A priori estimates for semistable solutions of semilinear elliptic equations

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound follows from a $L^p$-estimate for $f'(u)$ for every $p<3$ and $n\geq 2$. Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.


Introduction
In this note we consider semistable solutions of the boundary value problem (1.1) Lu + f (u) = 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R n is a smooth bounded domain with n ≥ 2, f ∈ C 2 , and Lu := ∂ i (a ij (x)u j ) is uniformly elliptic. More precisely, we assume that (a ij (x)) is a symmetric n × n matrix with bounded measurable coefficients, i.e., a ij = a ji ∈ L ∞ (Ω), for which there exist positive constants c 0 and C 0 satisfying (1.2) c 0 |ξ| 2 ≤ a ij (x)ξ i ξ j ≤ C 0 |ξ| 2 for all ξ ∈ R n , x ∈ Ω.
By semistability of the solution u, we mean that the lowest Dirichlet eigenvalue of the linearized operator at u is nonnegative. That is, we have the semistability inequality There is a large literature on a priori estimates, beginning with the seminal paper of Crandall and Rabinowitz [4]. In [4] and subsequent works, a basic and standard assumption is that u is positive in Ω and f ∈ C 2 is positive, nondecreasing, and superlinear at infinity: Note that, under these assumptions and with f (u) replaced by λf (u) with λ ≥ 0, semistable solutions do exist for an interval of parameters λ ∈ (0, λ * ); see [4].
Research of the first and second authors supported by MTM2011-27739-C04-01 (Spain) and 2009SGR345 (Catalunya). The second author is also supported by ERC grant 320501 (ANGEOM project). Research of the third author supported in part by the NSF and Simons Foundation.
In recent years there have been strong efforts to obtain a priori bounds under minimal assumptions on f (essentially (1.4)), mainly after Brezis and Vázquez [1] raised several open questions. The following are the main results in this direction. The important paper of Nedev [5] obtains the L ∞ bound for n = 2 and 3 if f satisfies (1.4) and in addition f is convex. Nedev states his result for L = ∆ but it is equally valid for general L. When 2 ≤ n ≤ 4 and L = ∆, Cabré [2] established that the L ∞ bound holds for arbitrary f if in addition Ω convex. Villegas [9] replaced the condition that Ω is convex in Cabré's result assuming instead that f is convex. For the radial case, Cabré and Capella [3] proved the L ∞ bound when n ≤ 9. On the other hand, it is well known that there exist unbounded semistable solutions when n ≥ 10 (for instance, for the exponential nonlinearity e u ).
For convex nonlinearities f and under extra assumptions involving the two numbers much more is known (see more detailed comments after Corollary 1.3). For instance, Sanchón [6] proved that u ∈ L ∞ (Ω) whenever τ − = τ + ≥ 0 and n ≤ 9. This hypothesis is satisfied by f (u) = e u , as well as by f (u) = (1 + u) m , m > 1. It is still an open problem to establish an L ∞ estimate in general domains Ω when n ≤ 9 under (1.4) as the only assumption on f .
Our purpose here is to prove the following results: Theorem 1.1. Let f ∈ C 2 be convex and satisfy (1.4). Assume in addition that for every ε > 0, there exist T = T (ε) and C = C(ε) such that Then if u is a positive semistable solution of (1.1), we have f ′ (u) ∈ L p (Ω) for all p < 3 and n ≥ 2, while f (u) ∈ L p (Ω) for all p < n n−4 and n ≥ 6. As a consequence, we deduce respectively: (a) If n ≤ 5, then u ∈ L ∞ (Ω). (b) If n ≥ 6, then u ∈ W 1,p 0 (Ω) for all p < n n−5 and u ∈ L p (Ω) for all p < n n−6 . In particular, if n ≤ 9 then u ∈ H 1 0 (Ω).
The main novelty of our results are twofold. On the one hand, we do not assume any lower bound on f ′ to obtain our estimates, nor any bound on f ′′ as in [4] or [6] (as commented below). On the other hand, we obtain L p estimates for f ′ (u). To our knowledge such estimates do not exist in the literature. In fact, using the L p estimate for f (u) established in Theorem 1.2 and standard regularity results for uniformly elliptic equations, it follows that u is bounded in L ∞ (Ω) whenever n < 6 + 2ε 1−ε . Note that the range of dimensions obtained in Theorem 1.2 (a), n < 6 + 4ε 1−ε , is bigger than this one. This will follow from the L p estimate on f ′ (u). Of course, in both results (and also in the rest of the paper), u ∈ L p or u ∈ W 1,p mean that u is bounded in L p or in W 1,p by a constant independent of u.
(ii) Setting s = f (t) and t = γ(s), (1.6) is equivalent to the condition γ ′ (s) ≥ θs −1−ε for some θ > 0 and for all s sufficiently large. This clearly shows that (1.6) does not follow from the convexity of f alone (which is equivalent to γ ′ being nonincreasing).
(iii) Note that by convexity, εf for all t sufficiently large.

Preliminary estimates
We start by recalling the following standard regularity result for uniformly elliptic equations.
Proposition 2.1. Let a ij = a ji , 1 ≤ i, j ≤ n, be measurable functions on a bounded domain Ω. Assume that there exist positive constants c 0 and C 0 such that (1.2) holds.
with c, g ∈ L p (Ω) for some p ≥ 1.
Then, there exists a positive constant C independent of u such that the following assertions hold: (i) If p > n/2 then u L ∞ (Ω) ≤ C( u L 1 (Ω) + g L p (Ω) ).
(ii) Assume c ≡ 0. If 1 ≤ p < n/2 then u L r (Ω) ≤ C g L p (Ω) for every 1 ≤ r < np/(n − 2p). Moreover, u W 1,r 0 (Ω) ≤ C for every 1 ≤ r < np/(n − p). Part (i) of Proposition 2.1 is established in Theorem 3 of [7] with the L 2 -norm of u instead of the L 1 -norm. However, an immediate interpolation argument shows that the result also holds with u L 1 (Ω) . Note also that in the right hand side of this estimate, u L ∞ (Ω) ≤ C( u L 1 (Ω) + g L p (Ω) ), some dependence of u must appear (think on the equation with g ≡ 0 satisfied by the eigenfunctions of the Laplacian). For part (ii) we refer to Theorems 4.1 and 4.3 of [8].
As an easy consequence of Proposition 2.1 (i) we obtain the following:  The following estimates involving are due to Nedev [5] when L = ∆. We give here a new proof of the estimates consistent with our own approach. Note that assumptions (1.6) and (1.7) in Theorems 1.1 and 1.2, respectively, also hold replacing f byf on their right hand side, since f (t) ≤ 2(f (t) − f (0)) = 2f (t) for t large enough. We will use this fact in the proof of both results. Lemma 2.3. Let f ∈ C 2 be convex and satisfy (1.4). If u is a positive semistable solution of (1.1), then there exists a positive constant C independent of u such that Multiplying the previous identity byf (u) and using the semistability condition (1.3), we obtain or equivalently, As a consequence, the second estimate in (2.1) follows by the first one.

respectively.
Combining (2.2) and (2.3), we obtain (2.5) Choose M (depending on f ) such that f (t) > 2f (0) + 2 for all t ≥ M. On the one hand, using (2.4), the convexity of f , and that (a ij ) is a positive definite matrix, we obtain On the other hand, for some constant C depending only on f (and M), there holds where the last inequality follows from multiplying equation (1.1) by min{u, M}. Combining the previous bounds with (2.5), it follows that and using that f ′ (t) → +∞ at infinity (see Remark 2.4 below), we conclude where C is independent of u.
Remark 2.4. Note thatf (t)/t ≤ f ′ (t) for all t ≥ 0 since f is convex. In particular, by condition (1.4), we obtain lim t→∞ f ′ (t) = ∞. Therefore, as a consequence of estimate (2.1) we obtain where C is a constant independent of u. As in [5], from this and Proposition 2.1 (ii), one deduces (2.8) u in L q (Ω) for all q < n/(n − 2).
Our results improve this estimate under the additional assumptions on f of Theorems 1.1 and 1.2.
The following is a sufficient condition on f to guarantee u ∈ H 1 0 (Ω). Note that by convexity of f , tf ′ (t) −f (t) ≥ 0 for all t ≥ 0. If we further assume that for some where in the last inequality we used the superlinearity of f and the second estimate in (2.1).

Proof of Theorems 1.1 and 1.2
Proof of Theorem 1.1. Assume (1.6). In fact, as we said before Lemma 2.3 we may assume that (1.6) holds replacing f byf : for every ε > 0, there exist T = T (ε) and C = C(ε) such that In the following, the constants C may depend on ε and T but are independent of u.
We start by proving that f ′ (u) ∈ L p (Ω) for all p < 3 and as a consequence the statement in part (a). Let α = 3+ε 1+ε (with ε as in (3.1)). Multiplying (1.1) by and integrating by parts we obtain In particular, by Lemma 2.3 and the bound (2.7), we obtain Therefore, by the arbitrariness of ε > 0, we obtain f ′ (u) ∈ L p (Ω) for all p < 3. As a consequence, by Corollary 2.2 and since u ∈ L 1 (Ω) (see Remark 2.4), we obtain the L ∞ estimate established in part (a), i.e., if n < 6 then u L ∞ (Ω) ≤ C.