On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities

TL;DR

This paper investigates elliptic equations with non-Lipschitz nonlinearities, revealing solutions that violate Hopf's maximum principle on parts of the boundary, thus expanding understanding of boundary behavior in such problems.

Contribution

It demonstrates the existence of boundary solutions that partially violate Hopf's maximum principle for elliptic equations with non-Lipschitz nonlinearities on star-shaped domains.

Findings

Existence of solutions violating Hopf's maximum principle on nonempty boundary subsets.

Solutions are nonnegative ground states with partial boundary violations.

The results apply to elliptic problems with specific non-Lipschitz nonlinearities.

Abstract

We show that the elliptic equation with a non-Lipschitz right-hand side, $-\Delta u = \lambda |u|^{\beta-1}u - |u|^{\alpha-1}u$ with $\lambda>0$ and $0<\alpha<\beta<1$, considered on a smooth star-shaped domain $\Omega$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $\Gamma$ of the boundary $\partial\Omega$ such that $\Gamma \neq \partial\Omega$.

Equations

$$\mathcal{P}(\lambda ,\Omega )$$

$$\frac{\partial u}{\partial \nu }=0\text{ on }\partial \Omega ,$$

$$\frac{\partial u}{\partial \nu }<0\text{ on }\partial \Omega .$$

$$\frac{\partial u}{\partial \nu }=0\text{ on }\Gamma \text{and}\frac{\partial u}{\partial \nu }\ne 0\text{ on }\partial \Omega \setminus \Gamma$$

$$N\ge 3,0<\alpha <\beta <1,2(1+\alpha )(1+\beta )-N(1-\alpha )(1-\beta )<0.$$

$$E_{\lambda }(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^{2}dx-\frac{\lambda }{\beta +1}{\int }_{\Omega }|u{|}^{\beta +1}dx+\frac{1}{\alpha +1}{\int }_{\Omega }|u{|}^{\alpha +1}dx.$$

$$\frac{\partial u}{\partial \nu }=0\text{ on }\partial B_{R_2}\text{and}\frac{\partial u}{\partial \nu }<0\text{ on }\partial B_{R_1}.$$

$${\int }_{\partial \Omega }{\left|\frac{\partial u_{\lambda }}{\partial \nu }\right|}^2\left(x,\nu \right)ds>0,$$

$$P_{\lambda }(u_{\lambda }):=E_{\lambda }(u_{\lambda })-\frac{1}{N}{\int }_{\Omega }|\nabla u_{\lambda }{|}^{2}dx=-\frac{1}{2N}{\int }_{\partial \Omega }{\left|\frac{\partial u_{\lambda }}{\partial \nu }\right|}^2\left(x,\nu \right)ds<0$$

$$\frac{s^{\alpha +1}}{\alpha +1}-\frac{{\lambda }_0{s}^{\beta +1}}{\beta +1}>0\text{for all }s\in (0,\delta ).$$

$$C=\frac{1}{\sqrt{2}}{\int }_0^{\delta }{\left(\frac{s^{\alpha +1}}{\alpha +1}-\frac{{\lambda }_0{s}^{\beta +1}}{\beta +1}\right)}^{-\frac{1}{2}}ds.$$

$$r=\frac{1}{\sqrt{2}}{\int }_{w(r)}^{\delta }{\left(\frac{s^{\alpha +1}}{\alpha +1}-\frac{{\lambda }_0{s}^{\beta +1}}{\beta +1}\right)}^{-\frac{1}{2}}ds.$$

$$-w^{\prime }(r)=\sqrt{2}{\left(\frac{w(r{)}^{\alpha +1}}{\alpha +1}-\frac{{\lambda }_{0}w(r{)}^{\beta +1}}{\beta +1}\right)}^{\frac{1}{2}}$$

$$-w^{\prime \prime }={\lambda }_0|w{|}^{\beta -1}w-|w{|}^{\alpha -1}w$$

$$-\Delta v-\left({\lambda }_0|v{|}^{\beta -1}v-|v{|}^{\alpha -1}v\right)=-w^{\prime \prime }-\frac{N-1}{|x|}{w}^{\prime }-\left({\lambda }_0|w{|}^{\beta -1}w-|w{|}^{\alpha -1}w\right)=-\frac{N-1}{|x|}{w}^{\prime }\ge 0,$$

$$-\Delta v\ge {\lambda }_0|v{|}^{\beta -1}v-|v{|}^{\alpha -1}v\ge \lambda |v{|}^{\beta -1}v-|v{|}^{\alpha -1}v\text{for all }\lambda \le {\lambda }_0.$$

$$u_{{\lambda }_n}<\delta \text{leavevmode}\text{leavevmode}\text{mbox}in{T}_{l_0},\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\forall n\ge n_0.$$

Full text

V. Bobkov and P. Drábek were supported by the grant 18-03253S of the Grant Agency of the Czech Republic. V. Bobkov was also supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. Y. Ilyasov wishes to thank the University of West Bohemia, where this research was conducted, for the invitation and hospitality. The authors would like to thank J. I. Díaz and J. Hernández for the encouraging and stimulating discussions.

1. Introduction and main result

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $N\geq 2$. Consider the boundary value problem

[TABLE]

where $\lambda>0$ and $0<\alpha<\beta<1$, that is, the nonlinearity in $\mathcal{P}(\lambda,\Omega)$ is non-Lipschitz at zero. The latter property prevents to conclude that nonnegative solutions of $\mathcal{P}(\lambda,\Omega)$ a priori obey the strong maximum principle or Hopf’s maximum principle (the boundary point lemma). In fact, as it follows from [9] (see also [5, 8]), there exists $\lambda^{*}>0$ such that problem $\mathcal{P}(\lambda,\Omega)$ possesses the so-called free boundary solution (equivalently, compact support solution), which is a nonzero solution $u$ such that

[TABLE]

where $\nu$ is the unit outward normal vector to $\partial\Omega$.

Let us note that a symmetry result of [13] (see also [10]) together with a uniqueness result of [9] force any connected component of the support of a nonnegative free boundary solution of $\mathcal{P}(\lambda,\Omega)$ to be a ball whose radius $R_{\lambda}$ is uniquely defined by $\lambda$. Moreover, $\lambda\mapsto R_{\lambda}$ is a decreasing function, as it follows from a simple scaling argument. Therefore, it is clear that for any $\lambda>\lambda^{*}$ problem $\mathcal{P}(\lambda,\Omega)$ has a continuum of free boundary solutions.

At the same time, in [7] there was proved the existence of $\underline{\lambda}>0$ such that for any $\lambda>\underline{\lambda}$ problem $\mathcal{P}(\lambda,\Omega)$ has a positive solution $u$ which does satisfy Hopf’s maximum principle at every point of the boundary $\partial\Omega$, i.e.,

[TABLE]

The existence of two types of solutions described above naturally leads to the problem of the existence of a complementary class of solutions. Namely, we consider the following question:

Is there a solution of problem $\mathcal{P}(\lambda,\Omega)$ which violates Hopf’s maximum principle only on a part of the boundary $\partial\Omega$?

The aim of our note is to give an affirmative answer to this question. More precisely, we are interested in finding so-called partially free boundary solution of $\mathcal{P}(\lambda,\Omega)$, i.e., a solution $u$ which satisfies

[TABLE]

for some nonempty subset $\Gamma\subsetneq\partial\Omega$. Let us state our main result.

Theorem 1.1**.**

Let $N$, $\alpha$, $\beta$ satisfy

[TABLE]

Then there exist a bounded, strictly star-shaped (i.e., $(x,\nu)>0$ on $\partial\Omega$) domain $\Omega\subset\mathbb{R}^{N}$ of class $C^{2}$ and a value $\lambda>0$ such that problem $\mathcal{P}(\lambda,\Omega)$ possesses a nonnegative partially free boundary ground state solution.

Here, under a ground state solution of $\mathcal{P}(\lambda,\Omega)$ we mean a solution $u$ with the least action property, namely, $E_{\lambda}(u)\leq E_{\lambda}(v)$ for any nonzero solution $v$ of $\mathcal{P}(\lambda,\Omega)$, where $E_{\lambda}$ is the energy functional associated with $\mathcal{P}(\lambda,\Omega)$:

[TABLE]

Remark 1.2**.**

The result of [3] on the existence of radial sign-changing solutions in combination with the compact support principle [12] implies that problem $\mathcal{P}(\lambda,\Omega)$, considered on an annulus $\Omega=B_{R_{2}}\setminus\overline{B_{R_{1}}}$, has for certain values of $R_{1}$, $R_{2}$, $R_{2}>R_{1}$, and $\lambda$ a radial positive solution $u$ such that

[TABLE]

That is, $u$ is a partially free boundary solution. However, apart from this example, the existence of other partially free boundary solutions of $\mathcal{P}(\lambda,\Omega)$, including the case in which $\Omega$ is a simply connected domain, was not known to the authors.

Here and below, $B_{R}\subset\mathbb{R}^{N}$ stands for the open ball of radius $R$ centred at the origin.

The proof of Theorem 1.1 is given in Section 3. We provide an explicit construction of the domain with required properties. To this end, in Section 2, we recall some auxiliary results based on investigations made in [5].

2. Preliminaries

Our approach to prove Theorem 1.1 relies on the idea of the proof of the compact support principle [12, Theorem 2] and the following result obtained in [5].

Theorem 2.1**.**

Let $N,\alpha,\beta$ satisfy (1.3). Let $\Omega\subset\mathbb{R}^{N}$ be a bounded strictly star-shaped domain of class $C^{2}$. Then there exists $\lambda^{*}=\lambda^{*}(\Omega)\in(0,+\infty)$ such that for any $\lambda\geq\lambda^{*}$ problem $\mathcal{P}(\lambda,\Omega)$ has a nonnegative ground state solution $u_{\lambda}$, whereas for $\lambda<\lambda^{*}$, $\mathcal{P}(\lambda,\Omega)$ has no nonzero solutions. Moreover,

1. (i)

for any $\lambda>\lambda^{*}$ there holds

[TABLE] 2. (ii)

$u_{\lambda^{*}}$* is a free boundary solution, $\text{supp}(u_{\lambda^{*}})=\overline{B_{R(\Omega)}}$ is a maximal ball inscribed in $\Omega$, and $u_{\lambda^{*}}$ is radially symmetric with respect to the centre of $B_{R(\Omega)}$,* 3. (iii)

any sequence $\{u_{\lambda_{n}}\}$, where $\lambda_{n}\downarrow\lambda^{*}$, converges (up to a subsequence) strongly in $H_{0}^{1}(\Omega)$ to some $u_{\lambda^{*}}$.

The existence and nonexistence parts of Theorem 2.1 were obtained in [5, Theorems 1.1]. The assertion (i) follows from the following inequality for the Pohozaev identity:

[TABLE]

for any $\lambda>\lambda^{*}$, see [5, Proposition 4.2 and Corollary 5.3]. The assertion (ii) is stated in [5, Theorem 1.2], and the assertion (iii) follows from [5, Lemma 8.1].

Remark 2.2**.**

Theorem 2.1 (ii) together with the uniqueness result of [9] imply that $\lambda^{*}(\Omega)=\lambda^{*}(R(\Omega))$. That is, $\lambda^{*}$ depends only on the radius of a maximal ball inscribed in $\Omega$.

Remark 2.3**.**

Let $\lambda>\lambda^{*}$. Evidently, (2.1) means that $u_{\lambda}$ cannot be a free boundary solution. At the same time, (2.1) does not provide more detailed information about the pointwise behaviour of $\frac{\partial u_{\lambda}}{\partial\nu}$ on $\partial\Omega$. In particular, it is not known a priori whether $u_{\lambda}$ is either a partially free boundary solution or it satisfies Hopf’s maximum principle on the whole of $\partial\Omega$.

In Section 3 below, we will also need the following refinement of Theorem 2.1 (iii).

Corollary 2.4**.**

Under the assumptions of Theorem 2.1, any sequence $\{u_{\lambda_{n}}\}$, where $\lambda_{n}\downarrow\lambda^{*}$, converges (up to a subsequence) in $C^{1}(\overline{\Omega})$ to some $u_{\lambda^{*}}$.

Proof.

Since $\lambda^{*}\in(0,+\infty)$ and $u_{\lambda_{n}}\to u_{\lambda^{*}}$ strongly in $H_{0}^{1}(\Omega)$ up to a subsequence, we see from Theorem 2.1 (iii) that $\{u_{\lambda_{n}}\}$ is bounded in $H_{0}^{1}(\Omega)$. Therefore, applying the standard bootstrap argument (see, e.g., [6, Lemma 3.2, p. 114]), we obtain that $\{u_{\lambda_{n}}\}$ is bounded in $L^{\infty}(\Omega)$. Thus, the regularity result [11] implies that $\{u_{\lambda_{n}}\}$ is bounded in $C^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in(0,1)$. Finally, the Arzelà-Ascoli theorem yields $u_{\lambda_{n}}\to u_{\lambda^{*}}$ in $C^{1}(\overline{\Omega})$ up to a subsequence. ∎

3. Construction

We will prove Theorem 1.1 in three steps.

Step 1. Taking the ball $B_{1}$ of radius $1$, we define the value $\lambda^{*}=\lambda^{*}(1)$, see Remark 2.2. Let us fix any $\lambda_{0}>\lambda^{*}$ and $\delta>0$ such that

[TABLE]

Let us also fix some $l_{0}>1$. We are going to show the existence of $l_{1}>l_{0}$ and a supersolution $v$ of $\mathcal{P}(\lambda,\Omega)$ which has the following properties:

1. (i)

$v$ is radial; 2. (ii)

$v$ is nonnegative and nonincreasing; 3. (iii)

$v=\delta$ on $\partial B_{l_{0}}$; 4. (iv)

$v(x)=0$ for all $x$ satisfying $|x|\geq l_{1}$.

The existence of such $v$ was obtained in the proof of [12, Theorem 2] in more general settings. We repeat some arguments from [12] applied to our particular case, for the sake of completeness. Let us define a constant

[TABLE]

Note that $C<+\infty$ due to the assumption $0<\alpha<\beta<1$ and the choice of $\delta$. Consider a function $w=w(r)$ for $r\in[0,C)$ given by the implicit formula

[TABLE]

Differentiating this equality, we get

[TABLE]

for all $r\in[0,C)$. Thus, we deduce from (3.1), (3.2), and (3.3) that $w^{\prime}<0$, $w\in(0,\delta]$, $w(0)=\delta$, and both $w(r)\to 0$ and $w^{\prime}(r)\to 0$ as $r\uparrow C$. Moreover,

[TABLE]

on the interval $r\in[0,C)$, as it follows from [12, Lemma 1 (ii)]. Recalling that $w(r),w^{\prime}(r)\to 0$ as $r\uparrow C$, we can extend $w$ by zero for $r\geq C$, and hence $w$ is a solution of (3.4) for all $r\geq 0$.

Let us now define $v(x)=w(|x|-l_{0})$ for $x\in\mathbb{R}^{N}\setminus\overline{B_{l_{0}}}$. Using (3.4), we obtain

[TABLE]

since $w^{\prime}\leq 0$. Thus, $v$ is a supersolution of $\mathcal{P}(\lambda,\Omega)$ with the desired properties (i)-(iv) stated above, where $l_{1}=l_{0}+C$. Moreover, since $v$ is nonnegative, we also have

[TABLE]

That is, $v$ is a supersolution of $\mathcal{P}(\lambda,\Omega)$ for all $\lambda\leq\lambda_{0}$.

Step 2. Let us fix any $l>l_{1}$ and define a strictly star-shaped domain $\Omega$ as $\Omega=B_{1}\cup C_{l}\cup V_{l}$ (see Figure 1), where

• –

$C_{l}=D_{r}\times(0,l)$ is an $N$-dimensional cylinder, $D_{r}\subset\mathbb{R}^{N-1}$ is the open ball of radius $r\in(0,1)$ centred at the origin;

• –

$V_{l}$ is an appropriate residual part which smooths the boundary of $B_{1}\cup C_{l}$ in such a way that $\Omega$ is strictly star-shaped and its boundary is of class $C^{2}$.

Note that, by construction, the radius $R(\Omega)$ of a maximal ball inscribed in $\Omega$ equals $1$. Consequently, $\lambda^{*}(R(\Omega))=\lambda^{*}(1)$ coincides with the value $\lambda^{*}$ defined above. Thus, for any $\lambda>\lambda^{*}$ there exists a corresponding ground state solution $u_{\lambda}$ of $\mathcal{P}(\lambda,\Omega)$ obtained by Theorem 2.1. Now, taking an arbitrary sequence $\lambda_{n}\downarrow\lambda^{*}$, we get from Corollary 2.4 that $u_{\lambda_{n}}\to u_{\lambda^{*}}$ in $C^{1}(\overline{\Omega})$ as $n\to+\infty$, up to a subsequence. Therefore, recalling that $u_{\lambda^{*}}=0$ in $\Omega\setminus B_{1}$ (see Theorem 2.1 (ii)) and $l_{0}>1$, we deduce the existence of $n_{0}>0$ such that

[TABLE]

Here, $T_{l_{0}}:=\Omega\setminus\overline{B_{l_{0}}}$ is the “tail” of $\Omega$ lying outside of $\overline{B_{l_{0}}}$.

Step 3. Let us now compare $u_{\lambda_{n}}$ and the supersolution $v$. From (3.5) and the property (iii) of $v$ we have $u_{\lambda_{n}}\leq\delta=v$ on $\partial T_{l_{0}}\cap\partial B_{l_{0}}$ whenever $n\geq n_{0}$. Thus, recalling that $u_{\lambda_{n}}=0$ on $\partial T_{l_{0}}\setminus\partial B_{l_{0}}$ and $v$ is nonnegative (see (ii)), we conclude that $u_{\lambda_{n}}\leq v$ on $\partial T_{l_{0}}$. Moreover, since $\lambda_{n}\downarrow\lambda^{*}$, we can take $n$ larger, if necessary, to get $\lambda_{n}\leq\lambda_{0}$. Therefore, in view of (3.5) and the fact that $v$ is a supersolution of $\mathcal{P}(\lambda,\Omega)$, we can apply the weak comparison principle stated in [12, Lemma 3] to deduce that $u_{\lambda_{n}}\leq v$ in $T_{l_{0}}$, which yields $u_{\lambda_{n}}=0$ in $\Omega\setminus B_{l_{1}}$ due to (iv). In particular, there exists a nonempty subset $\Gamma_{n}\subsetneq\partial\Omega$ such that $\frac{\partial u_{\lambda_{n}}}{\partial\nu}=0$ on $\Gamma_{n}$. Since (2.1) is satisfied for all $\lambda_{n}$, we conclude that $u_{\lambda_{n}}$ is a nonnegative partially free boundary ground state solution of $\mathcal{P}(\lambda,\Omega)$. This completes the proof of Theorem 1.1.

Remark 3.1**.**

Clearly, the construction of $\Omega$ from above can be substantially generalized. We conjecture that Theorem 1.1 holds true for any sufficiently smooth bounded strictly star-shaped domain $\Omega$, except maybe the case when $\Omega$ is a ball. More precisely, based on the results of [7, 1] and Theorem 2.1, we conjecture that for any such $\Omega$ nonnegative ground state solutions $u_{\lambda}$ of $\mathcal{P}(\lambda,\Omega)$ have the following behaviour with respect to $\lambda$: there exists $\underline{\lambda}>\lambda^{*}$ such that

1. (i)

$u_{\lambda}$ is a partially free boundary solution for any $\lambda\in(\lambda^{*},\underline{\lambda})$; 2. (ii)

$u_{\underline{\lambda}}$ is positive; 3. (iii)

$u_{\lambda}$ is positive and satisfies Hopf’s maximum principle on the whole of $\partial\Omega$ for any $\lambda>\underline{\lambda}$.

We refer the interested reader to [2, 4] for additional discussions on problem $\mathcal{P}(\lambda,\Omega)$.