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- Open Access
Laplace’s equation with concave and convex boundary nonlinearities on an exterior region
- Jinxiu Mao^{1}Email author,
- Zengqin Zhao^{1} and
- Aixia Qian^{1}
- Received: 22 July 2018
- Accepted: 1 March 2019
- Published: 13 March 2019
Abstract
This paper studies Laplace’s equation \(-\Delta u=0\) in an exterior region \(U\varsubsetneq {\mathbb{R}}^{N}\), when \(N\geq 3\), subject to the nonlinear boundary condition \(\frac{\partial u}{\partial \nu }=\lambda \vert u \vert ^{q-2}u+\mu \vert u \vert ^{p-2}u\) on ∂U with \(1< q<2<p<2_{*}\). In the function space \(\mathscr{H} (U )\), one observes that, when \(\lambda >0\) and \(\mu \in \mathbb{R}\) arbitrary, then there exists a sequence \(\{u_{k} \}\) of solutions with negative energy converging to 0 as \(k\to \infty \); on the other hand, when \(\lambda \in \mathbb{R}\) and \(\mu >0\) arbitrary, then there exists a sequence \(\{\tilde{u}_{k} \}\) of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation \(-\Delta _{p} u=0\), the exterior p-harmonic Steklov eigenvalue problems are described.
Keywords
- Exterior regions
- Laplace operator
- Concave and convex mixed nonlinear boundary conditions
- Fountain theorems
- Steklov eigenvalue problems
MSC
- 35J20
- 35J65
- 46E22
- 49R99
1 Introduction
A region is a nonempty, open, connected subset U of \({\mathbb{R}}^{N}\), and is said to be an exterior region provided that its complement \({\mathbb{R}}^{N}\setminus U\) is a nonempty, compact subset. Without loss of generality, we simply assume that \(0\notin U\). The boundary of a set A is denoted by ∂A.
Our general assumption on U is the following condition.
Condition B.1
\(U\varsubsetneq {\mathbb{R}}^{N}\) is an exterior region, with \(0\notin U\), whose boundary ∂U is the union of finitely many disjoint, closed, Lipschitz surfaces, each of finite surface area.
Theorem 1.1
- (a)
When \(\lambda \in \mathbb{R}\) and \(\mu >0\) arbitrary, then problem (1.1) has a sequence \(\{ \mathfrak{u}_{k} \}\) of solutions in \(E^{1} (U )\) such that \(\varphi ( \mathfrak{u}_{k} )>0\) and \(\varphi ( \mathfrak{u}_{k} )\to \infty \) as \(k\to \infty \).
- (b)
When \(\lambda >0\) and \(\mu \in \mathbb{R}\) arbitrary, then problem (1.1) has a sequence \(\{ \mathfrak{v}_{k} \}\) of solutions in \(E^{1} (U )\) such that \(\varphi ( \mathfrak{v}_{k} )<0\) and \(\varphi ( \mathfrak{v}_{k} )\to 0^{-}\) as \(k\to \infty \).
We recall in Sect. 2 some necessary results to carry out the proofs that are detailed in Sect. 3; Sect. 4 is devoted to the description of the p-harmonic Steklov eigenvalue problems on an exterior region U in a Banach space \(E^{1,p} (U )\) when \(1< p< N\).
We remark all solutions considered in this paper are weak or distributional solutions.
It is interesting to see that some nice properties of the first exterior p-harmonic Steklov eigenvalue problem are described in Han [12, 13, 15] and he [14] also studied an exterior harmonic boundary value problem with some oscillating boundary condition. However, there is no result for the sequence of p-harmonic Steklov eigenvalue problems on an exterior region U, so we will study this in Sect. 4. See Torné [27] for the bounded region case.
Finally, one notices that we only wants to simply present an application of some result in [3, 12]. Theorem 1.1 remains true when the special nonlinearity in (1.1) is replaced by more general ones as mentioned in [7]. On the other hand, it is very interesting to know more results like this using the fountain theorems of Yan and Yang [29], Zou [33], Du and Mao [8], Sun, Liu and Wu [26]. Other important results can be found in Polidoro and Ragusa [22], Han [16, 17], Feng, Li and Sun [9], and Phung and Minh [21], Mao and Zhao [18–20], Guan, Zhao and Lin [10, 11], Zhang [30], Zhang, Liu and Wu [31, 32].
2 The function space \(E^{1,p} (U )\)
First, let us fix the notations that will be used in this paper. Given \(p,q\in [1,\infty ]\), \(L^{p} (U )\) and \(L^{q} (\partial U, \,d\sigma )\) are the usual spaces of extended, real-valued, Lebesgue measurable functions on U and ∂U, with their standard norms written as \(\Vert \cdot \Vert _{p, U}\) and \(\Vert \cdot \Vert _{q, \partial U}\), respectively.
Auchmuty and Han [3, 4, 12] recently introduced a new function space \(E^{1,p} (U )\) suitable for the study of harmonic boundary value problems on an exterior region U which satisfies the boundary regularity condition (B1)—that is, each function \(u\in E^{1,p} (U )\) satisfies \(u\in L^{p^{*}} (U )\) and \(\vert \nabla u \vert \in L^{p} (U )\) with \(N\geq 2\) and \(p^{*}:=\frac{Np}{N-p}\) when \(1< p< N\).
Notice when \(p\geq N\), Auchmuty and Han [4] showed, with an interesting example, that \(E^{1,p} (U )\) is not complete with respect to the gradient \(L^{p}\)-norm in general.
Let us recall some results as regards the space \(E^{1,p} (U )\) which will be used later.
Lemma 2.1
Suppose that \(N\geq 2\), \(1< p< N\) and condition (B1) holds. Then the embedding of \(E^{1,p} (U )\) into \(L^{p^{*}} (U )\) is continuous, where \(p^{*}:=\frac{Np}{N-p}\) is the critical Sobolev index; besides, the embedding of \(E^{1,p} (U )\) into \(L^{q} (\partial U, \,d\sigma )\) is continuous when \(1\leq q\leq p_{*}\) and also compact when \(1\leq q< p_{*}\), where \(p_{*}:= \frac{ (N-1 )p}{N-p}\) is the trace critical Sobolev index.
Obviously, Lemma 2.1 shows us some concrete function spaces that are contained in the dual space of \(E^{1,p} (U )\). The preceding results can be found, with details, in [3, 4, 12].
Below, we give the fountain theorems. Given a compact group \(\mathfrak{G}\) and a normed vector space \(\mathcal{X}\) with norm \(\Vert \cdot \Vert \), we say \(\mathfrak{G}\) acts isometrically on \(\mathcal{X}\) provided \(\Vert gu \Vert = \Vert u \Vert \) for all \(g\in \mathfrak{G}\) and \(u\in \mathcal{X}\); also, a subset \(\tilde{\mathcal{X}}\subseteq \mathcal{X}\) is said to be invariant with respect to \(\mathfrak{G}\) provided \(gu\in \tilde{\mathcal{X}}\) for every \(u\in \tilde{\mathcal{X}}\) and \(g\in \mathfrak{G}\). On the other hand, given \(\mathfrak{G}\) and a finite dimensional space V, we say the action of \(\mathfrak{G}\) on V is admissible when each continuous, equivariant map \(\wp :\partial \mathbf{O}\to \mathbf{V}^{k}\) has a zero, where O is an open, bounded, invariant (with respect to \(\mathfrak{G}\)) neighborhood of 0 in \(\mathbf{V}^{k+1}\) for some \(k\geq 1\); here, the map ℘ is said to be equivariant provided \(g\circ \wp =\wp \circ g\) for all \(g\in \mathfrak{G}\), with \(g (v_{1},v_{2},\ldots ,v_{k} ):= (gv_{1},gv_{2},\ldots ,gv_{k} )\) for any \(v= (v_{1},v _{2},\ldots ,v_{k} )\in \mathbf{V}^{k}\).
Next, given a Banach space \(\mathcal{X}\), a functional \(\psi : \mathcal{X}\to \mathbb{R}\) is said to belong to \(C^{1} (\mathcal{X}, \mathbb{R} )\), provided its first Fréchet derivative exists and is continuous on \(\mathcal{X}\); when ψ has a continuous first Gateaux derivative \(\psi ^{\prime }\) on \(\mathcal{X}\), then one observes \(\psi \in C^{1} (\mathcal{X},\mathbb{R} )\). Clearly, the functional φ defined in (1.2) is in \(C^{1} (\mathscr{H} (U ),\mathbb{R} )\) and we shall assume this from now on. Also, \(\psi :\mathcal{X}\to \mathbb{R}\) is said to be invariant with respect to \(\mathfrak{G}\) provided \(\psi \circ g=\psi \) for every \(g\in \mathfrak{G}\).
Theorem 2.2
([1])
- (a1)
\(\mathfrak{G}\) acts isometrically on \(\mathcal{X}\), the spaces \(\mathcal{X}(j)\) are invariant and there is a finite dimensional space V such that, for all \(j\in \mathbb{N}\), \(\mathcal{X}(j)\simeq \mathbf{V}\) and the action of \(\mathfrak{G}\) on V is admissible;
- (a2)
\(\mathfrak{a}_{k} := \max_{u\in \mathcal{Y}_{k}, \Vert u \Vert =\rho _{k}} \psi (u) < 0\);
- (a3)
\(\mathfrak{b}_{k} := \inf_{u\in \mathcal{Z}_{k}, \Vert u \Vert =\varrho _{k}}\psi (u) \to \infty \) as \(k\to \infty \);
- (a4)
ψ satisfies the \((PS)_{c}\)-condition for every \(c\in (0,\infty )\).
Theorem 2.3
- (b1)
\(\tilde{\mathfrak{a}}_{k} := \max_{u\in \mathcal{Y}_{k}, \Vert u \Vert =\varrho _{k}}\psi (u) < 0\);
- (b2)
\(\tilde{\mathfrak{b}}_{k} := \inf_{u\in \mathcal{Z}_{k}, \Vert u \Vert =\rho _{k}} \psi (u) \geq 0\);
- (b3)
\(\tilde{\mathfrak{c}}_{k} := \inf_{u\in \mathcal{Z}_{k}, \Vert u \Vert \leq \rho _{k}}\psi (u) \to 0^{-}\) as \(k\to \infty \);
- (b4)
ψ satisfies the \((PS)^{*}_{c}\)-condition with respect to \(\mathcal{Y}_{k}\) for each \(c\in [\tilde{\mathfrak{c}} _{k_{1}},0 )\).
Remark
Notice \(\tilde{\mathfrak{c}}_{k}\leq \min_{u\in \mathcal{X}(k), \Vert u \Vert =\varrho _{k}} \psi (u)\leq \max_{u\in \mathcal{X}(k), \Vert u \Vert =\varrho _{k}} \psi (u)\leq \tilde{\mathfrak{a}}_{k}<0\) as \(\mathcal{Y}_{k}\cap \mathcal{Z}_{k}=\mathcal{X}(k)\)—this fact is used in conditions (b3) and (b4) presented above in Theorem 2.3.
A sequence \(\{u_{k} \}\) is said to be a Palais–Smale sequence for the functional \(\psi \in C^{1} (\mathcal{X},\mathbb{R} )\) at level c in \(\mathcal{X}\), \((PS)_{c}\)-sequence for short, if \(\psi (u_{k})\to c\) yet \(\psi ^{\prime }(u_{k})\to 0\) as \(k\to \infty \); ψ satisfies the \((PS)_{c}\)-condition provided each \((PS)_{c}\)-sequence has a strongly convergent subsequence in \(\mathcal{X}\). On the other hand, a sequence \(\{\tilde{u}_{k_{l}} \}\), with \(\tilde{u}_{k_{l}}\) in \(\mathcal{Y}_{k_{l}}\), is said to be a generalized Palais–Smale sequence for ψ at level c, \((PS)^{*}_{c}\)-sequence for short, if \(\psi (\tilde{u}_{k _{l}})\to c\) yet \(\psi |^{\prime }_{\mathcal{Y}_{k_{l}}}(\tilde{u} _{k_{l}})\to 0\) as \(l\to \infty \); ψ satisfies the \((PS)^{*}_{c}\)-condition with respect to \(\mathcal{Y}_{k}\) provided each \((PS)^{*}_{c}\)-sequence has a subsequence that converges strongly to a critical point of ψ in \(\mathcal{X}\).
More details on fountain theorems can be found in [6, 7, 28, 29, 33].
3 Existence results of (1.1)
In this section, we shall present the proofs of Theorem 1.1. Matching with the preceding notations, we can identify \(\mathfrak{G}= \mathbb{Z}_{2}\)—the second order quotient group, \(\mathcal{X}= \mathscr{H} (U )\)—the Hilbert subspace of \(E^{1} (U )\) of all finite energy harmonic functions, and \(\psi =\varphi \in C^{1} (\mathcal{X},\mathbb{R} )\). One result in [3] shows \(\mathcal{X}=\overline{\bigoplus_{j\in \mathbb{N}}\mathcal{X}(j)}\); here, \(\mathcal{X}(j)= \mathrm{span} \{s_{j} \}\simeq \mathbf{V}=\mathbb{R}\), with \(s_{j}\in \mathscr{H} (U )\) a finite energy harmonic Steklov eigenfunction associated with the jth harmonic Steklov eigenvalue \(\delta _{j}>0\). Noting that the functional φ is even, condition (a1) is trivially satisfied since a classical result of Borsuk–Ulam says that the antipodal action of \(\mathbb{Z}_{2}\) on \(\mathbb{R}\) is admissible.
In the following, we shall deduce conditions (a2)–(a4) and (b1)–(b4) to guarantee the conclusions of the first and the second part of Theorem 1.1, respectively.
All the above discussions finish the proof of Theorem 1.1 completely.
We do not know whether \(\mathfrak{v}_{k} \to 0\) as \(k\to \infty \); this is the case if 0 is the only solution of problem (1.1) with energy 0. However, we can derive the following result.
Proposition 3.1
- (a)When \(\lambda \in \mathbb{R}\) arbitrary yet \(\mu \leq 0\), then (1.1) has no solution with positive energy; also,$$ \inf \bigl\{ \Vert u \Vert :u \textit{ solves } \mbox{(1.1)}\textit{ with }\varphi (u)>0 \bigr\} \to \infty \quad \textit{as } \mu \to 0^{+}. $$
- (b)When \(\mu \in \mathbb{R}\) arbitrary yet \(\lambda \leq 0\), then (1.1) has no solution with negative energy; also,$$ \sup \bigl\{ \Vert v \Vert :v \textit{ solves }\mbox{(1.1)}\textit{ with }\varphi (v)< 0 \bigr\} \to 0 \quad \textit{as } \lambda \to 0^{+}. $$
Proof
Theorem 3.2
- (a)
When \(\lambda \in \mathbb{R}\) and \(\mu >0\) arbitrary, then problem (3.14) has a sequence \(\{ \mathfrak{u}_{k} \}\) of solutions in \(\mathscr{N} (U )\) such that \(\phi ( \mathfrak{u}_{k} )>0\) and \(\phi ( \mathfrak{u}_{k} )\to \infty \) as \(k\to \infty \).
- (b)
When \(\lambda >0\) and \(\mu \in \mathbb{R}\) arbitrary, then problem (3.14) has a sequence \(\{ \mathfrak{v}_{k} \}\) of solutions in \(\mathscr{N} (U )\) such that \(\phi ( \mathfrak{v}_{k} )<0\) and \(\phi ( \mathfrak{v}_{k} )\to 0^{-}\) as \(k\to \infty \).
4 p-Laplacian Steklov eigenvalue problems
As mentioned earlier, the beauty of the paper [3] is the discovery of the generalizations to high dimensions of the classical 3d Laplace’s spherical harmonics exterior to the unit ball: the exterior harmonic Steklov eigenvalue problems whose full spectra are derived there. This section is devoted to the description of the exterior p-harmonic Steklove eigenvalue problems in the function space \(E^{1,p} (U )\) when \(N\geq 3\) and \(1< p< N\). Similar results on bounded regions may be found in the interesting paper of Torné [27].
Lemma 4.1
Proof
Using this result, we can derive a version of deformation lemma.
Proposition 4.2
Given a constant \(\kappa >0\), suppose there are constants \(\varsigma >0\) and \(\tau \in (0,\kappa )\), such that \(\Vert u _{\mathcal{B}} \Vert \geq \varsigma \) on \(\texttt {V}_{\tau }:= \{u\in \mathbf{S}_{1}: \vert \varphi (u)-\kappa \vert \leq \tau \}\). Then, for every compact, symmetric subset G of \(\mathbf{S}_{1}\), one finds a constant \(\epsilon \in (0,\tau )\) and an associated odd map \(H_{\epsilon }: \mathbf{S}_{1}\to \mathbf{S}_{1}\) that is continuous on \(\texttt {V} _{\epsilon }\cap \mathbf{G}\) and \(H_{\epsilon }(\texttt {V}_{\epsilon }\cap \mathbf{G})\subseteq \varphi _{\kappa +\epsilon }\), where \(\varphi _{\kappa +\epsilon }:= \{u\in \mathbf{S}_{1}:\varphi (u) \geq \kappa +\epsilon \}\).
Proof
Then we can prove the following main results.
Theorem 4.3
For every \(n\geq 1\), \(\kappa _{n}>0\) and there exists a function \(s_{n}\in E^{1,p} (U )\) such that \(\varphi (s_{n})=\kappa _{n}\); in addition, \(s_{n}\) is a weak solution of (4.1) with \(\delta =\delta _{n}:=\frac{1}{p \kappa _{n}}>0\).
Proof
As \(\gamma (\mathbf{S}_{1})=\infty \), \(\kappa _{n}\) is well defined in the sense that \(\mathscr{G}_{n}\neq \emptyset \) for each \(n\in \mathbb{N}\). Select a set \(\mathbf{G}_{n}\in \mathscr{G}_{n}\) with \(u\neq 0\) σ a.e. on ∂U for all \(u\in \mathbf{G}_{n}\) to derive \(\kappa _{n}\geq \min_{u\in \mathbf{G}_{n}}\varphi (u)>0\).
Next, given \(n\geq 1\), there exists a sequence \(\{u_{n, k} \}\) in \(\mathbf{S}_{1}\) such that \(\varphi (u _{n, k})\to \kappa _{n}\). Using a subsequence if necessary, it implies \(u_{n, k}\rightharpoonup s_{n}\in E^{1,p} (U )\) yet \(u_{n, k}\to s_{n}\in L^{p} (\partial U, d\sigma )\) in view of Lemma 2.1, when \(k\to \infty \). Thus, one deduces that \(\varphi (u_{n, k})\to \varphi (s_{n})=\kappa _{n}\).
Moreover, as linear functionals on \(E^{1,p} (U )\), \(\mathcal{B}_{u_{n, k}}\to 0\) when \(k\to \infty \). First, by definition of \(\kappa _{n}\), one can find a set \(\tilde{\mathbf{G}}_{n}\in \mathscr{G}_{n}\), with \(\kappa _{n}-\epsilon \leq \varphi (u)\leq \kappa _{n}+\epsilon \), for each \(u\in \tilde{\mathbf{G}}_{n}\) and some suitably small \(\epsilon \in (0,\frac{\kappa _{n}}{4} )\); now, if we suppose on the contrary \(\Vert \mathcal{B}_{u} \Vert _{*}>\varsigma >0\) uniformly on \(\{u\in \mathbf{S}_{1}:\frac{ \kappa _{n}}{2}\leq \varphi (u)\leq \frac{3\kappa _{n}}{2} \}\), Proposition 4.2 provides us with a continuous, odd map \(H_{\epsilon }\) on \(\tilde{\mathbf{G}}_{n}\) such that \(H_{\epsilon } (\tilde{\mathbf{G}}_{n} )\in \mathscr{G}_{n}\) and \(H_{\epsilon } (\tilde{\mathbf{G}}_{n} )\subseteq \varphi _{\kappa _{n}+\epsilon }\)—a contradiction thus is arrived at.
Theorem 4.4
Proof
The conclusion (4.14) follows if we can show that \(\lim_{n\to \infty }\kappa _{n}=0\).
Now, define \(\tilde{\kappa }_{n}:=\sup_{\mathbf{G}\in \mathscr{G}_{n}}\min_{u\in \mathbf{G}\cap \mathcal{E}^{c}_{n}}\varphi (u)\) to give \(\tilde{\kappa }_{n}\geq \kappa _{n}>0\). Then one proves \(\lim_{n\to \infty }\tilde{\kappa }_{n}=0\). Actually, if not, there is a constant \(\varepsilon >0\) such that \(\tilde{\kappa }_{n}\geq \varepsilon \) for all \(n\geq 1\). Thus, a set \(\breve{\mathbf{G}}_{n} \in \mathscr{G}_{n}\) exists such that \(\tilde{\kappa }_{n}\geq \min_{u\in \breve{\mathbf{G}}_{n}\cap \mathcal{E}^{c}_{n}} \varphi (u)\geq \frac{\varepsilon }{2}>0\) for each \(n\in \mathbb{N}\), so that we find a sequence \(\{u_{n} \}\), with \(u_{n} \in \breve{\mathbf{G}}_{n}\cap \mathcal{E}^{c}_{n}\), satisfying \(\varphi (u_{n})\geq \frac{\varepsilon }{2}\) uniformly. Keep in mind \(\breve{\mathbf{G}}_{n}\varsubsetneq \mathbf{S}_{1}\); from (4.15) and resorting to a subsequence if necessary, one has \(u_{n}\to 0\in L^{p} (\partial U, \,d\sigma )\), and thus \(\varphi (u_{n})\to 0\), as \(n\to \infty \). A contradiction follows and thereby one finishes the proof. □
Declarations
Acknowledgements
The authors would like to thank the referee for his/her careful reading and valuable suggestions. Supported by the Natural Science Foundation of China (#11571197) and the Science Foundation of Qufu Normal University of China (XJ201112).
Availability of data and materials
Data sharing not applicable to this article as no data-sets were generated or analyzed during the current study.
Funding
Supported by the Natural Science Foundation of China (11571197) and the Science Foundation of Qufu Normal University of China (XJ201112).
Authors’ contributions
JXM conceived of the study, carried out the main studies and drafted the manuscript. ZQZ participated in the design of the study and performed the theory analysis. AXQ participated in the study and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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