A priori estimates for semistable solutions of p -Laplace equations with general nonlinearity

Abstract

In this paper we consider the p-Laplace equation $- Δ_{p} u = λ f (u)$ in a smooth bounded domain $Ω \subset R^{N}$ with zero Dirichlet boundary condition, where $p > 1$ , $λ > 0$ and $f : [0, \infty) \to R$ is a $C^{1}$ function with $f (0) > 0$ , $f^{'} ⩾ 0$ and ${lim}_{t \to \infty} \frac{f (t)}{t^{p - 1}} = \infty$ . For the sequence ${(u_{λ})}_{0 < λ < λ^{*}}$ of minimal semi-stable solutions, by applying the semi-stability inequality we find a class of functions E that asymptotically behave like a power of f at infinity and show that $‖ E (u_{λ}) ‖_{L^{1} (Ω)}$ is uniformly bounded for $λ < λ^{*}$ . Then using elliptic regularity theory we provide some new $L^{\infty}$ estimates for the extremal solution $u^{*}$ , under some suitable conditions on the nonlinearity f, where the obtained results require neither the convexity of f nor the strictly convexity of the domain. In particular, under some mild assumptions on f we show that $u^{*} \in L^{\infty} (Ω)$ for $N < p + 4 p / (p - 1)$ , which is conjectured to be the optimal regularity dimension for $u^{*}$ .

Keywords

semi-stability extremal solution regularity asymptotic behavior

1. Introduction and main results

The aim of this paper is to study the regularity of extremal solutions of quasilinear reaction–diffusion problem: $\begin{matrix} (1) & \{\begin{matrix} - Δ_{p} u = λ f (u), & in Ω, \\ u = 0, & on \partial Ω, \end{matrix} \end{matrix}$ where the diffusion is driven by the p-Laplace operator $- Δ_{p} u : = - div (| \nabla u |^{p - 2} \nabla u)$ , $p > 1$ , $Ω \subset R^{N}$ is a smooth bounded domain, λ is a positive real parameter and f is a $C^{1}$ function and satisfies: $\begin{matrix} (2) & f (0) > 0, f^{'} ⩾ 0 and lim_{t \to \infty} \frac{f (t)}{t^{p - 1}} = \infty . \end{matrix}$ The functions ${(1 + u)}^{m}$ with $m > p - 1$ and the exponential $e^{u}$ are typical examples for the non-linearity f with the above properties.

These reaction–diffusion problems are attracting considerable interest due to their applications in the fields of physics, chemistry and biology. Particularly, Problem (1) in the exponential case and when $p = 2$ , is referred to the Gelfand problem which arises as a very simplified model in combustion theory. For instance, up to dimension $N = 3$ , (1) can be derived from the thermal self-ignition model which describes the reaction process in a combustible material during the ignition period. Also, it appears as a model in astrophysical problems, it was considered by Emden and Fowler and studied by Hopf and Chandrasekar in [12]. Furthermore, Kazdan and Warner considered the problem in Riemannian geometry context, see [25].

Throughout the paper, we say that a non-negative function $u \in W_{0}^{1, p} (Ω)$ is a weak energy solution of (1) if $f (u) \in L^{1} (Ω)$ and u satisfies $\begin{matrix} (3) & \int_{Ω} | \nabla u |^{p - 2} \nabla u . \nabla φ d x = λ \int_{Ω} f (u) φ d x, for all φ \in C_{0}^{1} (Ω) . \end{matrix}$ These solutions, which may be unbounded, are p-superharmonic and if $u ≢ 0$ then by strong maximum principle $u > 0$ almost every where in Ω (see [34,35]). Also, by the semi-stability of a solution $u \in W_{0}^{1, p} (Ω)$ of (1) we mean that the second variation of the energy functional associated to (1) is non-negative definite [9,32]. That is, $\begin{matrix} (4) & \int_{{\nabla u \neq 0}} | \nabla u |^{p - 2} [(p - 2) {(\frac{\nabla u}{| \nabla u |} . \nabla ϕ)}^{2} + | \nabla ϕ |^{2}] d x - λ \int_{Ω} f^{'} (u) ϕ^{2} d x ⩾ 0 \end{matrix}$ for every $ϕ \in A_{u}$ where $\begin{matrix} A_{u} : = W_{0}^{1, p} (Ω) if p ⩾ 2, \end{matrix}$ and $\begin{matrix} A_{u} : = {ϕ \in W_{0}^{1, p} (Ω) : | ϕ | ⩽ C u and | \nabla ϕ | ⩽ C | \nabla u | in Ω, for some constant C}, if 1 < p < 2 . \end{matrix}$ A weak energy solution u of (1) is said to be regular if $f (u) \in L^{\infty} (Ω)$ and minimal if $u ⩽ v$ a.e in Ω for any other solution v, [31]. Moreover, standard regularity results for degenerate elliptic equations shows that every regular solution u belongs to $C^{1, β} (\bar{Ω})$ for some $β > 0$ (see [32]).

Cabré and Sanchón [9] proved that, under assumption (2), there exists a parameter $λ^{*} \in (0, + \infty)$ such that Problem (1) has a bounded minimal solution $u_{λ}$ for $λ < λ^{*}$ and no regular solution for $λ > λ^{*}$ . Furthermore, the minimal solutions are semi-stable. Using this property, it was established that $\begin{matrix} (5) & u^{*} : = lim_{λ ↑ λ^{*}} u_{λ} \end{matrix}$ is a weak energy solution of (1) for $λ = λ^{*}$ , whenever the non-linearity $f (u)$ makes its growth comparable to $u^{m}$ (see [9]). In the literature, in the case $p = 2$ , $u^{*}$ and $λ^{*}$ are known as the extremal solution and parameter; respectively.

The problem of asymptotic behavior of stable radial solutions of (1) when Ω is a ball or the whole space $R^{N}$ , as well as regularity of the extremal solution, global analysis and Liouville theorems for stable solutions and solutions which are stable outside a compact set, in particular for important nonlinearities $f (u) = e^{u}$ and $f (u) = {(1 + u)}^{q}$ or $\frac{1}{{(1 - u)}^{q}}$ , $q > 1$ , are extensively studied in the literature, where most of the results are obtained for the special case of Problem (1), or some related problems, when $p = 2$ ; see for example [1–11,13–23,25–34,36–38]. Moreover, in the case of singular non-linearities such as $f (u) = {(1 - u)}^{- 2}$ , Problem (1) is also relevant as a model equation to describe Micro Electro Magnetic System (MEMS) devices theory (see [11,30]), also see [18] where the authors, by the method of asymptotic analysis gave a rather complete description of asymptotic behavior of the radial solutions and positive entire solutions as well as the extremal parameter of $λ^{*}$ .

As a particular case, authors in [9] considered Problem (1) with the power non-linearity $f (u) = {(1 + u)}^{m}$ where $m > p - 1$ and obtained that $u^{*}$ is regular solution for $\begin{matrix} (6) & N < G (m, p) : = \frac{p}{p - 1} (1 + \frac{m p}{m - (p - 1)} + 2 \sqrt{\frac{m}{m - (p - 1)}}) . \end{matrix}$ Ferrero in [19] (independently of [9]) proved that the extremal solution is regular when $N < G (m, p)$ and obtained that if $N ⩾ G (m, p)$ and $Ω \subset R^{N}$ is the unit ball then $u^{*}$ is unbounded. Garcia-Azorero, Peral and Puel [20,21] studied Problem (1) when $f (u) = e^{u}$ and proved that $u^{*}$ is bounded independently of Ω when $\begin{matrix} N < N_{opt} : = p + \frac{4 p}{p - 1}, \end{matrix}$ which is the optimal range of regularity for $u^{*}$ . Furthermore, they proved that if $N ⩾ p + \frac{4 p}{p - 1}$ and $Ω \subset R^{N}$ is the unit ball, then $λ^{*} = p^{p - 1} (N - p)$ and $u^{*} = log (\frac{1}{| x |^{p}})$ which is unbounded. M. Sanchón in [32] proved that under assumptions (2) and the following convexity condition $\begin{matrix} (7) & {(f (t) - f (0))}^{\frac{1}{p - 1}} is convex in [0, + \infty) \end{matrix}$ when $p ⩾ 2$ , then $u^{*}$ defined by (5) is a solution of (1) with $λ = λ^{*}$ when $N < p (1 + p^{'})$ and it is regular if $N < p + p^{'}$ where $p^{'} = \frac{p}{p - 1}$ . Furthermore, defining $\begin{matrix} τ_{-} : = \underset{t ⟶ \infty}{lim inf} \frac{(f (t) - f (0)) f^{″} (t)}{{f^{'} (t)}^{2}} ⩽ τ_{+} : = \underset{t ⟶ \infty}{lim sup} \frac{(f (t) - f (0)) f^{″} (t)}{{f^{'} (t)}^{2}}, \end{matrix}$ Sanchón in [31] proved some regularity results whenever $τ_{-} > \frac{p - 2}{p - 1}$ and f satisfies (7).

Castorina [11] proved that, in the case $1 < p < 2$ , $u^{*}$ is regular up to dimension $N ⩽ p + 2$ whenever f just satisfies (2) and ${f (t)}^{\frac{1}{p - 1}}$ is convex for any $t ⩾ T$ (without any further assumption on the behaviour of $f^{'}$ and $f^{″}$ at infinity). Furthermore, very recently P. Miraglio in [27] establishes an $L^{\infty}$ a priori estimate for stable solutions under a new condition on N and p, which holds for every $f \in C^{1}$ . His condition is optimal in the radial case for $N ⩾ 3$ , but more restrictive in the non-radial case. However, interestingly, his results consists of an interior estimate for stable solutions which does not depend on the boundary values of the function and holds for every $C^{1}$ non-linearity and every bounded domain. He proved that if $4 ⩽ N < \frac{1}{2} (\sqrt{(p - 1) (p + 7)} + p + 5)$ or $p < N = 3$ then for every $δ > 0$ , $‖ u ‖_{L^{\infty} (K_{δ})} ⩽ C (‖ u ‖_{L^{1} (Ω)} + ‖ \nabla u ‖_{L^{p} (Ω ∖ K_{δ})})$ , where $K_{δ} : = {x \in Ω : dist (x, \partial Ω) > δ}$ and C is a constant depending only on Ω, δ and p. Although this interior estimate does not require strict convexity of the domain, but passing from it to the global bound requires some boundary estimates, which are available if the domain is strictly convex.

It is worth mentioning that, under the hypotheses (2) and (7), Problem (1) has been first extensively studied for the classical case $p = 2$ , see [1,2,4,5,13,16,24,26,29]. Very recently Cabré, Figalli, Ros-Oton, and Serra [7] completely solved two famous open problems posed by Brezis and Brezis-Vazquez in [5] regarding the boundedness of the extremal solution $u^{*}$ for $N ⩽ 9$ , and the $H_{0}^{1}$ regularity of $u^{*}$ in every dimension, when one of the nonlinearity f or the domain Ω assumed to be convex. We also point out that Mironescu and Rădulescu [28] studied Problem (1) in the cases of nonlinearities with linear growth at infinity.

In this article, we rule out the convexity assumption and prove some regularity results for function $u^{*}$ defined by (5) under a mild extra assumption on f, whenever the nonlinearity f just satisfies the typical condition (2). To this end, by applying the semi-stability (4) to some suitable test functions we find a class of functions E defined on $[0, \infty)$ and that asymptotically behave like a power of f at infinity, and show that the functions $E (u_{λ})$ are uniformly bounded in $L^{1} (Ω)$ for $λ < λ^{*}$ . Then using elliptic regularity theory along with Sobolev imbedding theorems we show that $u_{λ}$ is uniformly bounded in $L^{\infty} (Ω)$ for dimensions $N < N (p, f)$ where $N (p, f)$ depends on p and the asymptotic behavior of the nonlinearity f at infinity.

The rest of the paper is divided into two Sections. In Section 2 we state our main regularity results, clarify them by some examples and compare our findings with previous ones in the literature. In Section 3 we prove our main results.

2. Main results and examples

For the remainder of this paper, we set: $\begin{matrix} h (t) : = {f (t)}^{\frac{1}{p - 1}} . \end{matrix}$ We note that from (2), the above function h is an increasing, super-linear function (i.e., ${lim}_{t \to \infty} \frac{h (t)}{t} ⟶ \infty$ ) and $h (0) > 0$ but it is not necessarily convex. Now, we define the following new parameters $\begin{matrix} η_{-} : = \underset{t ⟶ \infty}{lim inf} \frac{h^{'} (t) H (t)}{{h (t)}^{2}} ⩽ η_{+} : = \underset{t ⟶ \infty}{lim sup} \frac{h^{'} (t) H (t)}{{h (t)}^{2}}, \end{matrix}$ where $H (t) : = \int_{0}^{t} h (s) d s$ , for $t ⩾ 0$ .

Remark 1.
It is worth mentioning here that for a general non-linearity f satisfying (2) we always have $η_{+} ⩾ \frac{1}{2}$ , also when h is convex we have $η_{-} ⩾ \frac{1}{2}$ (see [2]).

In the following, we state our main regularity results.
Theorem 1.
Let $u^{}$ be the function defined in (* 5 ), f satisfy ( 2 ) (not necessarily convex), $\frac{1}{2} < η_{-} ⩽ η_{+} < \infty$ and Ω be an arbitrary bounded smooth domain. Then $u^{} \in L^{\infty} (Ω)$ whenever* $\begin{matrix} (8) & N < \frac{p}{p - 1} (p + \frac{2 η_{-} - 1}{η_{+}} + 2 \sqrt{\frac{2 η_{-} - 1}{η_{+}}}) . \end{matrix}$ Moreover, if $η_{+} < 1$ and $\frac{\tilde{f} (t)}{t^{p - 1}}$ is bounded near zero, then $u^{}$ is regular provided* $\begin{matrix} (9) & N < \frac{p}{p - 1} (1 + \frac{(p - 2) η_{+} + 2 η_{-}}{2 η_{+} - 1} + \frac{2 \sqrt{η_{+} (2 η_{-} - 1)}}{2 η_{+} - 1}) . \end{matrix}$

Notice that, by Theorem (1) (see the following Corollary 1) one obtains the optimal dimension range $p + \frac{4 p}{p - 1}$ for any usual super-linear function $h (t)$ without using the extra convexity condition. Also note that the extra condition that $\frac{\tilde{f} (t)}{t^{p - 1}}$ to be bounded near zero always holds automatically for all $1 < p ⩽ 2$ (so we need it just for the case $p > 2$ ).
Corollary 1.
Let $u^{}$ be the function defined in (* 5 ) and f satisfy ( 2 ). We have
If $2 η_{-} - η_{+} = 1$ or $η_{-} = η_{+} = 1$ then $u^{}$ is regular whenever* $N < p + \frac{4 p}{p - 1}$ .

If $η_{+} = η_{-} : = η$ , $\frac{1}{2} < η < 1$ and there exists $M > 0$ such that $\frac{\tilde{f} (t)}{t^{p - 1}} < M$ for small t, then $u^{}$ is bounded whenever:* $\begin{matrix} (10) & N < \frac{p}{p - 1} (1 + \frac{p η}{2 η - 1} + 2 \sqrt{\frac{η}{2 η - 1}}) . \end{matrix}$

It is not difficult to see that, by the assumption $\frac{1}{2} < η < 1$ , the right hand side of (10) is always larger than $p + \frac{4 p}{p - 1}$ which is the optimal regularity dimension for $u^{}$ .

Now, to illustrate the above theorem and corollary, we present some examples. Example 1.
Consider Problem (1) with the well-known convex non-linearity $f (t) = e^{u}$ in an arbitrary smooth bounded domain Ω. Here we have $η_{-} = η_{+} = 1$ , therefore, by (8) in Theorem 1, $u^{}$ is bounded whenever $\begin{matrix} N < p + \frac{4 p}{p - 1} \end{matrix}$ which is the optimal regularity dimension for this problem.

Moreover, one can take the other well-known convex non-linearity $f (u) = {(1 + u)}^{m}$ where $m > p - 1$ in Problem (1) for $1 < p ⩽ 2$ . By a simple computation we have $η_{-} = η_{+} = \frac{m}{m + (p - 1)} < 1$ so by (9) in Theorem 1 $u^{}$ defined as in (5) is regular whenever $N < G (m, p)$ , where $G (m, p)$ defined in (6).
Example 2.
Consider Problem (1) with $\begin{matrix} f (u) = {(u^{2} + 3 u + 3 - 3 sin u)}^{p - 1}, 1 < p ⩽ 3 \end{matrix}$ in an arbitrary smooth bounded domain Ω. The function f satisfies (2) but $h (t) = {f (t)}^{\frac{1}{p - 1}}$ is not convex even at infinity (note that we have $h^{″} (u) = 2 + 3 sin u$ , which is negative for all u such that $sin u < - \frac{2}{3}$ ). So none of the previous regularity results apply in this case. However, its easy to see that $\frac{1}{2} < η_{+} = η_{-} = \frac{2}{3} < 1$ and $\frac{\tilde{f} (u)}{u^{p - 1}}$ is bounded near zero, hence by (9) in Theorem 1 or (10) in Corollary (1), $u^{}$ is regular whenever $\begin{matrix} N < \frac{p}{p - 1} (2 p + 1 + 2 \sqrt{2}) . \end{matrix}$
Example 3.
As another example, consider Problem (1) with $\begin{matrix} f (u) = e^{(p - 1) u} {(15 + 8 sin u)}^{p - 1}, 1 < p ⩽ 2 \end{matrix}$ in an arbitrary smooth bounded domain Ω. The function $f (t)$ satisfies (2) but is not convex. So we can not apply the previous regularity results. However, we have $\begin{matrix} \frac{h^{'} (t) H (t)}{h {(t)}^{2}} = \frac{(15 + 8 cos t + 8 sin t) (15 + 4 sin t - 4 cos t)}{{(15 + 8 sin t)}^{2}} : = η (u) \end{matrix}$ which is a periodic function with period $2 π$ . By Mathematica we can compute that $\begin{matrix} η_{-} = min_{[0, 2 π]} η (u) \approx 0.547593 and η_{+} = max_{[0, 2 π]} η (u) \approx 1.80247, \end{matrix}$ Now by (8) in Theorem 1, $u^{*} \in L^{\infty} (Ω)$ whenever $\begin{matrix} N < \frac{p}{p - 1} (p + \frac{0.095186}{1.80247} + 2 \sqrt{\frac{0.095186}{1.80247}}) \approx \frac{p}{p - 1} (p + 0.5124) . \end{matrix}$

3. Preliminaries and proof of main results

In this section we prove our main results.

We first recall some basic regularity results. Consider the problem $\begin{matrix} (11) & \{\begin{matrix} - Δ_{p} u = g (x), & in Ω, \\ u = 0, & on \partial Ω, \end{matrix} \end{matrix}$ where $g \in L^{q} (Ω)$ for some $q ⩾ 1$ . The following result can be found in [3] and [23].

Lemma 1.

Assume that $g \in L^{q} (Ω)$ for some $q ⩾ 1$ , and that u is a solution of ( 11 ). The following assertions hold:

If $q > \frac{N}{p}$ then $u \in L^{\infty} (Ω)$ . Moreover $‖ u ‖_{\infty} ⩽ C ‖ g ‖_{q}^{\frac{1}{p - 1}}$ , where C is a constant depending only on N, p, q, and $| Ω |$ .

If $q = \frac{N}{p}$ then $u \in L^{r} (Ω)$ for all $r \in [1, + \infty)$ . Moreover $‖ u ‖_{r} ⩽ C ‖ g ‖_{q}^{\frac{1}{p - 1}}$ , where C is a constant depending only on N, p, q, and $| Ω |$ .

If $1 ⩽ q < \frac{N}{p}$ then $u \in L^{r} (Ω)$ for all $r \in (0, r_{1})$ , where $r_{1} = \frac{N q (p - 1)}{(N - p q)}$ . Moreover $‖ | u |^{r} ‖_{r}^{1 / r} ⩽ C ‖ g ‖_{q}^{\frac{1}{p - 1}}$ , where C is a constant depending only on N, p, q, and $| Ω |$ .

We also need the following standard regularity result which is a consequence of Theorem 3 in [33].

Proposition 1.

Let $u \in W_{0}^{1, p} (Ω)$ be a weak energy solution of the following problem $\begin{matrix} \{\begin{matrix} Δ_{p} u + c (x) u^{p - 1} = g (x), & in Ω, \\ u = 0, & on \partial Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{N}$ is a smooth bounded domain, $c, g \in L^{q} (Ω)$ for some $q ⩾ 1$ , and $p > 1$ . Then there exists a positive constant C independent of u such that $\begin{matrix} ‖ u ‖_{L^{\infty} (Ω)} ⩽ C (‖ u ‖_{L^{1} (Ω)} + ‖ g ‖_{L^{q} (Ω)}) if q > \frac{N}{p} . \end{matrix}$

By the above proposition, we obtain the following result:

Proposition 2.

Let f satisfy ( 2 ), $u_{λ}$ be the non-negative minimal solution of Problem ( 1 ). If $\frac{\tilde{f} (t)}{t^{p - 1}}$ is bounded near zero and there exists a positive constant C independent of λ such that $\begin{matrix} (12) & ‖ u_{λ} ‖_{L^{1} (Ω)} ⩽ C and {‖ \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{σ (p - 1)}} ‖}_{L^{1} (Ω)} ⩽ C, for some 0 ⩽ σ ⩽ α (α ⩾ 1) \end{matrix}$ then $\begin{matrix} (13) & {‖ u^{*} ‖}_{L^{\infty} (Ω)} ⩽ C for N < α p . \end{matrix}$

Proof.

By the assumption $\frac{\tilde{f} (t)}{t^{p - 1}}$ is bounded near zero so one can take $M : = {sup}_{0 < t < ⩽ 1} \frac{{\tilde{f} (t)}^{α}}{t^{α (p - 1)}} < \infty$ and obtain that $\begin{matrix} \int_{Ω} \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{α (p - 1)}} d x = \int_{u_{λ} ⩽ 1} \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{α (p - 1)}} d x + \int_{u_{λ} > 1} \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{σ (p - 1)}} {u_{λ}}^{(σ - α) (p - 1)} d x ⩽ M | Ω | + {‖ \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{σ (p - 1)}} ‖}_{L^{1} (Ω)} . \end{matrix}$ Then from (12) we get $‖ \frac{{\tilde{f} (u_{λ})}^{α}}{{u_{λ}}^{α (p - 1)}} ‖_{L^{1} (Ω)} ⩽ C$ or equivalently $‖ \frac{\tilde{f} (u_{λ})}{{u_{λ}}^{(p - 1)}} ‖_{L^{α} (Ω)} ⩽ C$ , where C is a constant independent of λ. Now similar to the proof of Corollary 2.2 in [10] we rewrite Problem (1) as $- Δ_{p} u_{λ} - c (x) {u_{λ}}^{p - 1} = - f (0)$ where $c (x) : = \frac{\tilde{f} (u_{λ})}{{u_{λ}}^{p - 1}}$ , hence Proposition 1 gives the desired result. □

Now, we prove the following lemma and proposition which are crucial for the proof of our main results.

Lemma 2.

Let f be a general non-linearity satisfy ( 2 ), $u^{*}$ be defined in ( 5 ) and $g : [0, \infty) ⟶ [0, \infty)$ a $C^{1}$ function such that $g (0) = 0$ and satisfies: $\begin{matrix} (14) & H (s) : = {h (s)}^{p - 2} (h^{'} (s) {g (s)}^{2} - h (s) G (s)) ⩾ 0, for s sufficiently large, \end{matrix}$ where $G (s) : = \int_{0}^{s} {g^{'} (t)}^{2} d t$ , and $h (t)$ defined as before. Then $‖ H (u_{λ}) ‖_{L^{1} (Ω)} ⩽ C$ , where C is a constant independent of λ.

Proof.

Let $u_{λ}$ be the minimal semi-stable solution of Problem (1). Taking $ϕ = g (u_{λ})$ as a test function in the semi-stability inequality (4), we get $\begin{matrix} (15) & λ \int_{Ω} {h (u_{λ})}^{p - 2} h^{'} (u_{λ}) {g (u_{λ})}^{2} d x ⩽ \int_{Ω} | \nabla u_{λ} |^{p} {g^{'} (u_{λ})}^{2} d x . \end{matrix}$ Now, by using the integration by part formula, we compute $\begin{array}{l} \int_{Ω} | \nabla u_{λ} |^{p} {g^{'} (u_{λ})}^{2} d x & = \int_{Ω} | \nabla u_{λ} |^{p - 2} \nabla u_{λ} . \nabla u u_{λ} {g^{'} (u_{λ})}^{2} d x = \int_{Ω} | \nabla u_{λ} |^{p - 2} \nabla u_{λ} . \nabla G d x \\ = \int_{Ω} - div (| \nabla u_{λ} |^{p - 2} \nabla u_{λ}) G (u_{λ}) d x = λ \int_{Ω} {h (u_{λ})}^{p - 1} G (u_{λ}) d x, \end{array}$ and using the above equality in (15) we obtain that $\begin{matrix} \int_{Ω} {h (u_{λ})}^{p - 2} (h^{'} (u_{λ}) {g (u_{λ})}^{2} - h (u_{λ}) G (u_{λ})) d x ⩽ 0, \end{matrix}$ so $\begin{matrix} (16) & \int_{Ω} H (u_{λ}) d x ⩽ 0 . \end{matrix}$ From (14) there is an $M_{0} > 0$ such that $H (s) ⩾ 0$ for $s ⩾ M_{0}$ and hence using (16) we get: $\begin{array}{l} \int_{Ω} | H (u_{λ}) | d x & = \int_{u ⩽ M_{0}} | H (u_{λ}) | d x + \int_{u_{λ} ⩾ M_{0}} H (u_{λ}) d x \\ ⩽ \int_{u_{λ} ⩽ M_{0}} (| H (u_{λ}) | - H (u_{λ})) d x ⩽ C_{0} | Ω |, \end{array}$ where $| Ω |$ denotes the Lebesgue measure of Ω and $C_{0} : = {sup}_{s \in [0, M_{0}]} (| H (s) | - H (s))$ . Note that we have indeed $H (s) = \frac{f^{'} (s)}{p - 1} {g (s)}^{2} - h {(s)}^{p - 1} G (s)$ which is continuous in $[0, \infty)$ for $p > 1$ , hence $C_{0} < \infty$ is independent of λ that proves our claim. □

Proposition 3.

Let $u_{λ}$ be the minimal solution of ( 1 ) and $ζ : [0, \infty) ⟶ [0, \infty)$ be a $C^{1}$ function such that for some $t_{0} > 0$ we have $ζ (t) ⩽ \frac{h^{'} (t)}{h (t)}$ , $0 ⩽ ζ^{'} (t) + {ζ (t)}^{2}$ for $t ⩾ t_{0}$ and $\frac{E (t)}{h {(t)}^{p - 1}} ⟶ \infty$ as $t ⟶ \infty$ , where $\begin{matrix} (17) & E (t) : = {h (t)}^{p - 1} (\frac{h^{'} (t)}{h (t)} - ζ (t)) e^{2 \int_{t_{0}}^{t} ζ (s) + \sqrt{ζ^{'} (s) + {ζ (s)}^{2}} d s} . \end{matrix}$ Then $‖ E (u_{λ}) ‖_{L^{1} (Ω)} ⩽ C$ , where C is a constant independent of λ.

Proof.

Let $g : [0, \infty) \to [0, \infty)$ be a $C^{1}$ function with $g (0) = 0$ and $\begin{matrix} g (t) = e^{\int_{t_{0}}^{t} (ζ (s) + \sqrt{ζ^{'} (s) + ζ {(s)}^{2}}) d s}, for t ⩾ t_{0} . \end{matrix}$ Also, let $G (t) = \int_{0}^{t} g^{'} {(s)}^{2} d s$ as in Lemma 2. Then using the equality $\begin{matrix} g^{'} (t) = (ζ (t) + \sqrt{ζ^{'} (t) + ζ {(t)}^{2}}) g (t) for t ⩾ t_{0}, \end{matrix}$ we get $\begin{array}{l} \frac{d}{d t} (ζ (t) {g (t)}^{2} - G (t)) & = ζ^{'} (t) {g (t)}^{2} + 2 ζ (t) g (t) g^{'} (t) - {g^{'} (t)}^{2} \\ = g {(t)}^{2} (ζ^{'} (t) + 2 {ζ (t)}^{2} + 2 ζ (t) \sqrt{ζ^{'} (t) + {ζ (t)}^{2}}) - {g^{'} (t)}^{2} \\ = g {(t)}^{2} {(ζ (s) + \sqrt{ζ^{'} (t) + ζ {(t)}^{2}})}^{2} - {g^{'} (t)}^{2} \\ = {g^{'} (t)}^{2} - {g^{'} (t)}^{2} = 0, for t ⩾ t_{0}, \end{array}$ which implies that $\begin{matrix} (18) & G (t) = ζ (t) g {(t)}^{2} + C_{0}, where C_{0} : = G (t_{0}) - ζ (t_{0}) . \end{matrix}$ Now using (18), for $t ⩾ t_{0}$ and the definition of $H (t)$ in (14) we have $\begin{matrix} H (t) = {h (t)}^{p - 2} g {(t)}^{2} h^{'} (t) - {g (t)}^{2} ζ (t) {h (t)}^{p - 1} - C_{0} {h (t)}^{p - 1} = E (t) - C_{0} {h (t)}^{p - 1}, \end{matrix}$ which is positive for large $t ⩾ t_{0}$ (by the assumption), so by Lemma 2 we get $‖ H (u_{λ}) ‖_{L^{1} (Ω)} ⩽ C$ , where C is a constant independent of λ. Then, by the assumption that $\frac{E (t)}{h {(t)}^{p - 1}} ⟶ \infty$ as $t ⟶ \infty$ , we have $0 < E (t) < 2 H (t)$ for large t that also gives $‖ E (u_{λ}) ‖_{L^{1} (Ω)} ⩽ C_{1}$ , where $C_{1}$ is a constant independent of λ, which is the desired result. □

Proof of Theorem 1.

Take arbitrary $η_{1}$ , $η_{2}$ and $η_{3}$ such that $\frac{1}{2} < η_{1} < η_{2} < η_{-} ⩽ η_{+} < η_{3} < \infty$ . By the definition of $η_{-}$ , $η_{+}$ there exists a $t_{0} > 0$ so that for $t ⩾ t_{0}$ $\begin{matrix} (19) & η_{1} < η_{2} < \frac{h^{'} (t) H (t)}{{h (t)}^{2}} < η_{3} . \end{matrix}$ Let $η : [0, \infty] \to [0, \infty]$ be a $C^{1}$ function such that $ζ (t) = η_{1} \frac{h (t)}{H (t)}$ for $t ⩾ t_{0}$ , where $H (t) = \int_{0}^{t} h (s) d s$ . From (19) there exists a $t_{0} > 0$ so that for $t ⩾ t_{0}$ we have $ζ (t) ⩽ \frac{h^{'} (t)}{h (t)}$ and $\begin{matrix} (20) & ζ^{'} (t) + {ζ (t)}^{2} = η_{1} (\frac{h^{'} (t)}{H (t)} - (1 - η_{1}) \frac{{h (t)}^{2}}{{H (t)}^{2}}) ⩾ (2 η_{1} - 1) \frac{h^{'} (t)}{H (t)} ⩾ \frac{2 η_{1} - 1}{η_{3}} \frac{{h^{'} (t)}^{2}}{{h (t)}^{2}} . \end{matrix}$ Note that, in (20) we used the fact that $η_{1} < 1$ . Also, from (19) there exists a $t_{0} > 0$ so that for $t ⩾ t_{0}$ we have $\begin{matrix} \frac{h^{'} (t)}{h (t)} - ζ (t) ⩾ (η_{2} - η_{1}) \frac{h (t)}{H (t)} . \end{matrix}$ Now let the function $E (t)$ be given as in (17) in Proposition 3. By the above inequality, (19), (20) and the fact that $\int_{t_{0}}^{t} ζ (s) d s = η_{1} (ln H (t) - ln H (t_{0}))$ , we have $\begin{matrix} (21) & E (t) = {h (t)}^{p - 1} (\frac{h^{'} (t)}{h (t)} - ζ (t)) e^{2 \int_{t_{0}}^{t} (ζ (s) + \sqrt{ζ^{'} (s) + {ζ (s)}^{2}}) d s} ⩾ C H {(t)}^{2 η_{1} - 1} h {(t)}^{p + 2 \sqrt{\frac{2 η_{1} - 1}{η_{3}}}}, \end{matrix}$ for $t ⩾ t_{0}$ , where C is a positive constant depends only on h. Now, writing inequality (19) as $\frac{h^{'} (t)}{h (t)} < η_{3} \frac{h (t)}{H (t)}$ for $t_{0} > 0$ , then integration from $t_{0}$ to t gives $\begin{matrix} (22) & H (t) ⩾ C h {(t)}^{\frac{1}{η_{3}}} for t ⩾ t_{0} . \end{matrix}$ Using the above inequality in (21) we get $\begin{matrix} (23) & E (t) ⩾ h {(t)}^{γ}, where γ : = p + \frac{2 η_{1} - 1}{η_{3}} + 2 \sqrt{\frac{2 η_{1} - 1}{η_{3}}}, for t ⩾ t_{0} . \end{matrix}$ Since $\frac{E (t)}{h {(t)}^{p - 1}} \to \infty$ as $t \to \infty$ , then from Proposition 3 we get $‖ E (u_{λ}) ‖_{L^{1} (Ω)} ⩽ C$ , where C is a constant independent of λ. Then from (23), $‖ h (u_{λ}) ‖_{L^{γ} (Ω)} ⩽ C$ with C independent of λ, and since $u^{*}$ is the increasing limit of $u_{λ}$ we get $h (u^{*}) \in L^{γ} (Ω)$ or equivalently ${f (u^{*})}^{\frac{1}{p - 1}} \in L^{γ} (Ω)$ . Now the first part of Lemma 1 gives $u^{*} \in L^{\infty} (Ω)$ for $n < \frac{p}{p - 1} γ$ , and since $η_{1}$ , $η_{2}$ and $η_{3}$ were arbitrary in the intervals $(\frac{1}{2}, η_{-})$ and $(η_{+}, \infty)$ , respectively, thus $u^{*} \in L^{\infty} (Ω)$ for $\begin{matrix} (24) & N < \frac{p}{p - 1} (p + \frac{2 η_{-} - 1}{η_{+}} + 2 \sqrt{\frac{2 η_{-} - 1}{η_{+}}}), \end{matrix}$ that proves the first part.

Now, we prove the second part of the theorem. First, from $η_{+} < 1$ we can assume that $η_{3} < 1$ and from (22) there exists a $t_{0} > 0$ so that for $t ⩾ t_{0}$ we have $\begin{matrix} h (t) {H (t)}^{- η_{3}} ⩽ C, for t ⩾ t_{0} . \end{matrix}$ Now, by integration from $t_{0}$ to t of the above inequality, for some $t_{1} ⩾ t_{0}$ we get $\begin{matrix} (25) & H (t) ⩽ C t^{\frac{1}{1 - η_{3}}}, for t ⩾ t_{1} \end{matrix}$ Then (22) and (25) imply that $h (t) ⩽ C t^{\frac{η_{3}}{1 - η_{3}}}$ , for $t ⩾ t_{1}$ . So, for some $t_{2} ⩾ t_{1}$ we have $\begin{matrix} {h (t)}^{γ} ⩾ C \frac{{h (t)}^{γ_{1}}}{t^{γ_{1}}} = C \frac{{f (t)}^{\frac{γ_{1}}{p - 1}}}{t^{γ_{1}}} for t ⩾ t_{2}, where γ_{1} : = \frac{η_{3}}{2 η_{3} - 1} γ, \end{matrix}$ where γ is given in (23). Hence, by the boundedness of $\frac{\tilde{f} (t)}{t^{p - 1}}$ near zero, we conclude that $‖ \frac{{\tilde{f} (u^{*})}^{\frac{γ_{1}}{p - 1}}}{{(u^{*})}^{γ_{1}}} ‖_{L^{1} (Ω)} ⩽ C$ with C independent of λ. Now Proposition 2 with $α = σ = \frac{γ_{1}}{p - 1}$ gives $u^{*} \in L^{\infty} (Ω)$ for $N < \frac{p}{p - 1} γ_{1}$ . Since $η_{1}$ , $η_{2}$ and $η_{3}$ were arbitrary in the intervals $(\frac{1}{2}, η_{-})$ and $(η_{+}, 1)$ , respectively, we then conclude that $u^{*} \in L^{\infty} (Ω)$ whenever $\begin{matrix} N < \frac{p}{p - 1} (1 + \frac{(p - 2) η_{+} + 2 η_{-}}{2 η_{+} - 1} + \frac{2 \sqrt{η_{+} (2 η_{-} - 1)}}{2 η_{+} - 1}) . \end{matrix}$ which is the desired result. □

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