 Research
 Open Access
 Published:
Laplace’s equation with concave and convex boundary nonlinearities on an exterior region
Boundary Value Problems volume 2019, Article number: 51 (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 \(\frac{\partial u}{\partial \nu }=\lambda \vert u \vert ^{q2}u+\mu \vert u \vert ^{p2}u\) 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 pLaplacian equation \(\Delta _{p} u=0\), the exterior pharmonic Steklov eigenvalue problems are described.
Introduction
This paper discusses the existence of infinitely many harmonic functions in an exterior region \(U\varsubsetneq {\mathbb{R}}^{N}\) when \(N\geq 3\), subject to a nonlinear boundary condition on ∂U that combines concave and convex terms with \(1< q<2<p<2_{*}\), described as below
in the space \(E^{1} (U )\) of functions where \(u\in L^{2^{*}} (U )\) and \(\nabla u\in L^{2} (U;{\mathbb{R}}^{N} )\). Here, \(2^{*}:=\frac{2N}{N2}\) is the critical Sobolev index and \(\nabla u:= (D_{1}u,D_{2}u,\ldots ,D_{N}u )\) is the weak gradient of u.
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.
One may want to notice here that the prototypical problem
has originally been investigated by Ambrosetti, Brézis and Cerami [1] in 1994 and then in 1995 by Bartsch and Willem [7], in the function space \(H_{0}^{1} (\varOmega )\) on a bounded region Ω with a smooth boundary ∂Ω. Since then, there have been a large number of papers appearing on some related problems; nevertheless, the description of the existence of solutions to problem (1.1) is missing as a reasonable decomposition result of the associated Hilbert function space is required for application of the dual fountain theorem, as discussed in [7, 28].
The aim of this paper is to solve problem (1.1) using a recent decomposition result by Auchmuty and Han [3, 12]. To state our result, we first define the energy functional
Here, dx is the Lebesgue volume element of \({\mathbb{R}}^{N}\) while dσ is the Hausdorff \((N1 )\)dimensional surface element of ∂U. The main result of this paper is described as below.
Theorem 1.1
Assume condition (B1) holds and \(1< q<2<p<2_{*}\).

(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 pharmonic 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 pharmonic 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 pharmonic 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 [18,19,20], Guan, Zhao and Lin [10, 11], Zhang [30], Zhang, Liu and Wu [31, 32].
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, realvalued, 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 \(p^{*}:=\frac{Np}{Np}\) when \(1< p< N\).
The gradient \(L^{p}\)norm provides a norm to guarantee \(E^{1,p} (U )\) a Banach function space—that is, \(E^{1,p} (U )\) is a Banach function space with respect to the norm
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.
When \(N\geq 3\) and \(p=2\), they instead used the notation \(E^{1} (U )\) to denote the associated Hilbert function space with respect to the gradient \(L^{2}\)inner product
whose norm is thus written as \(\Vert u \Vert _{\nabla }\). In addition, one has the direct sum
where \(\mathscr{H} (U )\) denotes the Hilbert subspace of \(E^{1} (U )\) of all functions u satisfying
and \(E_{0}^{1} (U )\) denotes the closure of \(C_{c}^{1} (U )\) with respect to this ∇norm. Here, \(C_{c}^{1} (U )\) is the set of functions that are continuously differentiable and have compact support in U.
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 \(p^{*}:=\frac{Np}{Np}\) 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 \(p_{*}:= \frac{ (N1 )p}{Np}\) 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 \(g\in \mathfrak{G}\).
Now, let \(\mathcal{X}\) be a Banach space with \(\mathcal{X}=\overline{ \bigoplus_{j\in \mathbb{N}}\mathcal{X}(j)}\) and write, for each \(k\in \mathbb{N}\),
Then one has the following results: the fountain theorem and the dual fountain theorem.
Theorem 2.2
([1])
Let \(\mathfrak{G}\) be a compact group, \(\mathcal{X}=\overline{ \bigoplus_{j\in \mathbb{N}}\mathcal{X}(j)}\) a Banach space with norm \(\Vert \cdot \Vert \), and \(\psi \in C^{1} (\mathcal{X}, \mathbb{R} )\) an invariant functional; for each \(k\in \mathbb{N}\), let \(\mathcal{Y}_{k}\), \(\mathcal{Z}_{k}\) be defined as in (2.4), and \(\rho _{k}>\varrho _{k}>0\) some constants. For every \(k\geq k_{0}\) with a fixed \(k_{0}\in \mathbb{N}\), we also assume

(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 \(c\in (0,\infty )\).
Then ψ has a sequence of critical values \(\{ \mathfrak{u} _{k} \}\) with \(\psi ( \mathfrak{u}_{k} )>0\) and \(\psi ( \mathfrak{u}_{k} )\to \infty \) when \(k\to \infty \).
Theorem 2.3
[1] Under the hypotheses of Theorem 2.2, suppose again that condition (a1) holds. For every \(k\geq k_{1}\) with a fixed \(k_{1}\in \mathbb{N}\), we also assume

(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 \(c\in [\tilde{\mathfrak{c}} _{k_{1}},0 )\).
Then ψ has a sequence of critical values \(\{ \mathfrak{v} _{k} \}\) with \(\psi ( \mathfrak{v}_{k} )<0\) and \(\psi ( \mathfrak{v}_{k} )\to 0^{}\) when \(k\to \infty \).
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].
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.
To further simplify notations, set \(\Vert \cdot \Vert := \Vert \cdot \Vert _{\nabla }\) and \(\Vert \cdot \Vert _{s}:= \Vert \cdot \Vert _{s, \partial U}\) in this section. Take a nonzero \(u\in \mathcal{Y}_{k}\), and use tu and (1.2) to derive, for some sufficiently large \(t>0\),
in view of the fact that the space \(\mathcal{Y}_{k}\) is of finite dimension—so that all norms are equivalent. As such, condition (a2) is satisfied for every \(\rho _{k}\geq t \Vert u \Vert >0\) when \(\mu >0\).
Next, define
Then one observes \(\alpha _{k}\to 0\) when \(k\to \infty \). Actually, it is readily seen that \(0<\alpha _{k+1}\leq \alpha _{k}\), so that \(\alpha _{k}\to \alpha \geq 0\) as \(k\to \infty \). By hypotheses, there exists a \(u_{k}\in \mathcal{Z}_{k}\) satisfying \(\Vert u_{k} \Vert =1\) and \(\Vert u_{k} \Vert _{p}\geq \alpha _{k}/2\) for each k; by definition of \(\mathcal{Z} _{k}\), one sees \(u_{k}\rightharpoonup 0\), i.e., \(u_{k}\) converges weakly to 0, in \(\mathscr{H} (U )\). Lemma 2.1 then yields \(u_{k_{l}}\to 0\) in \(L^{p} (\partial U, \,d\sigma )\) as \(l\to \infty \), for a subsequence \(\{u_{k_{l}} \}\) of \(\{u_{k} \}\). That is, \(\alpha =0\). On each subspace \(\mathcal{Z}_{k}\) with a sufficiently large norm, we have
where \(c_{1}>0\) is such a constant that \(\Vert u \Vert _{q}\leq c_{1} \Vert u \Vert \) for all \(u\in E^{1} (U )\). Take \(\varrho _{k}:= (\mu \alpha ^{p}_{k} )^{\frac{1}{p2}}\). Then, via the property of \(\alpha _{k}\), there exists a \(k_{0}\in \mathbb{N}\) such that one can always choose \(\rho _{k}=2\varrho _{k}\) in the foregoing estimate (3.1) for every \(k\geq k_{0}\); in addition, for (3.3), one derives
when \(k\to \infty \). As a consequence, condition (a3) is ensured.
Finally, take a \((PS)_{c}\)sequence \(\{u_{k} \}\) for the functional φ at level \(c>0\) in \(\mathscr{H} (U )\); that is, \(\varphi (u_{k})\to c\) yet \(\varphi ^{\prime }(u_{k})\to 0\) as \(k\to \infty \). Then one has, for k sufficiently large,
from which one deduces immediately that
As a result, these \(u_{k}\) are bounded, and thus, without loss of generality, converge weakly to a function \(u\in \mathscr{H} (U )\); via result 2.1 again, we may simply suppose that \(u_{k}\to u\) in \(L^{p} (\partial U, \,d\sigma )\) and \(L^{q} (\partial U, \,d\sigma )\) when \(k\to \infty \); besides, using (1.2), a routine calculation leads to
when \(k\to \infty \). Thus, condition (a4) is also derived so that part (a) of Theorem 1.1 is proved.
On the other hand, apply a parallel idea as shown in (3.1) to prove that condition (b1) is satisfied for every \(0<\varrho _{k}\leq t \Vert u \Vert \), with a nonzero \(u\in \mathcal{Y}_{k}\) and some sufficiently small \(t>0\), when \(\lambda >0\)—since \(\mathcal{Y}_{k}\) is of finite dimension. Next, define
Similarly, one observes that \(\beta _{k}\to 0\) when \(k\to \infty \) again from Lemma 2.1. On each subspace \(\mathcal{Z}_{k}\) with a sufficiently small norm, we have
where \(c_{2}>0\) is such a constant that \(\Vert u \Vert _{p}\leq c_{2} \Vert u \Vert \) for any \(u\in E^{1} (U )\). Take \(\rho _{k}:= (\frac{4\lambda \beta ^{q}_{k}}{q} )^{\frac{1}{2q}}\) to derive \(\varphi (u) \geq 0\). Via the property of \(\beta _{k}\), there exists a \(k_{1}\in \mathbb{N}\) such that we may select the above \(\varrho _{k}=\frac{\rho _{k}}{2}\) to be sufficiently small for all \(k\geq k_{1}\). As such, condition (b2) is ensured; also, condition (b3) follows in view of the fact \(\rho _{k}\to 0\) as \(k\to \infty \). Finally, take a \((PS)^{*}_{c}\)sequence \(\{\tilde{u}_{k_{l}} \}\), with \(\tilde{u}_{k_{l}}\) in \(\mathcal{Y}_{k_{l}}\), for φ at level \(c\in [\tilde{\mathfrak{c}}_{k_{1}},0 )\); that is, \(\varphi (\tilde{u}_{k_{l}})\to c\) whereas \(\varphi ^{\prime }_{ \mathcal{Y}_{k_{l}}}(\tilde{u}_{k_{l}})\to 0\) as \(l\to \infty \). So, one infers \(\frac{1}{p} \vert \varphi ^{\prime }(\tilde{u}_{k_{l}})( \tilde{u}_{k_{l}}) \vert \leq \Vert \tilde{u}_{k_{l}} \Vert \) for sufficiently large l. Thus, (3.6) holds again. As a result, these \(\tilde{u}_{k_{l}}\) are bounded, and thus converge weakly without loss of generality to a function \(\tilde{u}\in \mathscr{H} (U )\); via Lemma 2.1 again, we may simply assume that \(\tilde{u}_{k_{l}}\to \tilde{u}\) in \(L^{p} (\partial U, \,d\sigma )\) and \(L^{q} (\partial U, \,d\sigma )\) as \(l\to \infty \); by use of (1.2) again, one observes, just like (3.7), \(\Vert \tilde{u}_{k_{l}}\tilde{u} \Vert \to 0\) when \(l\to \infty \). Thus, condition (b4) is also derived so that part (b) of Theorem 1.1 is proved.
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
Assume condition (B1) holds and \(1< q<2<p<2_{*}\).

(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
Take \(u\in \mathscr{H} (U )\) to be such that \(\varphi (u) \geq 0\) and \(\varphi ^{\prime }(u)=0\). Then one has
When \(\mu \leq 0\), then \(u=0\) follows immediately. Accordingly, we need to assume \(\mu >0\) in general. In this case, we correspondingly have
from which one can deduce that
when \(\mu \to 0^{+}\). This finishes the proof for part (a) of Proposition 3.1.
In addition, let \(v\in \mathscr{H} (U )\) be such that \(\varphi (v)\leq 0\) and \(\varphi ^{\prime }(v)=0\). Similarly, one has
When \(\lambda \leq 0\), then \(v=0\) follows immediately. Accordingly, we need to assume \(\lambda >0\) in general. In this case, we correspondingly have
from which one finds readily that
when \(\lambda \to 0^{+}\). This finishes the proof for part (b) of Proposition 3.1. □
Finally, we consider the following analogous problem:
in the standard Hilbert–Sobolev space \(H^{1} (U )\), where all \(u\in H^{1} (U )\) satisfy \(u, \vert \nabla u \vert \in L^{2} (U )\), and we define the associated energy functional
Note that in view of some result in [2], we have the following direct sum:
where \(\mathscr{N} (U )\) is the Hilbert subspace of \(H^{1} (U )\) of all functions u satisfying
and \(H_{0}^{1} (U )\) is the closure of \(C_{c}^{1} (U )\) with respect to the standard \(H^{1}\)norm. Applying a similar procedure to the proof of Theorem 1.1, we can obtain the following result.
Theorem 3.2
Assume condition (B1) holds and \(1< q<2<p<2_{*}\).

(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 \(k\to \infty \).
pLaplacian 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 pharmonic 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].
Recall the pLaplacian is defined as \(\Delta _{p} u:=\operatorname{div} ( \vert \nabla u \vert ^{p2}\nabla u )\). The exterior pharmonic Steklove eigenvalue problems are to seek weak solutions of the problem below
in \(E^{1,p} (U )\). This problem has been well developed on bounded regions over a century since Stekloff [23, 24], yet only been investigated recently in [3,4,5, 12, 13, 15].
Take \(\Vert \cdot \Vert := \Vert \cdot \Vert _{p, \nabla }\) in this section. Define two functionals on \(E^{1,p} (U )\) as
Accordingly, given \(u\in E^{1,p} (U )\), write two linear functionals on \(E^{1,p} (U )\) to be
Since \(E^{1,p} (U )\) is a reflexive, uniformly convex Banach space when \(p>1\) (see [4]), there exists a unique element, say, \(u_{\mathcal{B}}\) in \(E^{1,p} (U )\), from Riesz’s theorem, such that
therefore, one finds a homeomorphism \(\mathfrak{H}:E^{1,p} (U ) \to E^{1,p} (U )\) such as \(\mathfrak{H}(u):=u_{\mathcal{B}}\). Noticing \(\mathfrak{H}\) is odd, and bounded, uniformly continuous on the set \(\mathbf{S_{1}}:= \{u\in E^{1,p} (U ): \Vert u \Vert =1 \}\), there are constants \(t_{0},\upsilon _{1},\upsilon _{0}>0\) such that, for all \(t\in [t_{0},t_{0} ]\) and \(u\in \mathbf{S}_{1}\), one has
Now, define a flow \(H:\mathbf{S}_{1}\times [t_{0},t_{0} ] \to \mathbf{S}_{1}\) by
Then H is odd in u, with \(H(u,0)=u\), uniformly continuous and verifies the property below.
Lemma 4.1
There exists a map \(\ell (u,t):\mathbf{S}_{1}\times (t_{0},t _{0} )\to \mathbb{R}\) such that \(\ell (u,t)\to 0\) as \(t\to 0\), uniformly on \(\mathbf{S}_{1}\), and, for each \(u\in \mathbf{S}_{1}\) and \(t\in (t_{0},t_{0} )\), we have
Proof
Since
we can define, in view of (4.4),
and we have \(\ell (u,0)=0\) uniformly on \(\mathbf{S}_{1}\) via (4.5). Actually, a routine computation leads to
from which one deduces easily that, noticing that \(\Vert u \Vert =1\) on \(\mathbf{S}_{1}\),
this further implies that, remembering the fact that \(H(u,0)=u\) now,
which together with (4.3) and (4.9) gives the desired result as ℓ is bounded by (4.5). □
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
As \(\ell (u,0)=0\) uniformly on \(\mathbf{S}_{1}\), we can choose \(t_{1}\in (0,t_{0} )\) such that \(\vert \ell (u,t) \vert \leq \varsigma ^{2}/2\) for each \(u\in \mathbf{G}\varsubsetneq \mathbf{S}_{1}\) and \(t\in [t_{1},t_{1} ]\) since G is compact. Write \(\epsilon :=\min \{\tau , \varsigma ^{2}t_{1}/4 \}\). Then (4.4) and (4.7) implies that, for every \(u\in \texttt {V}_{\epsilon }\cap \mathbf{G} \subseteq \texttt {V}_{\tau }\), \(\varphi (H(u,t_{1}))\geq \kappa  \epsilon +\frac{\varsigma ^{2}t_{1}}{2}\geq \kappa +\epsilon \); that is, \(H(u,t_{1})\in \varphi _{\kappa +\epsilon }\) for all \(u\in \texttt {V} _{\epsilon }\cap \mathbf{G}\). As such, we define an odd map via (4.6) that is continuous on \(\texttt {V}_{\epsilon }\cap \mathbf{G}\) by (because φ is even and \(\mathbf{G}= \mathbf{G}\) so that \(\texttt {V}_{\epsilon }\cap \mathbf{G}\) again is symmetric)
and have \(H_{\epsilon }(\texttt {V}_{\epsilon }\cap \mathbf{G})\subseteq \varphi _{\kappa +\epsilon }\), as claimed. This in turn completes the proof. □
Next, define
Here, \(\mathscr{G}_{n}:= \{\mathbf{G}\varsubsetneq \mathbf{S} _{1}:\mathbf{G}\mbox{ compact, } \mathbf{G}=\mathbf{G} \mbox{ and } \gamma (\mathbf{G})\leq n \}\), with γ being the Krasnoselskii genus (see, for example, section ii.5 of [25] for more detailed descriptions).
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.
Since \(\mathcal{B}_{u_{n, k}}\to 0\) when \(k\to \infty \), one infers that, in view of (4.2) and (4.3),
where \(\mathcal{F}(v):=\frac{1}{p \kappa _{n}}\int _{\partial U} \vert s _{n} \vert ^{p2}s_{n}v \,d\sigma \) is a linear functional on \(E^{1,p} (U )\). Using Lemma 2.1 again, plus \(u_{n, k}\rightharpoonup s_{n}\in E^{1,p} (U )\) yet \(u_{n, k}\to s_{n}\in L^{p} (\partial U, \,d\sigma )\) as \(k\to \infty \), we obtain
Here, \(\delta _{n}:=\frac{1}{p \kappa _{n}}>0\). As such, \(s_{n}\in E ^{1,p} (U )\) is a weak solution of problem (4.1). □
Theorem 4.4
Define \(\kappa _{n}\) as in (4.12) and \(\delta _{n}\) by \(\frac{1}{p \kappa _{n}}\) for each \(n\in \mathbb{N}\). Then one has
Proof
The conclusion (4.14) follows if we can show that \(\lim_{n\to \infty }\kappa _{n}=0\).
For each \(n\geq 1\), choose \(\mathcal{E}_{n}\) to be a linear subspace of \(E^{1,p} (U )\) of dimension n such that \(u\neq 0\) σ a.e. on ∂U for every \(u\in \mathcal{E} _{n}\), and denote its complement in \(E^{1,p} (U )\) by \(\mathcal{E}^{c}_{n}\). Without loss of generality, assume further that \(\mathcal{E}_{1}\varsubsetneq \mathcal{E}_{2}\varsubsetneq \cdots \varsubsetneq \mathcal{E}_{n}\varsubsetneq \cdots \varsubsetneq E^{1,p} (U )\). Note that (4.13) guarantees our choice as clearly \(s_{n}_{\partial U}\neq 0\) σ a.e. on ∂U for all \(n\in \mathbb{N}\). Also, we have
where \(E^{1,p}_{0} (U )\) denotes the subspace of \(E^{1,p} (U )\) that is the closure of \(C_{c}^{1} (U )\) with respect to the gradient \(L^{p}\)norm (2.1) and the notation \(\dot{\cup }\) means disjoint union.
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. □
Finally, it is worth to mention here that the problem
can be studied analogically in the space \(W^{1,p} (U )\) by use of some results in [2, 5].
References
 1.
Ambrosetti, A., Brézis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)
 2.
Auchmuty, G., Han, Q.: Spectral representations of solutions of linear elliptic equations on exterior regions. J. Math. Anal. Appl. 398, 1–10 (2013)
 3.
Auchmuty, G., Han, Q.: Representations of solutions of Laplacian boundary value problems on exterior regions. Appl. Math. Optim. 69, 21–45 (2014)
 4.
Auchmuty, G., Han, Q.: pLaplacian boundary value problems on exterior regions. J. Math. Anal. Appl. 417, 260–271 (2014)
 5.
Auchmuty, G., Han, Q.: Wellposedness and approximation of solutions of linear divergenceform elliptic problems on exterior regions. Math. Methods Appl. Sci. 38, 1867–1875 (2015)
 6.
Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993)
 7.
Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995)
 8.
Du, X., Mao, A.: Existence and multiplicity of nontrivial solutions for a class of semilinear fractional Schrödinger equations. J. Funct. Spaces 2017, Article ID 3793872 (2017)
 9.
Feng, M., Li, P., Sun, S.: Symmetric positive solutions for fourthorder ndimensional mLaplace systems. Bound. Value Probl. 2018, Article ID 63 (2018)
 10.
Guan, Y., Zhao, Z., Lin, X.: On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques. Bound. Value Probl. 2016, Article ID 141 (2016)
 11.
Guan, Y., Zhao, Z., Lin, X.: On the existence of solutions for impulsive fractional differential equations. Adv. Math. Phys. 2017, Article ID 1207456 (2017)
 12.
Han, Q.: Positive solutions of Laplace’s equation involving both critical Sobolev nonlinearities on exterior regions. Monatshefte Math. 176, 107–141 (2015)
 13.
Han, Q.: Addendum to: Positive solutions of Laplace’s equation involving both critical Sobolev nonlinearities on exterior regions. Monatshefte Math. 177, 325–327 (2015)
 14.
Han, Q.: Infinitely many positive harmonic functions with an oscillating boundary condition on exterior regions. Complex Var. Elliptic Equ. 60, 1106–1113 (2015)
 15.
Han, Q.: On the first exterior pharmonic Steklov eigenvalue. J. Math. Anal. Appl. 434, 1182–1193 (2016)
 16.
Han, Q.: Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation. Proc. R. Soc. Edinb., Sect. A 146, 693–721 (2016)
 17.
Han, Q.: Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations. Bull. Sci. Math. 141, 46–71 (2017)
 18.
Mao, J., Zhao, Z.: On existence and uniqueness of positive solutions for integral boundary boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 16 (2010)
 19.
Mao, J., Zhao, Z.: The existence and uniqueness of positive solutions for integral boundary value problems. Bull. Malays. Math. Sci. Soc. 34, 153–164 (2011)
 20.
Mao, J., Zhao, Z., Wang, C.: The exact iterative solution of fractional differential equation with nonlocal boundary value conditions. J. Funct. Spaces 2018, Article ID 8346398 (2018)
 21.
Phung, P.D., Minh, H.B.: Existence of solutions to fractional boundary value problems at resonance in Hilbert spaces. Bound. Value Probl. 2017, Article ID 105 (2017)
 22.
Polidoro, S., Ragusa, M.A.: Hölder regularity for solutions of ultraparabolic equations in divergence form. Potential Anal. 14, 341–350 (2001)
 23.
Stekloff, W.: Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Éc. Norm. Supér. 19, 191–259 (1902)
 24.
Stekloff, W.: Sur les problèmes fondamentaux de la physique mathématique ((suite et fin)). Ann. Sci. Éc. Norm. Supér. 19, 455–490 (1902)
 25.
Struwe, M.: Variational Methods, 4th edn. Springer, Berlin (2008)
 26.
Sun, F., Liu, L., Wu, Y.: Infinitely many signchanging solutions for a class of biharmonic equation with pLaplacian and Neumann boundary condition. Appl. Math. Lett. 73, 128–135 (2017)
 27.
Torné, O.: Steklov problem with an indefinite weight for the pLaplacian. Electron. J. Differ. Equ. 2005, 87 (2005)
 28.
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
 29.
Yan, S., Yang, J.: Fountain theorem over cones and applications. Acta Math. Sci. Ser. B Engl. Ed. 30, 1881–1888 (2010)
 30.
Zhang, K.: On signchanging solution for some fractional differential equations. Bound. Value Probl. 2017, Article ID 59 (2017)
 31.
Zhang, X., Liu, L., Wu, H., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015)
 32.
Zhang, X., Liu, L., Wu, Y.: Variational structure and multiple solutions for a fractional advectiondispersion equation. Comput. Math. Appl. 68, 1794–1805 (2014)
 33.
Zou, W.: Variant fountain theorems and their applications. Manuscr. Math. 104, 343–358 (2001)
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 datasets 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).
Author information
Affiliations
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.
Corresponding author
Correspondence to Jinxiu Mao.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Abbreviations
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Received
Accepted
Published
DOI
MSC
 35J20
 35J65
 46E22
 49R99
Keywords
 Exterior regions
 Laplace operator
 Concave and convex mixed nonlinear boundary conditions
 Fountain theorems
 Steklov eigenvalue problems