following cases.

Although this distinction will be relevant in the proof, the critical case being more delicate, our main result applies for both classes of nonlinearities.

We recall that u ∈ W01,N (Ω) is a weak solution of problem (3) if

Notice that, unlike [1, 2], we are not prescribing that our solutions are in L^∞(Ω). Indeed, by [17, Proposition 6.2.2], every W01,N (Ω) weak solution of problem (3) is classical, namely it belongs to C^1,γ(Ω) for some γ ∈ (0, 1) (see also [9, Proposition 3.1]).

Once the uniform a-priori bound of Theorem 1.1 is established, the existence of a weak solution for problem (3) may be proved by means of the topological degree theory, provided a control of the behaviour of f in 0 is imposed.

We will not give the proof of Theorem 1.2, being rather standard once Theorem 1.1 is established, addressing the interested reader to [2, Theorem 1.5].

This paper is organized as follows: in Section 2 we collect some auxiliary results as well as boundary and energy estimates obtained in [1]; Section 3 contains the proof of Theorem 1.1 distinguishing between the subcritical and the critical case. In Section 4 we give a generalisation of our result for a class of nonhomogeneous problems and finally in Section 5 we provide an example of nonlinearities which, although being subcritical in the sense of Trudinger-Moser, do not satisfy assumption (A3) and for which one can find unbounded solutions.

2 Preliminary estimates

In this section, we state some known results which will be used in the sequel. We begin with a result by Serrin, improved by the regularity theory from [20]. Next, we state a Harnack-type inequality.

The next result is taken from [1, Propositions 2.1 - 3.1] and concerns the behaviour near the boundary of solutions of (3) and their energy.

Roughly speaking, the proof of the first inequality in (5) is obtained similarly to the analogous estimate in the second-order case (see [5]) by means of a Picone inequality originally proved in [23] (see also [2, Theorem 2.5]). This implies the second inequality in (5) by standard regularity estimates. Finally, (6) follows from (5) testing the equation with a suitable function with vanishing gradient outside Ω_r.

We point out that the uniform estimates (5)-(6) do not rely on assumption (A3), so they can be deduced for any superlinear growth (in the sense of assumption (A2)).

3 Uniform bounds inside the domain

The argument is based on a blow-up method inspired by [18] and developed in [19]: supposing the existence of an unbounded sequence of solutions, we define a rescaled sequence which locally converges to a solution of a Liouville’s equation on ℝ^N. Then, we have to distinguish two cases according to the growth of f. If f is subcritical, the right-hand side of the limit equation is constant and this readily yields the contradiction. In the critical framework, the Liouville’s equation is still nonlinear, so a more accurate analysis is needed: we will see that the energy concentrates around the blow-up points with a threshold which is too large for the energy bound of Proposition 2.3 to hold.

Let (u_k)_k be a sequence of solutions of (3) and suppose there exist points (x_k)_k ⊂ Ω such that

Since Ω is bounded, then, up to a subsequence, x_k → x_∞ ∈ Ω, where d(x_∞, ∂Ω) > 0 by Proposition 2.3. Moreover, we define v_k : Ω_k → ℝ^– as

where Ωk:=Ω−xkμk and

Notice that μ_k → 0 by (A1)-(A2) and this implies Ω_k ↗ ℝ^N, since x_∞ ∈ Ω. Moreover, each v_k satisfies (in a weak sense) the equation

Indeed, for any test function φ ∈ C0∞ (Ω) there holds

Moreover, fixing R > 0 and looking at the behaviour of the sequence (v_k)_k in B_R(0), applying the Harnack inequality given by Lemma 2.2 to w_k := –v_k ≥ 0, we find

Therefore, by the local estimate provided by Lemma 2.1, we infer that, up to a subsequence, v_k → v in Cloc1,α (ℝ^N). We claim that v is a weak solutions of the Liouville’s equation

where β is defined in (A3). Indeed, the equation (10) can be rewritten as

and thus, by a first-order Taylor expansion of log ∘ f around M_k, one finds

where z_k(x) := M_k + θ(x)(u_k(x_k + μ_kx) – M_k) = M_k + θ(x) v_k(x), θ(x) ∈ (0, 1). Since v_k → v uniformly on compact sets and M_k → +∞, then z_k(x) → +∞ uniformly on compact sets, so (12) follows by taking the limits as k → +∞ in (13).

Consequently, we split our analysis according to whether f is subcritical or critical in the sense of Definition 1.1.

4 Generalization to nonhomogeneous problems

So far, we studied the homogeneous quasilinear problem (3), which allows a clearer exposition and a more direct comparison with the results in [1, 2]. However, a similar analysis can be carried out also in the nonhomogeneous setting, provided some conditions at infinity and near the boundary are fulfilled. More precisely, we may consider the problem

where h : Ω × ℝ → ℝ⁺ ∪ {0} is a Carathéodory function satisfying the following conditions:

1. h ∈ L^∞(Ω × [0, τ]) for all τ ∈ ℝ⁺ and h(x, t) > 0 for any x ∈ Ω and t > 0;

2. there exist f ∈ C¹([0, +∞)) satisfying (A1)-(A3) and 0 < a ∈ L^∞(Ω) ∩ C(Ω) such that

   limt→+∞h(x,t)a(x)f(t)=1uniformly inΩ;

3. there exist r̄, δ̄ > 0 such that

   • h(⋅, t) ∈ C¹(Ω_r̄) for all t ≥ 0 and ∂h∂t (x, t) ≥ 0 in Ω_r̄ × ℝ⁺;

   • ∇_xh(x, t) ⋅ Ψ ≤ 0 for all x ∈ Ω_r, t ≥ 0 and unit vectors Ψ such that |Ψ – n(x)| ≤ δ̄.

We recall Ω_r := {x ∈ Ω | d(x, ∂Ω) ≤ r}.

The proof of Theorem 4.1 mainly follows the arguments of the previous sections, so we sketch here only the remarkable modifications.

5 A counterexample

As briefly mentioned in the Introduction, our assumption (A3) can be interpreted as a control from above for the growth at ∞ of the nonlinearity f by a suitable power of e^t. In the spirit of Brezis-Merle [14, Example 2], we show an example of nonlinearities which are still subcritical in the sense of the Trudinger-Moser inequality without satisfying (A3) and for which one may find a positive potential a(x) such that the problem (30) admits unbounded solutions, in a suitable distributional sense.

To this aim, let f(t) = e^tα with α∈(1,NN−1). Notice that f clearly satisfies (A1) and (A2), it is subcritical in the sense of Trudinger-Moser, but the condition (A3) is not fulfilled. Moreover, fix parameters δ:=(α−1)N+1αN>0 and γ:=α−1α∈(0,1) and consider the family of functions

where β > 0 is a positive parameter that will be chosen later. Notice that ϕ_β(t) > t – δ log t and ϕβ′(t)>1−dt. Hence, defining

where l(r):=log⁡(1r), there exists ρ₀ > 0 such that for any r ∈ (0, ρ₀) there holds u_β(r) > 0 and uβ′ (r) < 0. In the sequel, we adapt the strategy in [19, §6].

First, using an implicit function argument as in [19, Lemma 6.2], one can show (up to a smaller value of ρ₀) that for any ρ ∈ (0, ρ₀) there exists β=β(ρ)∼l(ρ)1α as ρ → 0 such that uβ(ρ)(ρ)−β(ρ)α=0 on ∂B_ρ(0). With this choice, one has uβ(ρ)(r)∼l(r)1α as r → 0. Then, for r ∈ (0, ρ) we compute

Using the relations 12≤ϕβ(ρ)′≤1 and ϕβ(ρ)″≤δl(r)−2 for r sufficiently small, one infers from (33) that there exists ρ₁ < ρ₀ such that for any ρ ∈ (0, ρ₁) and r ∈ (0, ρ) one has –Δ_Nu_β(ρ)(r) > 0.

Let us now fix ρ ∈ (0, ρ₁) and define

We have that w > 0 solves pointwise