Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $\Omega$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabr\'{e} \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<\beta_{-}:=\liminf_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}<\infty$$ where $F(t)=\int_{0}^{t}f(s)ds$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 6$. Also, if $\beta_{-}=\beta_{+}>\frac{1}{2}$ or $\frac{1}{2}<\beta_{-}\leq\beta_{+}<\frac{7}{10}$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 9$. Moreover, if $\beta_{-}>\frac{1}{2}$ then $u^{*}\in H^{1}_{0}(\Omega)$ for $n\geq 2$.

It is well known ( [3,11,13]) that there exists a finite positive extremal parameter λ * such that for any 0 < λ < λ * , problem (1.1) has a minimal classical solution u λ ∈ C 2 (Ω), while no solution exists, even in the weak sense for λ ≥ λ * . The function λ → u λ is increasing and the increasing pointwise limit u * (x) = lim λ↑λ * u λ (x) is a weak solution of (1.1) for λ = λ * which is called the extremal solution. If λ < λ * the solution u λ is obtained by the implicit function theorem and is stable in the sense that the first Dirichlet eigenvalue of the linearized problem at u λ , −∆ − λf ′ (u λ , is positive for all λ ∈ (0, λ * ). That is, 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 . When f is convex, Nedev in [16] proved that u * ∈ L ∞ (Ω) for n = 2, 3 in any domain Ω. When 2 ≤ n ≤ 4 the best known result was established by Cabré [5] who showed that u * ∈ L ∞ (Ω) for arbitrary nonlinearity f if in addition Ω is convex. Applying the main estimate used in the proof of the results of [5], Villegas [19] got the same replacing the condition that Ω is convex with f is convex. Cabré and Capella [8] proved that u * ∈ L ∞ (Ω) if n ≤ 9 and Ω = B 1 . Also, in [11], Cabré and Ros-Oton showed that u * ∈ L ∞ (Ω) if n ≤ 7 and Ω is a convex domain of double revolution (see [11] for the definition).
The case when f is not convex and Ω is arbitrary domain, is more challenging and there is nothing much in the literature about the boundedness of the extremal solution. Indeed, in this case, again the best result is due to Cabré [5] who showed that u * ∈ L ∞ (Ω) for arbitrary f and Ω in dimension n = 2.
In this work we consider problem (1.1) for the case when f is not necessarily convex and Ω is an arbitrary domain and prove the boundedness of the extremal solution in higher dimensions under some extra assumptions on f . Let f satisfy (H) and define The main results of this paper are as follows.
Theorem 1.1. Let f (not necessarily convex) satisfy (H) with 1 2 < β − ≤ β + < ∞ and Ω an arbitrary bounded smooth domain. Let u * be the extremal solution of problem (1.1). Then u * ∈ L ∞ (Ω) for n < 4 + 4 2β As consequences, by the assumption It is worth mentioning here that, for a convex nonlinearity f we always have β + ≥ β − ≥ 1 2 . Indeed in this case f ′ is a nondecreasing function, hence we have Also for general nonlinearities f (not necessarily convex) satisfy only (H) we always have β + ≥ 1 2 . To see this, 3 by contradiction assume that 0 ≤ β + < 1 2 and take a β ∈ (β + , 1 2 ). Then from the definition of β + there exists Example 1.1. Consider problem (1.1) in an arbitrary bounded smooth domain Ω with f (u) = u 2 + 3u + 3 cos u + 4. It is easy to see that f satisfies (H), but is not convex (even at infinity). Indeed, we have f ′′ (u) = 2 − 3 cos u, which is negative for all u such that cos u > 2 3 (so none of the previous results apply). However, by a simple computation we have 1 > β − = β + = 2 3 > 1 2 , hence by Theorem 1.1 we get u * ∈ L ∞ (Ω) for n ≤ 15.
To get the regularity of the extremal solution in low dimensions or proving that it is in the energy class (i.e., u * ∈ H 1 0 (Ω)) we can weaken the assumptions as follows.
Theorem 1.2. Let f (not necessarily convex) satisfy (H) and Ω an arbitrary bounded smooth domain in R n . Let u * be the extremal solution of problem (1.1). Then (i) if for some ǫ > 0 there exist t 0 > 0 such that we have then u * ∈ L ∞ (Ω) for n < 5. In particular this is true if β − > 1 2 .

Preliminaries and Auxiliary Results
To prove the main results we need the following simple technical lemma based on inequality (1.2), which is used frequently in the literature, for example [11,9,16,20]. It is also proved in [1] for the general semilinear elliptic equation −Lu = λf (u) with zero Dirichlet boundary condition, but for the convenience of the reader we sketch a proof here for the case L = ∆. Also, Proof. Let u λ ∈ C 2 (Ω) be the minimal classical solution of (1.1) where 0 < λ < λ * , and take ϕ = g(u λ ) in the semi-stability condition (1.2). Then we get By using the Green's formula one can show that Now from (2.1) there exists t 0 > 0 so that H(t) ≥ 0 for t ≥ t 0 , thus using (2.4) we obtain where |Ω| denotes the Lebesgue measure of Ω and C 0 := sup t∈[0,t0] (|H(t)| − H(t)). Now, since C 0 is independent of λ we get the desired result.
The following consequence of the above lemma is essential in the proof of the main results.
Proposition 2.1. Let u λ be the minimal solution of (1.1) and ξ : Proof. Let g : [0, ∞] → [0, ∞] be a C 1 function with g(0) = 0 and Also, let G(t) = t 0 g ′ (s) 2 ds as in lemma 2.1. Then using the equality implies that Now using (2.6), for t ≥ t 0 we have which is positive for large t ≥ t 0 (by the assumption), hence by Lemma 2.1 we get ||H(u λ )|| L 1 (Ω) ≤ C 1 , where C 1 is a constant independent of λ. However, again by the assumption we have 0 < E(t) < 2H(t) for large t that also gives ||E(u λ )|| L 1 (Ω) ≤ C 2 , where C 2 is a constant independent of λ, which is the desired result. 6 To prove Theorem 1.2 in the next section, we also need the following rather standard result. For a simple proof see [1].
Proposition 2.2. Let f satisfy (H) and u λ be the minimal solution of problem (1.1). If there exists a positive constant C independent of λ such that whereC is a positive constant independent of λ.
and C is a constant independent of λ. Now, from Proposition 2.2 gives u * ∈ L ∞ (Ω) for n < 2γ 2 , that proves the second part.
To prove part (a), note that in the case β + ≥ 1, it is easy to see that the right hand side of (1.5) is larger than 6 and when β + < 1 we can use (1.6).

Acknowledgement
This research was in part supported by a grant from IPM (No. 93340123).