A-priori bounds for a quasilinear problem in critical dimension

We establish uniform a-priori bounds for solutions of the quasilinear problem $-\Delta_Nu=f(u)$ in $\Omega$, with $u=0$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^N$ is a bounded smooth and convex domain, and $f$ is a positive superlinear and subcritical function in the sense of the Trudinger-Moser inequality. The typical growth of $f$ is thus exponential. Finally, a generalization of the result for nonhomogeneous nonlinearities is given. Using a blow-up approach, this paper completes the results in [Damascelli-Pardo, Nonlinear Anal. Real World Appl. 41 (2018)] and [Lorca-Ruf-Ubilla, J. Differential Equations 246 no. 5 (2009)], enlarging the class of nonlinearities for which the uniform a-priori bound applies.


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 di erent techniques have been developed to get such results for problems of the kind −∆m u = f (x, u) in Ω, where Ω ⊂ R N is typically a smooth bounded domain, ∆m u := div(|∇u| m− ∇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 = , namely for second-order semilinear problems, uniform a-priori bounds were rstly 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 * − = N+ N− , * 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 < 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 in nity with respect to u should be less than a subcritical power. Originally, most of these results concerned the case < m ≤ 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 > 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 W ,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´Ω g(u)dx < +∞ for any u ∈ W ,N (Ω) is g(t) = exp{α N t N N− }, the constant α N > 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 = (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 apriori 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 t α for some α ∈ ( , ), or which are comparable to t → e t . To prove a uniform a-priori bound in the rst 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 ll the gap between these growths. Indeed, Passalacqua improved the results of [1] in his Ph.D. Thesis [17] by means of a subtle modi cation of the argument therein, but the gap was still not completely lled 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 = .
We note that a similar approach has been also applied by Mancini and the author in [19] to study higherorder problems.
Throughout all the paper, Ω is a bounded and strictly convex domain in R N with C ,α boundary and the nonlinearity f satis es the following conditions: The second assumption is standard (cf. [1,2]) and in the linear case N = 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 > there are constants Cε , Dε > 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.
On the other hand, we say that f is critical if Remark 1. In view of (2), an example of a subcritical nonlinearity f (t) = log τ ( + t)t p e t α with τ ≥ , p ≥ and α ∈ [ , ), and for a critical nonlinearity is f (t) = e γt (t+ ) q with γ > . Notice that the second nonlinearity was excluded by the previous results [1,17] e.g. when γ = , q ≥ .
Although this distinction will be relevant in the proof, the critical case being more delicate, our main result applies for both classes of nonlinearities. Theorem 1.1. Let Ω ⊂ R N , N ≥ , be a bounded strictly convex domain of class C ,α for some α ∈ ( , ) and let f satisfy (A1)-(A3). Then, there exists a constant C > such that all weak solutions u ∈ W ,N (Ω) of the problem satisfy u L ∞ (Ω) ≤ C.
Remark 2. By the regularity theory of quasilinear problems by [20,21] the C ,α assumptions on ∂Ω are sucient to guarantee that the uniform a-priori estimate in Theorem 1.1 is actually in C ,α (Ω) for someα ∈ ( , ).
Remark 3. We may set our problem in the framework of entropy solutions and obtain the same result. Indeed, one may prove that, under conditions (A1)-(A3), such solutions are indeed weak and in turn classical. For further details, we refer to Section 5, in particular to Remark 6.
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.

Proposition 1.2.
In addition of the assumptions of Theorem 1.1, suppose that where λ denotes the rst eigenvalue of −∆ N . Then, problem (3) admits a positive weak solution.
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 nally 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 nd unbounded solutions.

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.
where the constants C, K depending on R and on h 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.

Proposition 2.3.
There exist positive constants r, C such that every solution u ∈ W ,N (Ω) of (3) veri es where Ωr Roughly speaking, the proof of the rst 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 Ωr. 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)).

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 de ne 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, where Ω k := Ω−x k µ k and µ k := (f (M k )) /N .
Notice that µ k → by (A1)-(A2) and this implies Ω k ↗ R N , since x∞ ∈ Ω. Moreover, each v k satis es (in a weak sense) the equation Indeed, for any test function φ ∈ C ∞ (Ω) there holdŝ Moreover, xing R > and looking at the behaviour of the sequence (v k ) k in B R ( ), applying the Harnack inequality given by Lemma 2.2 to w k : Therefore, by the local estimate provided by Lemma 2.1, we infer that, up to a subsequence, v where β is de ned in (A3). Indeed, the equation (10) can be rewritten as and thus, by a rst-order Taylor expansion of log •f around M k , one nds where 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 De nition 1.1.

. The subcritical case (β = )
In this case the equation (12) satis ed by the limit function v reduces to Recalling the energy estimate (6) of Proposition 2.3, it is su cient to integrate (14) on R N to get the desired contradiction, which proves Theorem 1.1 for such nonlinearities: . The critical case (β > ) With the same steps as in (15), one actually getŝ Therefore, we can characterize the limit pro le v by standard computations from a Liouville-type result by where . Furthermore, we claim that the energy concentrates around the blow-up points where θ is characterized by with ω N denoting the volume of the unit ball in R N . Indeed, We can quantify the constant θ as follows. Let us de ne the auxiliary function w := βv + (N − ) log β. Then, it is easy to show that w satis es −∆ N w = e w and´R N e w < ∞. Therefore, by [ In other words, if we apply the blow-up technique around a maximum point, we see that we can characterize the limit pro le and we get the concentration of the energy as in (18)- (19). This is indeed what happens around any blow-up point of the sequence (u k ) k , meaning points belonging to the set Lemma 3.1. From the sequence (u k ) k one can extract a subsequence still denoted by (u k ) k for which the following holds. There are P ∈ N and sequences (x k,i ) k for ≤ i ≤ P such that lim k→+∞ u k (x k,i ) = +∞ for any i and, setting where v is de ned in (17) and (18) Proof. The proof is inspired by [18,. We say that the property Hp holds whenever there exist p sequences (x k,i ) k for ≤ i ≤ p such that u k (x k,i ) → +∞ as k → +∞ and such that (i) and (ii) hold. From our previous investigation, we know that H holds. Our aim is to prove that the number of such sequences is nite by showing the following: if Hp holds then either H p+ holds too or there exists a constant C > such that (iii) holds. Indeed, if so, supposing by contradiction that Hp holds for any p, then we would be able to nd a sequence of disjoint balls B Rµ k,i (x k,i ) such that (18) holds, and thus by Lemma 2.3, an upper bound for p, a contradiction. As a consequence, Hp does not hold for any p and, by the claimed alternative, the proof is completed.
Now the claim has to be proved. Suppose that Hp for some p ∈ N but not (iii). Hence, de ning w k (x) := inf ≤i≤p |x − x k,i | N f (u k (x)), one can nd a sequence of points (y k ) k such that w k (y k ) = w k ∞ ↗ +∞. We show that for these points (y k ) k together with (x k,i ) k for ≤ i ≤ P, H p+ holds. First we get u k (y k ) → +∞, since and by the local boundedness of f on [ , +∞). Moreover, de ning Then, suppose by contradiction that a contradiction, so (i) is proved. Now, it remains to prove that u k has the same blow-up behaviour also for the sequence (y k ) k . To this aim, we rst show that, for any R > f (u k (y k + γ k x)) f (u k (y k )) ≤ C for some C = C(R) > and for any x ∈ B R ( ). Notice that this is not obvious as in (10), since (y k ) k do not have to be necessarily maximum points for (u k ) k . Rewriting in terms of the sequence (u k ) k the inequality w k (y k + γ k x) ≤ w k (y k ), which holds for any x ∈ B R ( ) for k large enough, one nds Fix now ε ∈ ( , ). By (20), there existsk =k(R, ε) > such that for any k >k we have Rγ k ≤ ε|y k − x k,i |.
In order to complete the proof of the local compactness of the sequence (ũ k ) k and consequently get (ii), we need to have its local boundedness in the L N norm according to Lemma 2.1. This time, unlike (11), before applying the Harnack inequality of Lemma 2.2, we need to know that our sequence is uniformly bounded from above, which is not immediate. So, suppose by contradiction that for any sequence of positive constants (C k ) k ↗ +∞ there exist points (x k ) k ⊂ B R ( ) such thatũ k (x k ) ≥ C k holds, thus, since f is increasing, Together with (21), this implies f (C k + u k (y k )) ≤ Cf (u k (y k )).
Choosing now C k = e u k (y k ) and setting t k := u k (y k ), (23) rereads as Then, by superlinearity of f and by the fact that f (s) ≤ e γs + D for some γ, D > for any s ∈ R + by (A3), we have which yields a contradiction since t k ↗ +∞. Hence, M(R) := max B R ( )ũk ∈ [ , +∞) and we can consider the function U k := M(R) −ũ k . We have U k ≥ , min B R ( ) U k = and furthermore U k satis es where the bound from below follows from (21). Therefore, by Lemma 2.2, we get Therefore, either max B R ( )ũk ≥ min B R ( )ũk , and in this case we easily infer ũ k L ∞ (B R ( )) ≤ C(R), or we deduce from (25) that min This readily implies min B R ( )ũk ≥ −C(R), so again we nd ũ k L ∞ (B R ( )) ≤ C(R). Applying Lemma 2.1, we infer that (ũ k ) k is locally compact in C ,α loc (R N ) and, with similar steps as in (13)- (17), we nally infer (ii). Consequently, the property Hp is satis ed and the proof is concluded.

Corollary 3.2. The set of blow-up points is nite.
Proof. We show the niteness of S by proving that S = {x (i) , ≤ i ≤ P}, where the points x (i) are de ned as in Lemma 3.1. Indeed, letx ∉ {x (i) , ≤ i ≤ P} for which one can nd a sequencex k →x such that, up to a subsequence, u k (x k ) → +∞. We have inf ≤i≤P, k∈N |x k − x (i) | ≥d > and, by Proposition 2.3, d(x k , ∂Ω) ≥ η > for some constantsd, η. Then, by (iii) of Lemma 3.1 and the superlinearity of f , we infer where µ is a positive measure which is singular at x . Hence, we can decompose µ as where ≤ A ∈ L (Br(x )), a is a positive constant and δx denotes the Dirac distribution centered at x . We rst claim that a ≥ θ with θ as in (18)- (19). Indeed, for any t ∈ ( , r) we havê by (18), where here x k is a blow-up sequence converging to x . As t is arbitrary, we thus nd a ≥ θ. Let now w be the distributional solution of Then by [25, Theorem 2.1] w ∈ C ,α (Br(x ) \ {x }) and has the explicit form We claim that w ≤ũ in Br(x ).
If so, choosing ε ∈ ( , β N ), in view of (2) on the one hand by Proposition 2.3 we get On the other hand, by means of the explicit expressions for w and θ: a contradiction. Therefore the proof of Theorem 1.1 is completed once we prove (29). Here, we follow the strategy in [1]. For any j ∈ N, let us de ne the maps B j : R → R + by Notice that z (j) k may be extended to 0 in Ω \ Br(x ). Since w and u k solve respectively problems (28) and (3), thenˆB Recalling f (u k ) µ and (26) and thusˆB which holds for any choice of j ∈ N. Then, the well-known inequality for any X ≠ Y ∈ R N , the constant d N > depending only on N (see [27,Proposition 4.6]), implies which nally proves our claim w −ũ ≤ in Br(x ), since (w −ũ) + ≤ on ∂Br(x ).

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 in nity and near the boundary are ful lled. More precisely, we may consider the problem where h : Ω × R → R + ∪ { } is a Carathéodory function satisfying the following conditions: for all τ ∈ R + and h(x, t) > for any x ∈ Ω and t > ; H2) there exist f ∈ C ([ , +∞)) satisfying (A1)-(A3) and < a ∈ L ∞ (Ω) ∩ C(Ω) such that H3) there existr,δ > such that • h(·, t) ∈ C (Ωr) for all t ≥ and ∂h ∂t (x, t) ≥ in Ωr × R + ; • ∇x h(x, t) · Ψ ≤ for all x ∈ Ωr, t ≥ and unit vectors Ψ such that |Ψ − n(x)| ≤δ.
We recall Ωr := {x ∈ Ω | d(x, ∂Ω) ≤ r}. Remark 4. In (H2) we assume a > only in the interior of Ω, but it may vanish on the boundary. Moreover, assumption (H3) is imposed to uniformly control solutions near the boundary by a moving-planes technique, prescribing in broad terms that h should be decreasing in x along suitable outer directions and increasing in t in a suitable neighbourhood of ∂Ω.
Remark 5. From (H1)-(H2) it follows that for each ε > there exists a constant dε ≥ such that where d is de ned in (A2), C is independent of u and Φ ∈ W ,N (Ω) is the rst (positive) eigenfunction of −∆ N on Ω. In order to prove that (32) yields a uniform bound near ∂Ω, we have to show that here our solutions are decreasing with respect to outer directions. We apply a moving-planes technique in the spirit of [28,Lemma 3.2]. Let us rst x some notation. For any direction ν, set the latter being nonempty for λ > a(ν) := inf x∈Ω x · ν. Moreover, for any x ∈ Ω, we denote by x ν λ the symmetric point with respect to the hyperplane T ν λ , namely Similarly, we de ne u ν λ := R ν λ (u), which is well-de ned on (Ω ν λ ) := R ν λ (Ω ν λ ). Our aim is to compare u and u ν λ near the boundary in the case ν = n(x) (the outer normal) and λ − a(ν) small, in order to conclude that u is decreasing along these directions in a small neighborhood of ∂Ω. First notice that by convexity of Ω, as long as λ − a(ν) is small, we have (Ω ν λ ) ⊂ Ω and Ω ν λ ∪ (Ω ν λ ) ⊂ Ωr. Therefore, for such λ, in Ω ν λ one has for some real number η(x, λ) lying between u ν λ (x) and u(x) by the mean value theorem, hence M ∈ L ∞ (Ω ν λ ).
on ∂Ω ν λ , and the weak comparison principle in small domains [12,Theorem 1.3] yields u ν λ ≥ u in Ω ν λ . We point out that the mentioned result is originally stated for M positive constant and for homogeneous nonlinearities, but it can be easily adapted to our setting (here assumption (H1) is required). By arbitrariness of λ and x ∈ Ωr, one deduces that u is decreasing along the outer normal direction and then, by a compactness argument, that there exist δ ∈ ( ,δ] and r ∈ ( ,r] independent of u such that ∇u(x) · Ψ ≤ for every x ∈ Ωr and for every direction Ψ such that |Ψ − n(x)| < δ. The boundedness of u and ∇u in a neighbourhood of the boundary now follows by a standard argument by [5], and moreover one inferŝ with Λ independent of u. We refer to [1, Propositions 2.1-3.1] for the details.
Let us now address to the blow-up analysis. We shall see that the argument carried out in Section 3 applies to the problem (30) with only minor modi cations. In particular, the rescaled functions v k de ned in (8)- (9) turn out to be weak solutions of and one shows that v k → v in C ,α loc (R N ) which satis es We recall that d(x∞, ∂Ω) > , as we have already excluded boundary blow-up, thus a(x∞) > by (H2).
In the subcritical case, the same argument as in Section 3.1 holds and, in the critical framework, Lemma 3.1 easily adapts, showing that near blow-up points the energy concentrates: h(y, u k (y))dy ≥ θ > , with the same constant θ de ned in (19). Then, the conclusion of the proof is analogous to the one for the homogeneous case.

A counterexample
As brie y 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 nd a positive potential a(x) such that the problem (30) admits unbounded solutions, in a suitable distributional sense.
Now we want to prove that w is an entropy solution of the problem (30). This kind of solutions has been introduced in the context of quasilinear elliptic problems in [29] in order to weaken the notion of weak solution for problems with L -data. Here we recall the precise meaning.
De nition 5.1. We say that u ∈ T ,N (Ω) if u is measurable and T k u ∈ W ,N (Ω) for any k > , where T k u is de ned as the truncation of u at level k, namely We start proving T k w ∈ W ,N (Bρ( )) for any k > and w de ned in (34). Indeed, T k w is radial and T k w(r) = w(r) for r ∈ [r k , ρ], w(r k ) for r ∈ [ , r k ), for some suitable r k ∈ ( , ρ), so it is easy to see that In order to show that u satis es (35), let φ ∈ C ∞ (Bρ( )) and notice that T k (w − φ) ∈ W ,N (Bρ( )). By our choice of a, the problem (30) is pointwise satis ed, thus we may test it in Bρ( ) by T k (w − φ): Note that the right-hand side is well-de ned as we proved that a ∈ L ∞ (Bρ( )) and e w α ∈ L (Bρ( )). Integrating by parts the left-hand side and recalling T k (w − φ) | ∂Bρ ( ) = , we havê which is the case of the equality in (35).
Remark 6. This counterexample shows that the class of the nonlinearities considered by assumption (A3) is sharp in order to have the property of (uniform) boundedness within the class of entropy solutions. Indeed, coming back to problem (30) under assumption (A3) and looking for entropy solutions in the sense of De nition 5.2, since this time (A3) implies a control of the nonlinearity by a suitable power of e t , say p, one hasˆΩ |h(x, u(x))| α dx ≤ CˆΩ e αpu < +∞, where the last inequality is due to [15,Corollary 1.7]. Therefore entropy solutions are classical, and our analysis and thus Theorem 1.1 apply.