Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0<a_{f} \leqslant \infty)$ is a smooth, increasing, convex nonlinearity such that $ f(0)>0 $ and which blows up at $ a_{f} $. Let $$0<\tau_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq \tau_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $\lambda_{m}$ satisfy the stability inequality $$ \sqrt{\lambda_{m}}\int_{\Omega}\sqrt{f'(u_{m})}\phi^{2}dx\leq \int_{\Omega}|\nabla\phi|^{2}dx, ~~\text{for all}~\phi\in H^{1}_{0}(\Omega),$$ then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n<\frac{4\alpha_{*}(2-\tau_{+})+2\tau_{+}}{\tau_{+}}\max \{1, \tau_{+}\},$ where $\alpha^{*}$ is the largest root of the equation $$(2-\tau_{-})^{2} \alpha^{4}- 8(2-\tau_{+})\alpha^{2}+4(4-3\tau_{+})\alpha-4(1-\tau_{+})=0.$$ In particular, if $\tau_{-}=\tau_{+}:=\tau$, then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n\leq12$ when $\tau\leq 1$, and for $n\leq7$ when $\tau\leq 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x)=\lim_{\lambda\uparrow\lambda^{*}}u_{\lambda}(x),$ where $\lambda^*$ is the extremal parameter of the eigenvalue problem.


Introduction and main results
In this article, we consider the problem where Ω ⊂ R n is a smooth bounded domain, n ≥ 1, λ > 0 is a real parameter, and the nonlinearity f satisfies (H) f : [0, a f ) → R + (0 < a f ∞) is a smooth, increasing, convex function such that f (0) > 0 and lim t→a f f (t) = ∞. Also, when a f = ∞ we assume that f is superlinear, i.e., lim t→∞ We call the nonlinearity f regular if a f = ∞ and singular when a f < ∞.
By a semistable solution of N λ we mean a solution u satisfies Also, we say that a smooth solution u of N λ is minimal provided u ≤ v a.e. in Ω for any solution v of N λ (see [6,7]. When f satisfies (H) is a regular, or f (t) = (1 − t) −p (p > 1), it is well known [2,3,17] that there exists a finite positive extremal parameter λ * > 0 depending on f and Ω such that for any 0 < λ < λ * , problem (N λ ) has a minimal smooth solution u λ , which is semistable and unique among the semistable solutions, while no solution exists for λ ≥ λ * . The function λ → u λ is strictly increasing on (0, λ * ), the increasing pointwise limit u * (x) = lim λ↑λ * u λ (x) is called the extremal solution. For 0 < λ < λ * the minimal solution u λ of problem (N λ ) satisfies the following stability inequality, for the proof see Corollary 1 in [7] or Lemma 6.1 in [11], for all φ ∈ H 1 0 (Ω).
The regularity and properties of the extremal solutions have been studied extensively in the literature [2][3][4][5][6][7][8][9][10][11][12]15,19] and it is shown that it depends strongly on the dimension n, domain Ω and nonlinearity f . Cowan, Esposito and Ghoussoub in [6] showed that for general nonlinearity f satisfies (H), u * is bounded for n ≤ 5. When f (u) = e u , in [6] it is shown that u * is bounded for n ≤ 8. This result improved by Cowan and Ghoussoub to n ≤ 10 in [7], and by Dupaigne, Ghergu and Warnault in [11] to n ≤ 12 which is the optimal dimension as we know on the unit ball u * is bounded if and only if n ≤ 12. As we shall see, in this paper we prove the same for a large class of nonlinearities including e u . When f (u) = (1 + u) p (p > 1) in [6] it is proved that u * is bounded if n < 8p p−1 that improved in [7] for to n < 4h(p) > 8p p−1 (for the definition of h(p) which is a decreasing function on (1, ∞) see [7]) with lim p→∞ 4h(p) ≈ 10.718. Recently, Hajlaoui, Harrabi and Ye in [18] improved this result by showing that u * is bounded for any p > 1 and n ≤ 12.
For the singular nonlinearity f (u) = (1 − u) −p (p > 1), in [6] it is proved that sup Ω u * < 1 if n ≤ 8p p+1 . In particular, when p = 2, u * is bounded away from 1 for n ≤ 5. The later result (and also the general case 1 < p = 3) is improved in [7] to n ≤ 6, and further improved by Guo and Wei in [14] to n ≤ 7. However, for p = 2 the expected optimal dimension is n = 8, holds on the ball, see [15].
By imposing extra assumptions on the general nonlinearity f satisfies (H), the authors in [6] obtained more regularity results in higher dimensions on general domains. Let f satisfy (H) and define In [6] the authors also show that for a regular and superlinear nonlinearity f with τ − > 0, u * is bounded for n ≤ 7 (see [6], Theorem 4.1). As we shall see here in Corollary 2.4, with a minor change in their proof, the same holds with a weaker condition. Also, they showed that if τ + < ∞ then u * is bounded for n < 8 τ+ , see Theorem 5.1 in [6].
The main results of this paper are as follows.

Preliminaries and Auxiliary Results
The following standard regularity result is taken from [8], for the proof see Theorem 3 of [19].
with c, g ∈ L q (Ω) for some q > n 2 . Then there exists a positive constant C independent of u such that: Consider problem (N λ ). By the elliptic regularity we know that, if for some q ≥ 1 we have ||f (u λ )|| L q (Ω) ≤ C, where C is a constant independent of λ, then u * is bounded, (hence smooth when f is regular), whenever n < 4q. Using the above proposition we show that, a similar result holds (for regular or singular nonlinearity) if f ′ (u λ ) is uniformly bounded in L q (Ω). For the proof we need the following two lemmas, the first one gives pointwise estimate on ∆u for a solution u of problem (N λ ), for the proof see [6]. Then where C is a constant independent of u.
Proof. Let ψ be the unique positive smooth function such that Let u be a semistable solution of problem (N λ ). By multiplying the equation ∆ 2 u = λf (u) in ψ and then an integration we get (using Green's formula) This gives that The inequality above and the uniform L 1 (Ω) boundedness of f (u) for semistable solutions (proved in Lemma 3.5 in [6]) gives the desired result.
In the sequel we will frequently use the following simple lemma. where C is a constant independent of u m , then the same holds for Ω g 2 (u m )dx.
Proof. Indeed, we have Proposition 2.2. Let f satisfy (H) (when f is singular we additionally assume that lim t→a f F (t) = ∞). Let u m be a sequence of semistable solutions of problem  Indeed, f ′ is a nondecreasing function by the convexity of f , thus we have, for now the fact that f (t) → ∞ as t → a f gives (2.6).

. Again an integration gives
Now the facts that lim t→a f f (t) = ∞ and τ < 2 imply that lim t→a f F (t) = ∞.
For example, take the singular nonlinearity f (t) = (1 − t) −p (p > 1) on [0, 1). We have τ − = p+1 p ∈ (0, 2) and Then, as a corollary of Proposition 2.2 , we have the next regularity result for problem (N λ ). It is proved in [7,6] by a different proof with the restriction that p = 3. Proof. Notice that we have

Hence, by the assumption sup
As an application of Proposition 2.2, consider problem (N λ ) with a convex nonlinearity f satisfies (H) such that f (t) = t ln t for t large. Then, for every ǫ > 0 there exist Now if u ≥ 0 is a semistable solution of problem (1.1), from Lemma 3.5 [6] we have Ω f (u)dx ≤ C with C independent of λ and u. This together (2.7) and Lemma 2.3 give f ′ (u) ∈ L 1 ǫ (Ω) uniformly, hence by Proposition 2.2, u * is bounded for n < 4 ǫ , and since ǫ > 0 was arbitrary, u * is bounded in every dimension n. Indeed, the same result is true for every regular nonlinearity f satisfies (H) with τ + = 0 or equivalently Indeed, (2.8) implies (2.7) and we can proceed as above.
The following lemma is a special case of an interesting result of [6].
where C is a constant independent of λ and u.
When f is regular, in [6] the authors used the above lemma to prove that u * is bounded for n < 8 τ+ . In a completely similar manner and using Proposition 2.2, we can prove a similar result when f is singular. Proof. Take an arbitrary number τ > τ + , then from the definition of τ + there exists a Thus, for a T > T 2 sufficiently close to a f we have τ , and since τ > τ + was arbitrary we get (2.10).
As we have mentioned before, another main result of [6] is that if τ − > 0 then u * is bounded for n ≤ 7. Using the same proof of this in [6] we can prove it by a weaker assumption as follows: Corollary 2.6. Consider problem (N λ ) with a regular nonlinearity f satisfies (H) such that for some 0 ≤ ǫ < 1 Then u * is bounded for n ≤ 7. 2 , t ≥ T , for some T > 0. Hence, using the inequality (2.6) and the fact that F is a nondecreasing function we get, for a T ′ > T sufficiently large, Thus, from Lemmas 2.3 and 2.4 we have ||f (u)|| L 2− ǫ 4 (Ω) < C where C is independent of u. Now the elliptic regularity implies u * is bounded for n ≤ 8 − ǫ > 7, that gives the desired result.

Proof of the main results
Following the idea of Dupaigne, Ghergu and Warnault in [7], we prove the following lemma which is crucial for the proof of the main results.
Proof. Let u be a positive smooth solution of (N λ ) satisfy (1.2) and set v := −∆u.
Up to rescaling, we may assume that λ = 1. Take φ = θ(u) as a test function in the stability inequality (1.2). Then we get Also, taking φ = v α (α > 1 2 ) as a test function in the stability inequality (1.2), we get Using Hölder inequality (with two conjugate numbers 2α and 2α 2α−1 ) on the righthand side of inequality (3.2) we get Similarly, form ( 3.3) and Hölder inequality we get that gives (3.5) Plugging (3.5) in (3.4) we arrive at which is the desired result.