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Volume 9, Issue 1

On the moving plane method for boundary blow-up solutions to semilinear elliptic equations

Annamaria Canino
/ Berardino Sciunzi
• Corresponding author
• Dipartimento di Matematica, UNICAL, Ponte Pietro Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy
• Email
• Other articles by this author:
/ Alessandro Trombetta
Published Online: 2018-07-07 | DOI: https://doi.org/10.1515/anona-2017-0221

Abstract

We consider weak solutions to $-\mathrm{\Delta }u=f\left(u\right)$ on ${\mathrm{\Omega }}_{1}\setminus {\mathrm{\Omega }}_{0}$, with $u=c\ge 0$ in $\partial {\mathrm{\Omega }}_{1}$ and $u=+\mathrm{\infty }$ on $\partial {\mathrm{\Omega }}_{0}$, and we prove monotonicity properties of the solutions via the moving plane method. We also prove the radial symmetry of the solutions in the case of annular domains.

MSC 2010: 35B01; 35J61; 35J75

1 Introduction

Let ${\mathrm{\Omega }}_{0}$ and ${\mathrm{\Omega }}_{1}$ be two bounded smooth domains of ${ℝ}^{N}$, $N\ge 2$, such that ${\mathrm{\Omega }}_{0}\subset {\mathrm{\Omega }}_{1}$. Moreover, let $f\in {C}^{1}\left(\left[0,+\mathrm{\infty }\right)\right)$ and $c\ge 0$. We consider weak solutions to the problem

(1.1)

i.e., we consider $u\in {C}^{1}\left(\overline{{\mathrm{\Omega }}_{1}}\setminus \overline{{\mathrm{\Omega }}_{0}}\right)$ such that

and

When $c=0$, actually we deal with the case of positive solutions. Necessary and sufficient conditions for the existence of solutions to (1.1) are provided by the classical results of Keller [18] and Osserman [19], under suitable assumptions on the nonlinearity. The literature regarding boundary blow-up solutions is really wide (see, for example, [1, 2, 3, 4, 6, 5, 7, 13, 14, 12, 15, 16, 17, 20, 21]). Here we exploit an adaptation of the celebrated moving plane technique (see [8]) in order to obtain monotonicity properties of the solutions to (1.1). The domain that we consider is not convex and the solutions are not in ${H}_{0}^{1}\left({\mathrm{\Omega }}_{1}\setminus {\mathrm{\Omega }}_{0}\right)$ as in the classical case. This is the same difficulty that occurs when dealing with the study of the uniqueness, symmetry and monotonicity properties of solutions to singular semilinear elliptic equations, see [9, 10, 11]. These problems exhibit in fact some similarities with problem (1.1) although the proofs cannot be adapted to our case.

In the second part of the paper, we prove the radial symmetry of the solutions on annular domains, under suitable assumptions. In our setting, this cannot be done just using the moving plane method, since the domain is not convex, and we prove that the solution is radially symmetric showing directly that the angular derivative is zero. The technique is based on a refined maximum principle for the linearized equation.

Let us introduce some notations. Let ν be a direction in ${ℝ}^{N}$, with $|\nu |=1$. Given a real number λ, we set

${T}_{\lambda }^{\nu }=\left\{x\in {ℝ}^{N}:x\cdot \nu =\lambda \right\}$

and

${x}_{\lambda }^{\nu }={R}_{\lambda }^{\nu }\left(x\right)=x+2\left(\lambda -x\cdot \nu \right)\nu ,$

that is, the reflection of x trough the hyperplane ${T}_{\lambda }^{\nu }$. We will make the following assumption throughout the paper:

• (A)

${\mathrm{\Omega }}_{0}$ and ${\mathrm{\Omega }}_{1}$ are strictly convex with respect to the ν-direction and symmetric with respect to ${T}_{0}^{\nu }$.

Moreover, we set

${\mathrm{\Omega }}_{\lambda }^{\nu }=\left\{x\in {\mathrm{\Omega }}_{1}:x\cdot \nu <\lambda \right\}\setminus {R}_{\lambda }^{\nu }\left({\mathrm{\Omega }}_{0}\right)\mathit{ }\text{and}\mathit{ }{\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right)}^{\prime }={R}_{\lambda }^{\nu }\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right).$

Also we set

$a\left(\nu \right)=\underset{x\in {\mathrm{\Omega }}_{1}}{inf}x\cdot \nu \mathit{ }\text{and}\mathit{ }b\left(\nu \right)=\underset{x\in {\mathrm{\Omega }}_{0}}{inf}x\cdot \nu .$

Observe that, by assumption (A), it follows that ${\mathrm{\Omega }}_{\lambda }^{\nu }$ is nonempty and ${\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right)}^{\prime }\subset {\mathrm{\Omega }}_{1}\setminus {\mathrm{\Omega }}_{0}$ for any $a\left(\nu \right)<\lambda \le 0$. We define

and for any $a\left(\nu \right)<\lambda \le 0$.

We are now ready to state our main results.

Theorem 1.1.

Let $u\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{{\mathrm{\Omega }}_{\mathrm{1}}}\mathrm{\setminus }\overline{{\mathrm{\Omega }}_{\mathrm{0}}}\mathrm{\right)}$ be a weak solution to (1.1). Then

(1.2)

Consequently, it follows that u is strictly increasing with respect to the ν-direction in the set

In order to get symmetry results for the solution to (1.1), we restrict our attention to annular domains. We denote by ${B}_{R}$ the open ball of center 0 and radius $R>0$ in ${ℝ}^{N}$. By Theorem 1.1, we immediately deduce the following.

Corollary 1.2.

Let $\mathrm{0}\mathrm{<}{R}_{\mathrm{0}}\mathrm{<}{R}_{\mathrm{1}}$ and $u\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{{B}_{{R}_{\mathrm{1}}}}\mathrm{\setminus }\overline{{B}_{{R}_{\mathrm{0}}}}\mathrm{\right)}$ be a weak solution to (1.1) in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus }{B}_{{R}_{\mathrm{0}}}$. Then $\frac{\mathrm{\partial }\mathit{}u}{\mathrm{\partial }\mathit{}r}\mathrm{<}\mathrm{0}$ in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus }{B}_{{R}_{\mathrm{0}}}$.

In the following we state sufficient conditions in order to deduce the radial symmetry of the solution, once we prove the monotonicity. We set

$v=-x\cdot \nabla u,$

and we denote by ${u}_{\theta }$ the angular derivative of u.

Theorem 1.3.

Let $\mathrm{0}\mathrm{<}{R}_{\mathrm{0}}\mathrm{<}{R}_{\mathrm{1}}$ and let $u\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{{B}_{{R}_{\mathrm{1}}}}\mathrm{\setminus }\overline{{B}_{{R}_{\mathrm{0}}}}\mathrm{\right)}$ be a weak solution to (1.1) in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus }{B}_{{R}_{\mathrm{0}}}$ that satisfies

(1.3)

If $f\mathrm{\le }\mathrm{0}$, then u is radially symmetric and radially decreasing in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus }{B}_{{R}_{\mathrm{0}}}$.

For the reader’s convenience, let us point out that the condition in (1.3) is inspired by the results in [20, 21].

In Section 2 we give the proof of Theorem 1.1. We prove Theorem 1.3 in Section 3.

2 Proof of Theorem 1.1

Let $a\left(\nu \right)<\lambda . We define

${w}_{\lambda }^{\nu }=u-{u}_{\lambda }^{\nu }.$

We need to prove that ${w}_{\lambda }^{\nu }<0$ in ${\mathrm{\Omega }}_{\lambda }^{\nu }$. We have

so that ${w}_{\lambda }^{\nu }$ weakly satisfies

$-\mathrm{\Delta }{w}_{\lambda }^{\nu }={c}_{\lambda }\left(x\right){w}_{\lambda }^{\nu },$

where

Observe that ${c}_{\lambda }\left(x\right)\in {L}_{\mathrm{loc}}^{\mathrm{\infty }}\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right)$ and ${c}_{\lambda }\left(x\right)\in {L}^{\mathrm{\infty }}\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right)$ for $\lambda -a\left(\nu \right)$ small. Moreover, since ${w}_{\lambda }^{\nu }=0$ on $\partial {\mathrm{\Omega }}_{\lambda }^{\nu }\cap {T}_{\lambda }^{\nu }$, it follows that ${w}_{\lambda }^{\nu }=c-{u}_{\lambda }^{\nu }<0$ on $\partial {\mathrm{\Omega }}_{\lambda }^{\nu }\cap \partial {\mathrm{\Omega }}_{1}$ and $u<{u}_{\lambda }^{\nu }$ on a neighborhood of $\partial {\mathrm{\Omega }}_{\lambda }^{\nu }\cap \partial {R}_{\lambda }^{\nu }\left({\mathrm{\Omega }}_{0}\right)$, thanks to the fact that

Then

${\left({w}_{\lambda }^{\nu }\right)}^{+}\in {H}_{0}^{1}\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right).$

Let $\lambda -a\left(\nu \right)$ be sufficiently small so that the weak maximum principle in small domains works for the operator $\mathrm{\Delta }+{c}_{\lambda }\left(x\right)$ in ${\mathrm{\Omega }}_{\lambda }^{\nu }$, recalling that ${c}_{\lambda }\left(x\right)\in {L}^{\mathrm{\infty }}\left({\mathrm{\Omega }}_{\lambda }^{\nu }\right)$ for $\lambda -a\left(\nu \right)$ small. We get ${w}_{\lambda }^{\nu }\le 0$ in ${\mathrm{\Omega }}_{\lambda }^{\nu }$ and, by the strong maximum principle, we obtain

Set now

We need to prove that $\mu \ge b\left(\nu \right)$. On the contrary, suppose $\mu . By continuity, it follows that ${w}_{\mu }^{\nu }\le 0$ in ${\mathrm{\Omega }}_{\mu }^{\nu }$. Moreover, it is possible to apply the same argument of the first part of the proof to obtain, by the strong maximum principle, that

Take now $\epsilon >0$ sufficiently small such that $\mu +\epsilon . Given $\delta >0$, we fix a compact set $\mathcal{𝒦}\subset {\mathrm{\Omega }}_{\mu }^{\nu }$ so that $\mathcal{ℒ}\left({\mathrm{\Omega }}_{\mu }^{\nu }\setminus \mathcal{𝒦}\right)<\frac{\delta }{2}$. Since ${w}_{\mu }^{\nu }<0$ in K, by the continuity of u, it follows that there exists $M>0$ such that ${w}_{\mu }^{\nu }\le M<0$ in K. Therefore, we can find ${\epsilon }_{0}>0$ such that

whenever $0<\epsilon \le {\epsilon }_{0}$. Moreover, choosing $\epsilon >0$ sufficiently small such that $\mathcal{ℒ}\left({\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}\right)<\delta$ and using ${\left({w}_{\mu +\epsilon }^{\nu }\right)}^{+}$ as test function in the weak formulation of (1.1), since f is locally Lipschitz continuous, an application of the Poincaré inequality gives

${\int }_{{\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}}|\nabla {\left({w}_{\mu +\epsilon }^{\nu }\right)}^{+}{|}^{2}={\int }_{{\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}}\frac{f\left(u\right)-f\left({u}_{\mu +\epsilon }^{\nu }\right)}{u-{u}_{\mu +\epsilon }^{\nu }}{\left(u-{u}_{\mu +\epsilon }^{\nu }\right)}^{2}$$\le {C}_{p}^{2}\left({\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}\right)C{\int }_{{\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}}|\nabla {\left({w}_{\mu +\epsilon }^{\nu }\right)}^{+}{|}^{2},$

where

$C=\underset{s,t\in \left[0,{\parallel u\parallel }_{{L}^{\mathrm{\infty }}\left({\mathrm{\Omega }}_{\mu +{\epsilon }_{0}}^{\nu }\right)}\right]}{sup}\frac{f\left(t\right)-f\left(s\right)}{t-s}.$

Choosing δ sufficiently small, we have ${C}_{p}^{2}\left({\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}\right)C<1$ so that ${\left({w}_{\mu +\epsilon }^{\nu }\right)}^{+}=0$ in ${\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }\setminus \mathcal{𝒦}$. Hence, ${w}_{\mu +\epsilon }^{\nu }\le 0$ in ${\mathrm{\Omega }}_{\mu +\epsilon }^{\nu }$. By the strong maximum principle, we get

This gives a contradiction with the definition of μ and shows that $\mu \ge b\left(\nu \right)$, namely, (1.2) is proved.

Now, let x and ${x}^{\prime }$ such that $a\left(\nu \right). Setting $\lambda =\left(x\cdot \nu +{x}^{\prime }\cdot \nu \right)/2$, we get ${x}_{\lambda }^{\nu }={x}^{\prime }$ and

$u\left(x\right)<{u}_{\lambda }^{\nu }\left(x\right)=u\left({x}^{\prime }\right).$

Therefore, u is strictly increasing with respect to the ν-direction. To conclude the proof, fix $x\in {\mathrm{\Omega }}_{b\left(\nu \right)}^{\nu }$ and let $a\left(\nu \right)<\lambda be such that $x\in {T}_{\lambda }^{\nu }\cap {\mathrm{\Omega }}_{1}$. We have that $-\mathrm{\Delta }{w}_{\lambda }^{\nu }={c}_{\lambda }\left(x\right){w}_{\lambda }^{\nu }$, ${w}_{\lambda }>0$ in ${\mathrm{\Omega }}_{\lambda }^{\nu }$ and ${w}_{\lambda }=0$ in ${T}_{\lambda }^{\nu }\cap {\mathrm{\Omega }}_{1}$. By the Hopf lemma, we obtain

$0<\frac{\partial {w}_{\lambda }^{\nu }}{\partial \nu }=2\frac{\partial u}{\partial \nu }\left(x\right).$

3 Proof of Theorem 1.3

We start by proving the following proposition.

Proposition 3.1.

Let ${R}_{\mathrm{0}}\mathrm{<}{R}_{\mathrm{1}}$ and $u\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{\left(}\overline{{B}_{{R}_{\mathrm{1}}}}\mathrm{\setminus }\overline{{B}_{{R}_{\mathrm{0}}}}\mathrm{\right)}$ be a weak solution to (1.1) in ${B}_{{R}_{\mathrm{1}}}\mathrm{\setminus }{B}_{{R}_{\mathrm{0}}}$. If $f\mathrm{\le }\mathrm{0}$, then

Proof.

Let $x=\left({x}_{1},\mathrm{\dots },{x}_{N}\right)\in {ℝ}^{N}$. Set ${u}_{i}=\frac{\partial u}{\partial {e}_{i}}$ for each $i=1,\mathrm{\dots },N$. We have

$\nabla v=-\nabla \left(\sum _{i=1}^{N}{x}_{i}{u}_{i}\right)=-\nabla u-\sum _{i=1}^{N}{x}_{i}\nabla {u}_{i}.$

Since $-\mathrm{\Delta }u=f\left(u\right)$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}}$ and $f\le 0$, we obtain

$-\mathrm{\Delta }v=\text{div}\left(\nabla u+\sum _{i=1}^{N}{x}_{i}\nabla {u}_{i}\right)=2\mathrm{\Delta }u+\sum _{i=1}^{N}{x}_{i}\mathrm{\Delta }{u}_{i}$$=-2f\left(u\right)-\sum _{i=1}^{N}{x}_{i}{f}^{\prime }\left(u\right){u}_{i}=-2f\left(u\right)+{f}^{\prime }\left(u\right)v\ge {f}^{\prime }\left(u\right)v,$

so that $-\mathrm{\Delta }v\ge {f}^{\prime }\left(u\right)v$. ∎

In the following we will also exploit the fact that

as it follows by direct computation.

Proof of Theorem 1.3.

We shall actually show that ${u}_{\theta }\ge 0$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}}$. By assumption (1.3), choosing $\sigma >0$ sufficiently small, we have ${u}_{\theta }+tv\ge 0$ in ${B}_{{R}_{0}+\sigma }\setminus {B}_{{R}_{0}}$.

Furthermore, by Corollary 1.2, since v is continuous, we have

$v=-\frac{x}{|x|}\cdot |x|\nabla u=|x|\frac{\partial u}{\partial \nu }>0,$

where $\nu =-x/|x|$ with $x\in {B}_{{R}_{1}}\setminus {B}_{{R}_{0}+\sigma }$. Moreover, $v>0$ on $\partial {B}_{{R}_{1}}$, by the Hopf lemma. Then $v>\theta \left(\sigma \right)>0$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}+\sigma }$. Since ${u}_{\theta }$ is bounded in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}+\sigma }$, for t sufficiently large, we have ${u}_{\theta }+tv>0$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}+\sigma }$. Hence,

Set now

We need to prove that ${t}_{0}=0$. Conversely, suppose ${t}_{0}>0$. By the definition of ${t}_{0}$ and Proposition 3.1, we obtain

$-\mathrm{\Delta }\left({u}_{\theta }+{t}_{0}v\right)\ge {f}^{\prime }\left(u\right)\left({u}_{\theta }+{t}_{0}v\right)$${u}_{\theta }+{t}_{0}v\ge 0$

By the strong maximum principle, since ${u}_{\theta }=0$ and $v>0$ on $\partial {B}_{{R}_{1}}$, we get

Since (1.3) is in force, there exists ${\delta }_{0}>0$ such that

Moreover, we have

By continuity, for $\epsilon >0$ small, we have that

Resuming, we have that

This contradicts the definition of ${t}_{0}$, showing that actually ${t}_{0}=0$ and ${u}_{\theta }\ge 0$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}}$. This is possible only if ${u}_{\theta }=0$ in ${B}_{{R}_{1}}\setminus {B}_{{R}_{0}}$, namely, if the solution is radial. We conclude that the solution is also radially decreasing by Theorem 1.1. ∎

References

• [1]

S. Alarcón, G. Díaz and J. M. Rey, Large solutions of elliptic semilinear equations in the borderline case. An exhaustive and intrinsic point of view, J. Math. Anal. Appl. 431 (2015), no. 1, 365–405.

• [2]

C. Bandle, Asymptotic behavior of large solutions of elliptic equations, An. Univ. Craiova Ser. Mat. Inform. 32 (2005), 1–8.  Google Scholar

• [3]

C. Bandle and M. Chipot, Large solutions in cylindrical domains, Adv. Math. Sci. Appl. 23 (2013), no. 2, 461–476.  Google Scholar

• [4]

C. Bandle and E. Giarrusso, Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Adv. Differential Equations 1 (1996), no. 1, 133–150.  Google Scholar

• [5]

C. Bandle and M. Marcus, “Large” solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992), 9–24.

• [6]

C. Bandle and M. Marcus, Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curvature of the boundary, Complex Var. Theory Appl. 49 (2004), no. 7–9, 555–570.  Google Scholar

• [7]

C. Bandle, V. Moroz and W. Reichel, “Boundary blowup” type sub-solutions to semilinear elliptic equations with Hardy potential, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 503–523.

• [8]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37.

• [9]

A. Canino, M. Grandinetti and B. Sciunzi, Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities, J. Differential Equations 255 (2013), no. 12, 4437–4447.

• [10]

A. Canino, L. Montoro and B. Sciunzi, The moving plane method for singular semilinear elliptic problems, Nonlinear Anal. 156 (2017), 61–69.

• [11]

A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 2, Paper No. 8.

• [12]

F. C. Cîrstea and V. Rǎdulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3275–3286.

• [13]

O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: asymptotics, uniqueness and symmetry, J. Differential Equations 249 (2010), no. 4, 931–964.

• [14]

O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions, J. Math. Anal. Appl. 395 (2012), no. 2, 806–812.

• [15]

S. Dumont, L. Dupaigne, O. Goubet and V. Rǎdulescu, Back to the Keller–Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud. 7 (2007), no. 2, 271–298.  Google Scholar

• [16]

L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, Entire large solutions for semilinear elliptic equations, J. Differential Equations 253 (2012), no. 7, 2224–2251.

• [17]

F. Gladiali and M. Squassina, On explosive solutions for a class of quasi-linear elliptic equations, Adv. Nonlinear Stud. 13 (2013), no. 3, 663–698.  Google Scholar

• [18]

J. B. Keller, On solutions of $\mathrm{\Delta }u=f\left(u\right)$, Comm. Pure Appl. Math. 10 (1957), 503–510.  Google Scholar

• [19]

R. Osserman, On the inequality $\mathrm{\Delta }u\ge f\left(u\right)$, Pacific J. Math. 7 (1957), 1641–1647.  Google Scholar

• [20]

A. Porretta and L. Véron, Asymptotic behaviour of the gradient of large solutions to some nonlinear elliptic equations, Adv. Nonlinear Stud. 6 (2006), no. 3, 351–378.  Google Scholar

• [21]

A. Porretta and L. Véron, Symmetry of large solutions of nonlinear elliptic equations in a ball, J. Funct. Anal. 236 (2006), no. 2, 581–591.

Revised: 2018-04-04

Accepted: 2018-05-14

Published Online: 2018-07-07

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 1–6, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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