A priori bounds for positive solutions of subcritical elliptic equations

Abstract

We provide a-priori $L^\infty$ bounds for positive solutions to a class of subcritical elliptic problems in bounded $C^2$ domains. Our analysis widens the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded.

Appendices

Appendix A: A-priori bounds in a neighborhood of the boundary: the convex case

In this Appendix, we collect some well known results on the moving planes method, see Proposition A.1. Next, we state results concerning a-priori bounds in a neighborhood of the boundary for the convex case: Theorem A.3. All those results are essentially well known, see [8], we include them here in order to clarify which hypotheses are used in the convex case and which in the non-convex case, see Theorem B.3 in Appendix B.

We will be moving planes in the $x_1$-direction to fix ideas. Let us first define some concepts and notations.

• The moving plane is defined in the following way: $T_\lambda := \{ x\in {\mathbb {R}}^N : x_1 = \lambda \} ,$

• the cap: $\varSigma _\lambda := \{ x=(x_1,x')\in {\mathbb {R}}\times {\mathbb {R}}^{N-1}\cap \varOmega \ :\ x_1 < \lambda \} ,$

• the reflected point: $x^\lambda := (2\lambda - x_1,x') ,$

• the reflected cap: $\varSigma '_\lambda := \{x^\lambda \ :\ x\in \varSigma _\lambda \},$ see Fig. 1a.

• the minimum value for $\lambda$ or starting value: $\lambda _0 := \min \{ x_1\ :\ x\in \overline{\varOmega }\},$

• the maximum value for $\lambda$: $\lambda ^\star := \max \{ \lambda \ :\ \varSigma '_{\mu } \subset \overline{\varOmega } \quad \text{ for } \text{ all }\quad \mu \le \lambda \},$

• the maximal cap: $\varSigma :=\varSigma _{\lambda ^\star }.$

Fig. 1

a A cap $\varSigma _\lambda$ and its reflected cap $\varSigma '_\lambda$ in the $e_1$ direction. b A cap $\varSigma _\lambda (-e_1)$ and its reflected cap $\varSigma '_\lambda (-e_1)$ (in the $-e_1$ direction). c A maximal cap $\varSigma (-e_1)$

We will need the moving plane method for a nonlinearity $f=f(x,u)$.

Proposition A.1

Suppose $u\in C^2(\overline{\varOmega })$ is a positive solution of

$$\begin{aligned} -\varDelta u=f(x,u), \quad \text{ in } \,\, \varOmega , \qquad u= 0, \quad \text{ on } \,\, \partial \varOmega . \end{aligned}$$

(A.1)

Assume $f=f(x,s)$ and its first derivative $f_s$ are continuous, for $(x,s)\in \overline{\varOmega }\times {\mathbb {R}}.$

Assume that

$$\begin{aligned} f(x^\lambda ,s)\ge f(x,s)\quad \text{ for } \text{ all }\quad x\in \varSigma ,\quad \text{ for } \text{ all }\quad s>0. \end{aligned}$$

(A.2)

Then for any $\lambda \in (\lambda _0,\lambda ^\star )$ the following conclusion holds

• (C)

  $$\begin{aligned} u (x) < u(x^\lambda ) \quad \text{ and }\quad \dfrac{\partial u\ \ }{\partial x_1}(x) > 0 \quad \text{ for } \text{ all }\quad x\in \varSigma _\lambda . \end{aligned}$$

Furthermore, if $\ \frac{\partial u\ \ }{\partial x_1}(x) = 0$ at some point in $\varOmega \cap T_{\lambda ^\star },$ then u is symmetric with respect to the plane $T_{\lambda ^\star },$ and $\varOmega =\varSigma \cup \varSigma '\cup (T_{\lambda ^\star } \cap \varOmega ).$

Proof

It is a Corollary of Theorem 2.1’ in [10]. $\square$

Remark A.2

Set $x_0\in \partial \varOmega \cap T_{\lambda _0},$ see Fig. 1a. Let us observe that by definition of $\lambda _0$, $T_{\lambda _0}$ is the tangent plane to the graph of the boundary at $x_0$, and the inward normal at $x_0,$ is $n_i(x_0)=e_1.$ The above Theorem says that the partial derivative following the direction given by the inward normal at the tangency point is strictly positive in the whole maximal cap. Consequently, there are no critical points in the maximal cap.

Now, we can apply the above result in any direction. First, let us fix the notation for a general $\nu \in {\mathbb {R}}^N$ with $|\nu |=1.$ We set

• the moving plane defined as: $T_\lambda (\nu )= \{ x\in {\mathbb {R}}^N\ :\ x\cdot \nu = \lambda \} ,$

• the cap: $\varSigma _\lambda (\nu )= \{ x\in \varOmega \ :\ x\cdot \nu < \lambda \} ,$

• the reflected point: $x^\lambda (\nu )= x+2(\lambda - x\cdot \nu )\nu ,$

• the reflected cap: $\varSigma '_\lambda (\nu )= \{x^\lambda \ :\ x\in \varSigma _\lambda (\nu )\},$ see Fig. 1b, for $\nu =-e_1,$

• the minimum value of $\lambda$: $\quad \lambda _0 (\nu )= \min \{ x\cdot \nu \ :\ x\in \overline{\varOmega } \},$

• the maximum value of $\lambda$: $\ \ \lambda ^\star (\nu )= \max \{ \lambda \, :\, \varSigma '_{\mu }(\nu ) \subset \overline{\varOmega } \ \ \text{ for } \text{ all } \mu \le \lambda \},$

• and the maximal cap: $\varSigma (\nu )=\varSigma _{\lambda ^\star (\nu )}(\nu ),$ see Fig. 1c, for $\nu =-e_1.$

Set $x_0\in \partial \varOmega \cap T_{\lambda _0}.$ The above Proposition says that the partial derivative following the direction given by the inward normal, $n_i(x_0)$, at the tangency point $x_0$, is strictly positive in the whole maximal cap $\varSigma =\varSigma (n_i(x_0))$, consequently the function $g(t):=u (x_0+t\, n_i(x_0))$ is non-decreasing for $t\in [0,t_0]$ for some $t_0=t_0(x_0)>0.$

Now consider a neighborhood of $x_0,$ denoted by $B_{\delta _0}(x_0).$ We can observe that for any $x\in \partial \varOmega \cap B_{\delta _0}(x_0)\cap \varSigma$, also the function $g(t):=u (x+t\, n_i(x_0))$ is non-decreasing for $t\in [0,t_0]$ for some $t_0=t_0(x_0,x)>0.$ By choosing points x such that $dist(x,T_{\lambda ^*} (n_i(x_0)))>\delta ,$ we see that the function $g(t):=u (x+t\, n_i(x_0))$ is non-decreasing for $t\in [0,\delta ]$ for any $x\in \partial \varOmega \cap \varSigma (n_i(x_0)): dist(x,T_{\lambda ^*} (n_i(x_0)))>\delta .$

Now, let us move to a different cap, in a neighborhood of $x_0$. We can apply this idea, to their corresponding maximal caps $\varSigma$, with their corresponding vectors $\nu$. Then, choosing points in the intersection of the maximal caps, such that $dist(x,T_\lambda (\nu ))>\delta ,$ also the function $g(t):=u (x+t\nu )$ is increasing for $t\in [0,\delta ]$.

From now, the arguments split into two ways, depending on the convexity of the domain. If $\varOmega$ is convex, we observe that, reasoning as in [8], any positive solution u is locally increasing in the maximal cap following directions close to the normal direction, which provides $L^\infty$ bounds locally in a neighborhood of the boundary. This is the statement of the following Theorem.

Theorem A.3

Assume that $\varOmega \subset {\mathbb {R}}^N$ is a bounded, convex domain with $C^{2}$ boundary. Assume that the nonlinearity f satisfies (H4).

If $u\in C^2(\overline{\varOmega })$ satisfies (1.1) and $u>0$ in $\varOmega$, then there exists a constant $\delta >0$ depending only on $\varOmega$ and not on f or u, and a constant C depending only on $\varOmega$ and f but not on u, such that

$$\begin{aligned} \max _{\varOmega {\setminus } \varOmega _{\delta }} u\le C \end{aligned}$$

(A.3)

where $\varOmega _{\delta }:=\{x\in \varOmega \ : \ d(x,\partial \varOmega )>\delta \}.$

Proof

See Step 2 in the proof of [8, Theorem 1.1]. $\square$

Appendix B: A-priori bounds in a neighborhood of the boundary: the general case

Finally, we treat the general case, applying the moving plane method on the Kelvin transform, see below for a precise definition. First, we fix regions where a Kelvin transform of the solution has no critical points, see Theorem B.2. This result will imply a priori bounds in a neighborhood of the boundary for any solution of the elliptic equation, see Theorem B.3.

Let us recall that every $C^2$ domain $\varOmega$ satisfies the following condition, known as the uniform exterior sphere condition,

• (P) there exists a $\rho >0$ such that for every $x\in \partial \varOmega$ there exists a ball $B=B_{\rho } (y)\subset {\mathbb {R}}^N{\setminus }\varOmega$ such that $\partial B\cap \partial \varOmega =x.$

Let $x_0\in \partial \varOmega ,$ and let $\overline{B}$ be the closure of a ball intersecting $\overline{\varOmega }$ only at the point $x_0.$ Let us suppose $x_0=(1,0,\ldots ,0),$ and B is the unit ball with center at the origin. The inversion mapping

$$\begin{aligned} x\rightarrow h(x)=\frac{x}{|x|^2}, \end{aligned}$$

(B.1)

is an homeomorphism from ${\mathbb {R}}^N{\setminus }\{0\}$ into itself; and observe that $h(h(x))=x$. We perform an inversion from $\varOmega$ into the unit ball B,  in terms of the inversion map $h\left| _\varOmega \right. ,$ see Fig. 2a.

Let $\widetilde{\varOmega }=h(\varOmega )$ denote the image through the inversion map into the ball B. For any $x_0\in \partial \varOmega$, let $\tilde{n}_i(x_0)$ be the normal inward at $x_0$ in the transformed domain $\widetilde{\varOmega },$ and let $\widetilde{\varSigma }= \widetilde{\varSigma } (\tilde{n}_i(x_0))$ be its maximal cap, see Fig. 2b.

Fig. 2

a The exterior tangent ball and the inversion of the boundary into the unit ball. b A maximal cap $\widetilde{\varSigma }$ in the transformed domain $h(\varOmega )$. c The set $h^{-1}(\widetilde{\varSigma })$ (i.e. the inverse image of the maximal cap $\widetilde{\varSigma }$) in the original domain $\varOmega$

The following Lemma B.1 states a well known geometrical result, for any boundary point of a $C^2$ domain, the maximal cap in the transformed domain is nonempty. This result could seem surprising in presence of highly oscillatory boundaries. For example, suppose the boundary of $\varOmega$ includes $\varGamma _2 =\{(x,f(x)):f(x):= 1+ x^5\sin (\frac{1}{x}), \ x\in [-0.01,0.01]\},$ to visualize the scale, see in Fig. 3b $\{(x,x^5\sin (\frac{1}{x}))$, $\ \ \ x\in [-0.01,0.01]\}$. Let $h(\varGamma _2)$ be the image through the inversion map into the unit ball B,  and let $\varGamma _3$ be the arc of the boundary $\partial B$ given by $\varGamma _3=\{(x,g(x)) : g(x):=\sqrt{1-x^2}, \ x\in [-0.01,0.01]\},$ see Fig. 3c. At this scale, the oscillations are not appreciable. We plot in Fig. 3d the derivative of the ‘vertical’ distance between the boundary $\varGamma _2$ and the ball, concretely we plot $f'(x)-g'(x)$ for $x\in [-0.01,0.01]$. We plot in Fig. 3e the second derivative of the ‘vertical’ distance between the boundary and the ball, which is $f''(x)-g''(x)$ for $x\in [-5\cdot 10^{-4},5\cdot 10^{-4}]$. Let us observe that this second derivative is strictly positive, and that $f''(0)-g''(0)=1.$ Consequently, the first derivative is strictly increasing, and therefore the ’vertical’ distance $f(x)-g(x)$ does not oscillate.

Moreover, let us consider the image through the inversion map of the straight line $y=1,$ i.e. $h(x,1) =h(\{(x,1),\ x\in [-0.01,0.01]\}) .$ In Fig. 3f and g we plot the second coordinate of the difference $h(\varGamma _2)-h(x,1).$ The oscillation phenomena is present here. In Fig. 3h we plot the second coordinate of the difference $h(\varGamma _2)-h(\partial B).$ This difference does not oscillate.

Fig. 3

a An inflection point at the boundary $\varGamma _1$ joint with the inversion $h(\varGamma ),$ and the unit circumference, b a degenerated critical point at the boundary $\varGamma _2$, c $\varGamma _2$ joint with its inversion into the unit ball, $h(\varGamma _2),$ and the arc of circumference, $\varGamma _3$, d $f'(x)-g'(x)$ for $x\in [-0.01,0.01]$, e $f''(x)-g''(x)$ for $x\in [-5\cdot 10^{-4},5\cdot 10^{-4}]$, f second coordinate of the difference $h(\varGamma _2 )-h(x,1)$ where h(x, 1) is the image of the straight line $y=1$, g a zoom of the same graphic and h second coordinate of the difference $h(\varGamma _2 )-h(\varGamma _3)$

In Fig. 3a we draw the inversion of the boundary into the unit ball at an inflexion point; more precisely we set $\varGamma _1: =\{(x,f(x)):f(x)= \frac{x^3}{2}+1,\ x\in [-\pi /4,\pi /4]\},$ which has an inflexion point at $x=0.$

The following Lemma states the local convexity of the transformed domain.

Lemma B.1

If $\varOmega \subset {\mathbb {R}}^N$ is a bounded domain with $C^{2}$ boundary, then for any $x_0\in \partial \varOmega ,$ there exists a maximal cap in the transformed domain $\widetilde{\varSigma }=\widetilde{\varSigma } (\tilde{n}_i(x_0))$ non empty.

Let u solve (1.1). The Kelvin transform of u at the point $x_0\in \partial \varOmega$ is defined in the transformed domain $\widetilde{\varOmega }:=h(\varOmega )$ by

$$\begin{aligned} v(y):=\left( \frac{1}{|y|}\right) ^{N-2}\ u(h(y))=\left( \frac{1}{|y|}\right) ^{N-2}\ u\left( \frac{y}{|y|^2}\right) ,\quad \text{ for }\quad y\in \widetilde{\varOmega }. \end{aligned}$$

(B.2)

Next, we fix regions where a Kelvin transform of the solution has no critical points. This is the statement of the following Theorem.

Theorem B.2

Assume that $\varOmega \subset {\mathbb {R}}^N$ is a bounded domain with $C^{2}$ boundary. Assume that the nonlinearity f satisfies (H1).

If $u\in C^2(\overline{\varOmega })$ satisfies (1.1) and $u>0$ in $\varOmega$, then for any $x_0\in \partial \varOmega$ its maximal cap in the transformed domain $\widetilde{\varSigma }$ is nonempty, and its Kelvin transform v,  defined by (B.2), has no critical point in the maximal cap $\widetilde{\varSigma }.$

Consequently, for any $x_0\in \partial \varOmega ,$ there exists a $\delta >0$ only dependent of $\varOmega$ and $x_0,$ and independent of f and u such that its Kelvin transform v has no critical point in the set $B_\delta (x_0)\cap h(\varOmega )$.

Finally, we observe that, reasoning on the Kelvin transform, the Kelvin transform of u at $x_0\in \partial \varOmega$ is locally increasing in the maximal cap of the transformed domain, which provides $L^\infty$ bounds for the Kelvin transform locally. By a compactification process, we then translate this into $L^\infty$ bounds in a neighborhood of the boundary for any solution of the elliptic equation. This is the statement of the following Theorem.

Theorem B.3

Assume that $\varOmega \subset {\mathbb {R}}^N$ is a bounded domain with $C^{2}$ boundary. Assume that the nonlinearity f satisfies (H1) and (H4).

If $u\in C^2(\overline{\varOmega })$ satisfies (1.1) and $u>0$ in $\varOmega$, then there exists a constant $\delta >0$ depending only on $\varOmega$ and not on f or u, and a constants C depending only on $\varOmega$ and f but not on u, such that

$$\begin{aligned} \max _{\varOmega {\setminus } \varOmega _{\delta }} u\le C \end{aligned}$$

(B.3)

where $\varOmega _{\delta }:=\{x\in \varOmega \ : \ d(x,\partial \varOmega )>\delta \}.$

Proof

See the proof of [8, Theorem 1.2]. $\square$