Equivalence Between Uniform $L^{p^*}$ A Priori Bounds and Uniform $L^{\infty }$ A Priori Bounds for Subcritical p-Laplacian Equations

Abstract

We establish sufficient conditions for a uniform $L^{p^\star }(\Omega )$ bound to imply a uniform $L^\infty (\Omega )$ bound for positive weak solutions of subcritical p-Laplacian equations. We also provide an equivalent result for sequences of boundary-value problems. As consequences, we prove that any set of solutions with finite energy is $L^\infty (\Omega )$ a priori bounded, and also obtain an alternative proof of the existence of a priori bounds for subcritical power like nonlinearities.

Appendix

In this section, we recall all the auxiliary results that are used to prove Corollary 4.1 on existence of a priori $L^\infty (\Omega )$ bounds for power like nonlinearities, which include monotonicity of the solutions and moving planes method, first eigenvalue of the p-Laplace operator, Picone’s identity and inequality, and Pohozaev’s identity.

1.1 Monotonicity of the Solutions and Moving Planes Method

Let us recall some results on the monotonicity of solutions of the p-Laplace equations. We consider the following problem:

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} { - \Delta _{p} (u) = f(u),} &{} {\qquad {\text {in}}\, \Omega } \\ {u > 0,} &{} {\qquad {\text {in}}\, \Omega } \\ { u = 0,} &{} {\qquad {\text {on}}\, \partial \Omega ,} \\ \end{array} } \right. \end{aligned}$$

(6.1)

where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^N$, $N\geqslant 2$, $1<p<\infty$, and we have the following hypotheses on f:

• (*) $f:[0,\infty )\rightarrow {\mathbb {R}}$ is a continuous function which is locally Lipschitz continuous in $(0, \infty )$.

The results that we are going to recall can be briefly rephrased saying that all the conclusions of the moving planes method of Gidas, Ni, and Nirenberg’s Theorem (see [25]) hold for the p-Laplacian, assuming that f is locally Lipschitz continuous, and if $p>2$, assuming moreover that $f(s)>0$ if $s>0$ (see [13,14,15, 17, 18]). Concerning weak and strong comparison principles for the p-Laplacian, see, e.g., [12, 13, 17, 18, 27, 36, 41].

To state more precisely some known result about the monotonicity and symmetry of solutions of p-Laplace equations, we introduce the following notations.

Let $\nu$ be a direction in ${\mathbb {R}}^N$. For a real number $\lambda$, we define the moving plane:

$$\begin{aligned} T^{\nu }_{\lambda }=\{x \in {\mathbb {R}} : x\cdot \nu = \lambda \}, \end{aligned}$$

(6.2)

the cap:

$$\begin{aligned} \Omega ^{\nu }_{\lambda }= \{x \in \Omega :x \cdot \nu < \lambda \}, \end{aligned}$$

(6.3)

the reflected point:

$$\begin{aligned} x^{\nu }_{\lambda }=R^{\nu }_{\lambda }(x)=x+2(\lambda - x \cdot \nu )\nu , \quad \quad x \in {\mathbb {R}}^N, \end{aligned}$$

(6.4)

and

$$\begin{aligned} a(\nu )=\inf _{x\in \Omega }x\cdot \nu . \end{aligned}$$

(6.5)

If $\lambda > a(\nu )$, then $\Omega ^{\nu }_{\lambda }$ is nonempty, and thus, we set the reflected cap:

$$\begin{aligned} (\Omega ^{\nu }_{\lambda })'= R^{\nu }_{\lambda }(\Omega ^{\nu }_{\lambda }). \end{aligned}$$

(6.6)

Following [25, 39], we observe that for $\lambda - a(\nu )$ small, then $(\Omega ^{\nu }_{\lambda })'$ is contained in $\Omega$ and will remain in it, at least until one of the following occurs:

• (iv) $(\Omega ^{\nu }_{\lambda })'$ becomes internally tangent to $\partial \Omega$.

• (ii) $\ T^{\nu }_{\lambda }$ is orthogonal to $\partial \Omega$.

Let $\Lambda _1 (\nu )$ be the set of those $\lambda >a(\nu )$, such that for each $\mu < \lambda$, none of the conditions (iv) and (ii) holds and defines:

$$\begin{aligned} \lambda _{1}(\nu )= \sup \, \Lambda _1 (\nu ). \end{aligned}$$

(6.7)

Moreover, let:

$$\begin{aligned} \Lambda _2(\nu )=\{\lambda > a(\nu ): (\Omega ^{\nu }_\mu )'\subseteq \Omega \quad \forall \mu \in (a(\nu ),\lambda ]\}, \end{aligned}$$

(6.8)

and

$$\begin{aligned} \lambda _2 (\nu )=\sup \,\Lambda _2(\nu ). \end{aligned}$$

(6.9)

Since $\Omega$ is assumed to be smooth, note that neither $\Lambda _1(\nu )$ nor $\Lambda _2(\nu )$ are empty, and $\Lambda _1(\nu ) \subseteq \Lambda _2(\nu )$, so that $\lambda _1 (\nu ) \leqslant \lambda _2 (\nu )$ (in the terminology of [25], $\Omega ^{\nu }_{\lambda _1 (\nu )}$ and $\Omega ^{\nu }_{\lambda _2 (\nu )}$ correspond to the ‘maximal cap’, and the ‘optimal cap’, respectively). Finally, define:

$$\begin{aligned} \Lambda _0(\nu )=\{\lambda > a(\nu ): u\leqslant u^{\nu }_{\lambda }\quad \forall \mu \in (a(\nu ),\lambda ]\}, \end{aligned}$$

(6.10)

and

$$\begin{aligned} \lambda _0 (\nu )=\sup \,\Lambda _0(\nu ). \end{aligned}$$

(6.11)

Theorem 6.1

Let $\Omega$ be a bounded smooth domain in ${\mathbb {R}}^N$, $N\geqslant 2$, $1<p<\infty$, $f:[0,\infty )\rightarrow {\mathbb {R}}$, a continuous function which is locally Lipschitz continuous in $(0, \infty )$ and strictly positive in $(0, \infty )$ if $p>2$. Let $u\in C^1({\overline{\Omega }})$ be a weak solution of (6.1).

For any direction $\nu$ and for $\lambda$ in the interval $(a(\nu ),\lambda _1 (\nu )]$, we have:

$$\begin{aligned} u(x)\leqslant u(x^{\nu }_{\lambda })\quad \forall x \in \Omega ^{\nu }_{\lambda }. \end{aligned}$$

(6.12)

If f is locally Lipschitz continuous in the closed interval $[0, \infty )$, then (6.12) holds for any $\lambda$ in the interval $(a(\nu ),\lambda _2 (\nu ))$.

Corollary 6.2

If f is locally Lipschitz continuous in the closed interval $[0, \infty )$ and strictly positive in $(0, \infty )$, and the domain $\Omega$ is convex with respect to a direction $\nu$ and symmetric with respect to the hyperplane $T_{0}^{\nu }= \left\{ x \in \mathbb {R}^{N} : x \cdot \nu =0 \right\}$, then u is symmetric, i.e., $u(x)=u(x_0^{\nu })$, and nondecreasing in the $\nu$-direction in $\Omega _0 ^{\nu }$. In particular, if $\Omega$ is a ball, then u is radially symmetric and radially decreasing.

Remark 6.3

In this paper, we assume f positive for $p>2$ only, because we are going to exploit the monotonicity results stated in the previous theorem, obtained in [14, 15, 17, 18], and in these papers the positivity of f is assumed when $p>2$.

In any case, in all the results that we prove, we always assume that f satisfies (A).

As a consequence of the previous theorem, solutions are monotone increasing from the points on the boundary along directions that belong to a neighborhood of directions close to the inner boundary.

As a further consequence, we have the following property, as observed in [19] for $p=2$, which can be deduced by contradiction using the strict convexity of the domain and the monotonicity of the solutions provided by the previous cited papers (see [3] for a related geometric discussion).

Lemma 6.4

Let $\Omega$ be a strictly convex bounded smooth domain, and define $\Omega _{\delta }= \{ x \in \Omega : \text { dist }(x , \partial \Omega ) > \delta \}$ for $\delta >0$.

Then, the following holds for a weak solution $u \in C^1 ({\overline{\Omega }})$ of the problem (1.1), where f satisfies the condition (A):

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} {\exists \;\gamma ,\varepsilon > 0,\,{\text {depending\,only\,on}}\,\Omega , {\text {such\, that}}} \\ {\forall \;x \in \Omega {\setminus }\Omega _{\varepsilon }\, {\text {there\,is\,a\,part\,of\,a\,cone\,I}}_{{\text {x}}} {\text { with}}} \\ {(i)\quad u(\xi ) \ge u(x)\;\quad \forall \;\xi \in I_{x} ,} \\ {(ii)\quad I_{x} \subset \Omega _{{\frac{\varepsilon }{2}}} ,} \\ {(iii)\quad {\text {meas }}(I_{x} ) \ge \gamma .} \\ \end{array} } \right. \end{aligned}$$

(6.13)

Geometrically, $I_x$ is a part of a cone $K_x$ with vertex in x, where all the $K_x$ are congruent to a fixed cone K, and if $x \in \Omega {\setminus }\Omega _{ \frac{\varepsilon }{2}}$, then $I_x=K_x\cap \Omega _{ \frac{\varepsilon }{2}}$.

Let us emphasize that $\varepsilon$ and $\gamma$ depend only on the geometry of the strictly convex, bounded, smooth domain $\Omega$.

We will use this conditions to get $L^{\infty }$ a priori bounds in a neighborhood of the boundary, for the solutions on a strictly convex, bounded, smooth domain $\Omega$.

1.2 First Eigenvalue and Eigenfunction

Let $\Omega$ be a bounded domain and $1<p< \infty$. A real number $\lambda$ is a (nonlinear) eigenvalue of the p-Laplacian, with associated eigenfunction u if $u \in W_0^{1,p}(\Omega )$, $u \not \equiv 0$, solves the equation $- \Delta _p u = \lambda \, |u|^{p-2}u \text { in } \Omega$.

Although the general theory of eigenvalues for the p-Laplacian is far from complete (see [24] and the survey [35]), the properties of the first eigenvalue are known and are the same as in the case $p=2$. Namely, the following result holds (see [2, 31, 35]).

Theorem 6.5

Let us define:

$$\begin{aligned} \lambda _1 = \lambda _1 (p, \Omega ) := \inf _{v \in W_0^{1,p}(\Omega )} \left\{ \int _{\Omega } |Dv|^p \, dx \, : \, \int _{\Omega } |v|^p \, dx =1 \right\} . \end{aligned}$$

(6.14)

Then, $\lambda _1$ is the first eigenvalue, it is simple, and it is isolated.

Moreover, a first eigenfunction does not change sign in $\Omega$, and by the strong maximum principle, it is in fact either strictly positive or strictly negative in $\Omega$. Therefore, we can select a unique eigenfunction $\phi _1$, such that $\int _{\Omega } |\phi _1|^p \, dx =1$ and $\phi _1 >0$ in $\Omega$.

1.3 Picone’s Identity and Inequality

The following extension of the Picone’s identity for the p-Laplacian has been proved by Allegretto and Huang, see [1].

Theorem 6.6

Let $v_1,v_2 \ge 0$ be differentiable functions in an open set $\Omega$, with $v_2 >0$ and $p>1$. Put:

$$\begin{aligned} L(v_1, v_2)= & {} |\nabla v_1 |^p +(p-1) \frac{v_1^p}{v_2^p} |\nabla v_2 |^p - p\frac{v_1^{p-1}}{v_2^{p-1}} \nabla v_1 \cdot |\nabla v_2 |^{p-2} \nabla v_2, \\ R(v_1, v_2)= & {} |\nabla v_1 |^p - \nabla \left( \, \frac{v_1^p}{v_2^{p-1}}\, \right) \cdot |\nabla v_2 |^{p-2} \nabla v_2. \end{aligned}$$

Then, $R(v_1, v_2)=L(v_1, v_2)$ and $L(v_1, v_2) \ge 0$.

As a consequence, we have:

$$\begin{aligned} \nabla \left( \, \frac{v_1^p}{v_2^{p-1}}\, \right) \cdot |\nabla v_2 |^{p-2} \nabla v_2 \le |\nabla v_1 |^p. \end{aligned}$$

(6.15)

1.4 Pohozaev’s Identity for the p-Laplacian

The following extension of the Pohozaev’s identity for the p-Laplacian has been proved by Xu (see [37, Theorem 4.1]; see also [27] for $f=f(u)$).

Theorem 6.7

Let $u \in W_0^{1,p}(\Omega ) \cap L^{\infty } ( \Omega )$ be a weak solution of the problem:

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} { - \Delta _{p} (u) = f(x,u)} &{} {\quad {\text {in }} \Omega } \\ {u = 0} &{} {\quad {\text {on }} \partial \Omega ,} \\ \end{array} } \right. \end{aligned}$$

(6.16)

where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^N$, $N\geqslant 2$, $p>1$, and $f:[0,\infty )\rightarrow {\mathbb {R}}$ is a continuous function.

Let $F(x,t)= \int _0 ^t f(x,s) \, ds$, be a primitive of the function f. Then:*****

$$\begin{aligned}&\frac{N-p}{p} \int _{\Omega } \big [p^* F(x,u) - f(x,u)\, u\big ] \, dx + \sum _{i=1}^N \int _{\Omega } F_{x_i}(x,u) x_i \, dx\nonumber \\&\quad =\frac{p-1}{p} \int _{\partial \Omega }\left| \frac{\partial u}{\partial \nu }\right| ^p (x \cdot \nu ) \, ds, \end{aligned}$$

(6.17)

where $\nu$ is the unit exterior normal on $\partial \Omega$.

We will need also a local $W^{1, \infty }$ result at the boundary, namely the following.

Theorem 6.8

Let $\Omega$ be a smooth bounded domain in ${\mathbb {R}}^N$, $N \ge 2$, $g \in L^{(p^*)'}(\Omega )$, and let $u \in C^1 ({\overline{\Omega }})$ be a solution of the problem:

$$\begin{aligned} \left\{ {\begin{array}{*{20}l} { - \Delta _{p} (u) = f(x,u)} &{} {\quad {\text {in }}\Omega } \\ {u = 0} &{} {\quad {\text {on }}\partial \Omega ,} \\ \end{array} } \right. \end{aligned}$$

(6.18)

For $\delta >0$, let $\Omega _{\delta }= \{ x \in \Omega : \text { dist }(x ,\partial \Omega ) > \delta \}$, and assume that $u \; , \; g \in L^{\infty }(\Omega {\setminus }\Omega _{ \delta })$ with $\Vert g \Vert _ {L^{\infty } (\Omega {\setminus }\Omega _{\delta }) } \le M$, $\Vert u \Vert _ {L^{\infty } (\Omega {\setminus }\Omega _{\delta }) } \le M$.

Then, there exists a constant $C>0$ only depending on M and $\delta$, such that:

$$\begin{aligned} \Vert \nabla u \Vert _{L^{\infty } (\partial \Omega ) } \le C. \end{aligned}$$

This results follows from the global estimates by Lieberman ([30]) extending the local interior estimates by DiBenedetto ([20]). A direct simple proof can be read in [16].