Abstract
The existence of a positive solution for a class of asymptotically linear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.
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1 Introduction
In this paper, we look for a positive solution for the following class of elliptic problems
where \(\Omega \) is an exterior domain, i.e., a unbounded smooth domain in \(\mathbb {R}^{N}\), \(N\ge 2\), whose complement \(\mathbb {R}^{N}{\setminus }\Omega \) is bounded and not empty. The nonlinearity f is super-linear at zero and asymptotically linear at infinity, and the model example we have in mind is
This kind of equations arises in the study of the propagation of a light beam in saturable media, as photorefractive crystals, which are largely exploited in experimental observations because they require low optical power to exhibit very strong nonlinear effects [3, 26, 32, 35]. When the medium has some additional properties [2], the nonlinearity given in (1.2) is approximated by putting \(s=0\) and obtaining \(f(t)=t^{3}\), that is the more treated nonlinearity, also associated with the study of the propagation of a light beam in the so-called Kerr media [3]. From a mathematical point of view, this kind of problems has been extensively studied in the last thirty years, especially when \(f(t)=|t|^{p-2}t\), and \(p>2\) is subcritical. The main difficulty relies in the lack of compactness that can be overcome if \(\Omega \) is radially symmetric, and one looks for radially symmetric solutions [10, 22]. While, if \(\Omega \) has no symmetry properties, the lack of compactness becomes prevalent, moreover, it has been observed in [9] that there are no least action solutions, so that a higher action level solution has to be searched. As a consequence, a careful analysis of a general Palais–Smale sequence, even not minimizing, is needed and this has been done in [9] introducing the nowadays so-called splitting lemma. This analysis permits to locate the action levels where the Palais–Smale condition holds so that the difficulty becomes to construct a minimax level where this compactness property is satisfied. In [9], suitable subspaces of the \(L^{p}(\Omega )\) sphere have been considered, with additional prescriptions, in order to increase the minimax level. This approach led to the existence of a constrained minimization point, which, thanks to the homogeneity property of \(f(t)=|t|^{p-2}t\), gives a positive solution when \(\mathbb {R}^{N}{\setminus } \Omega \) has small diameter. This kind of argument has been developed in several directions: in [7], the assumption on the size of \(\mathbb {R}^{N}{\setminus } \Omega \) has been removed; in [17, 18], multiplicity results have been obtained, in dependence on \(\lambda \) and on the topology of \(\Omega \) (see also the interesting survey [14]); in [15, 20], sign-changing solutions have been found.
When f has not a polynomial dependence with respect to t, there are only a few contributions in the literature. In particular, in the case in which f is asymptotically linear, in [19] the existence of a solution is proved for f depending on the spatial variable and satisfying some assumptions corresponding, in the autonomous case, to a convexity property. More recently, Problem (1.1) has been studied in [24] where it is found a positive solution under some suitable hypotheses on f, among which is again the convexity for \(t\in (0,+\infty )\), and in [27], where it is proved the existence of a radial positive solution in the exterior of a ball.
Notice that f given in (1.2) is not convex so that the result in [19, 24] does not cover the physical meaningful case. Moreover, we do not suppose any symmetry property on the domain so that there is no reason to expect a radially symmetric solution. With this respect, our result in the model case (1.2) is related to the interest in the physical context of asymmetric solutions that are observed in the interaction between the crystal and the spatial solitons or in the collision of symmetric solitons [3, 26].
Our existence result in the model case can be stated as follows.
Theorem 1.1
Assume \(N\ge 2\) and that f is given in (1.2). Then, Problem (1.1) has at least a positive solution.
This theorem will follow as a consequence of a more general Theorem 2.2, proved for a general nonlinearity f(t) (see Sect. 2).
As in the super-linear case, when f(t) is given in (1.2), one expects that the least action level of the associated functional \(I_{\Omega }:H^{1}_{0}(\Omega )\mapsto \mathbb {R}\)
is equal to the least action level of the functional associated with the problem in the whole \(\mathbb {R}^{N}\)
which is shown to be attained by a unique, positive, radially symmetric solution [10, 11, 34]. But, when dealing with asymptotically linear f(t), even in the model case (1.2), one loses the precious homogeneity property, crucial in order to reabsorb the Lagrange multiplier when finding the least action solution as a constrained minimum point on a \(L^{p}(\Omega )\) sphere. So that, working on \(L^{p}(\Omega )\) spheres is not suitable in this context. In the light of these considerations, we will work on the well-known Nehari manifolds associated with the functionals \(I_{\Omega }\) and I [29, 30]
realizing the least action levels as minimum values of \(I_{\Omega }\) and I on these manifolds
This approach yields some new difficulties and some new advantages; concerning the formers, notice that not every function in \(H^{1}_{0}(\Omega )\) can be projected on \({\mathcal N}_{\Omega }\), so that one has to be careful when defining the projection of a function on \({\mathcal N}_{\Omega }\). But, thanks to a monotonicity condition of f ((2.3)), the projection turns out to be unique when it exists. With this approach, we will show that \(m_{\Omega }=m\) so that a higher-energy critical level has to be searched also in this case. On the other hand, we have some benefits when looking for a compactness property; indeed, recall that, usually when dealing with asymptotically linear nonlinearity, the suitable notion of compactness property is the Cerami condition introduced in [13]; here, strongly exploiting the features of \({\mathcal N}_{\Omega }\), we will be able to show that the usual Palais–Smale condition on \({\mathcal N}_{\Omega }\) holds. However, in doing this, we will again have to handle the lack of homogeneity of f.
Finally, we will be able to choose linking sets as proper subsets of \(\mathcal N_{\Omega }\). At this stage, it will be important to show proper estimates of \(I_{\Omega }\) on these subsets. This will be done by introducing a suitable abstract asymptotic threshold (in Lemma 3.2) that will be crucial in order to compare all the terms in the functional \(I_{\Omega }\) (defined in (1.3)). This threshold generalizes the commonly used one in the context of polynomial nonlinearity. Let us observe that, in order to highlight the main novelties and difficulties one encounters when facing this kind of problems, we have preferred to study the existence of a solution, but we believe that with the same tools introduced here, it could be possible to face the question of multiplicity of solutions as done in [16, 18] for the polynomial case. Moreover, our argument can be used to prove an analogous result for a general, not necessarily homogeneous, super-linear f (see for more details Remark 2.4).
Concluding this introduction, let us point out that the use of the constraint of Nehari manifold has extensively been used in the related context of the problem in the whole space with the presence of a potential (see, for instance, [5, 38, 39] and the references therein). In particular, in [23, 38] the properties of a generalized Nehari manifold are deeply exploited in order to study indefinite problems; here, we choose to assume all the regularity conditions needed to use the classical Nehari manifold, but it would be interesting to use the analysis of [23, 38] to weaken our regularity assumptions on the nonlinearity.
The paper is organized as follows: in Sect. 2, we settle our general context and state our main existence result (Theorem 2.2); in Sect. 3, we prove some useful technical propositions, while in Sect. 4, we deal with the compactness property. Finally, in Sect. 5 we conclude the proofs of Theorems 2.2 and 1.1.
2 Setting of the problem and main results
We will work in \(H^{1}_{0}(\Omega )\) and in \(H^{1}(\mathbb {R}^{N})\). The norms on these spaces will be denoted, respectively, by
where \(\Vert \cdot \Vert _{p.\Omega }\) (\(\Vert \cdot \Vert _{p}\)) is the usual norm in \(L^{p}(\Omega )\) (in \(L^{p}(\mathbb {R}^{N})\)).
Every solution of (1.1) is a critical point of the \(C^{1}\) functional \(I_\Omega \), defined in (1.3). We have introduced the model example in (1.2); however, in general, we will assume that f has the following properties
there exists a positive constant D and \(p_{1}\le p_{2}\in (1,\bar{p})\) where \(\bar{p}=+\infty \) if \(N=2\) and \(\bar{p}= (N+2)/N-2\) if \(N\ge 3\), such that
The following assumption will be crucial in all our arguments:
Moreover, we suppose that f is asymptotically linear at infinity; namely, we assume that there exists a positive number \(l_{\infty }\) such that
where the second inequality is assumed in order to have a solution of the problem in the whole \(\mathbb {R}^{N}\) (see [10, 11, 36]). When dealing with this kind of nonlinearities, it is often assumed a non-quadraticity-type condition (introduced in [21])
Finally, in order to let the Nehari manifold be a natural constraint, we will assume that
Remark 2.1
Notice that all the previous hypotheses are satisfied in the model case (1.2). Moreover, note that, as a consequence of (2.3) and (2.4), the general nonlinearity f(t) is always below the line \(l_{\infty } t\), for every \(t>0\).
In order to prove our existence result, we will make a comparison with the following problem in the whole \(\mathbb {R}^{N}\) which will be called problem at infinity
The problem (2.7) has a positive, radially symmetric, least action solution (whose existence is proved in [10, 36] for \(N\ge 3\) and in [11] for \(N= 2\).), which we denote with w. In [34], it is shown the uniqueness of w when f satisfies the additional hypothesis
where b is the (unique thanks to (2.3)) positive number given by \(f(b)=\lambda b\). Notice that this condition is satisfied in the case of (1.2). Since we will also deal with a more general nonlinearity and will not assume this hypothesis, we will suppose that
Our main result is the following.
Theorem 2.2
Assume \(N\ge 2\) and hypotheses (2.1), (2.2), (2.3), (2.4), (2.5), (2.6) and (U). Then Problem (1.1) has at least a positive solution.
Remark 2.3
In [19], Problem (1.1) is studied for a non-autonomous f satisfying some suitable hypotheses including one that becomes, in the autonomous case, a convexity condition. In [24], Problem (1.1) is studied assuming (2.1), (2.3), (2.4), (2.5), (2.6), (U) plus other conditions among which it is assumed that f is a convex function. Note that f given in (1.2) is not convex, so that the existence of a positive solution in exterior domains for this kind of nonlinearity cannot be deduced from the result of [19, 24]. Obviously, our result applies also to nonlinearities that have infinitely many flex points.
Remark 2.4
Our argument can be exploited to deal with super-linear, not necessarily homogeneous, nonlinearities assuming conditions (2.1), (2.2), (2.3), (2.6) and, in place of (2.4) and (2.5), supposing the classical Ambrosetti–Rabinowitz condition
Indeed, also under these hypotheses, it is possible to exploit the properties of the Nehari manifold \({\mathcal N}_{\Omega }\) as explained in [33].
As already said in the introduction, we will work on the Nehari manifold introduced in (1.4). Nowadays, this has become a classical tool in variational methods thanks to its useful features as it has been also highlighted in the recent contribution [31]. The following remark clarifies the role of (2.6) in order to use \({\mathcal N}_{\Omega }\).
Remark 2.5
Let us first observe that \({\mathcal N}_{\Omega }\) is the set of non-trivial zeroes of the function \(N_{\Omega }:H^{1}_{0}(\Omega ){\setminus }\{0\}\mapsto \mathbb {R}\) given by
Notice that \({\mathcal N}_{\Omega }\) is actually a manifold and it is a natural constraint. Namely, for every \(u\in {\mathcal N}_{\Omega }\) it results \( \langle N^{\prime }_{\Omega }(u),u\rangle <0. \) Indeed, consider \(u\in {\mathcal N}_{\Omega }\) and use (2.2) and (2.6) to obtain
Now, suppose that \(u\in {\mathcal N}_{\Omega }\) is a constrained critical point of \(I_{\Omega }\), then there exists a real number \(\mu \) such that \(I_{\Omega }^{\prime }(u)-\mu N_{\Omega }^{\prime }(u)=0\); taking u as test function, one gets \(\mu \langle N_{\Omega }^{\prime }(u),u\rangle =0\) and then yields \(\mu =0\), i.e., u is a free critical point.
3 Asymptotic estimates
In the sequel, when we compute the \(H^{1}(\mathbb {R}^{N})\) norm of a function \(u\in H^{1}_{0}(\Omega )\), it is implicitly assumed that u is extended as zero outside \(\Omega \) so that \(u\in H^{1}(\mathbb {R}^{N})\). Let us introduce a cutoff \(C^\infty \) function \(\xi : \mathbb {R}^N \rightarrow [0,1]\) defined by
with \(\rho \) being the smallest positive number such that \(\mathbb {R}^N \backslash \Omega \subset B_\rho (0)\) and \(\tilde{\xi }: \mathbb {R}^+ \cup \{0\} \rightarrow [0,1]\) being a non-decreasing function such that
Recall that w, the least action positive solution of Problem (2.7), enjoys the following asymptotic behavior (see [10])
We will often use the following lemma proved in Lemma II.2 in [7] (see also Proposition 1.2 in [6]).
Lemma 3.1
Let \(\varphi _{1}\in C(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N}),\,\varphi _{2}\in C(\mathbb {R}^{N})\) satisfy for some \(\alpha ,\,\beta \ge 0\), \(c\in \mathbb {R}\)
then
In what follows, we will use the notation
As a consequence of the previous lemma, it is easy to prove the following result.
Lemma 3.2
Assume (2.2), and let \(x_{0}\in \mathbb {R}^{N}\) with \(|x_{0}|=1\) and \(y\in \partial B_{2}(x_{0})\). Let us define the quantity
Then, \(\varepsilon _{R}\) satisfies
Remark 3.3
Let us observe that in [23], an analogous asymptotic threshold has been introduced in order to deduce precise energy estimates, for an indefinite problem in the whole space with super-linear, but not necessarily homogeneous, nonlinearities.
Proof
The proof can be done following [6] or [7]; indeed, by a change of variable, it results
In order to obtain the conclusion, it is sufficient to apply Lemma 3.1 with \(\varphi _{1}=w,\,\varphi _{2}=f(w)\) and \(\zeta =-R(x_{0}-y)\). Let us check that all the hypotheses are satisfied. Using (3.3), we obtain that \(\varphi _{1}\) satisfies the hypothesis of Lemma 3.1 with \(\alpha =\sqrt{\lambda }\) and \(\beta =(N-1)/2\). Let us see what happens with \(\varphi _{2}\). The first limit in (3.3) and condition (2.2) imply that there exists \(R_{1}\) such that
Then it holds, with C a positive constant
so that the hypotheses of Lemma 3.1 are satisfied since \(p_{i}>1\). \(\square \)
Moreover, for every \(q>1\), for every \(\theta \in \mathbb {R}^{N}{\setminus }\{0\}\) and for every compact set K, Lemma 3.1 implies
Indeed, denote with a the modulus of \(\theta \) and observe that (3.3) implies that there exists \(\rho _{0}\) such that it results
Let us fix \(r_{0}\) such that \(K\subseteq B_{r_{0}}(0)\) and put \(z=x-R\theta \). We have
Moreover, \(|z|\ge Ra-r_{0}\), so that, for every R such that \(Ra-r_{0}>\rho _{0}\), inequality (3.6) implies that there exists a positive constant C such that
Therefore, for every \(q\ge 2\), for every \(\theta \in \mathbb {R}^{N}{\setminus }\{0\}\) and for every compact set K, it results
where the second inequality follows by an analogous argument for \(w^{\prime }\). For every \(\theta \in \mathbb {R}^{N}{\setminus }\{0\}\), let us define the map \( \Phi ^{\theta }:\mathbb {R}^{+} \rightarrow H^1_0(\Omega )\) by
In the sequel, for simplicity, we will use the notation
Lemma 3.4
Let us assume (2.1) and (2.2). Let \(\theta \in \mathbb {R}^{N}{\setminus } \{0\}\). Then, the function \(\Phi ^{R\theta }\) is continuous in R and
Moreover, for every \(\tau \ge 0\) it results
Proof
The continuity with respect to R of \(\Phi ^{R\theta }\) follows from the regularity properties of w. Moreover, taking into account (3.2), (3.4) and using Young inequality, it follows
with C a positive constant. Then, (3.9) follows from (3.7).
In order to show the second conclusion in (3.9), note that (3.2) and (3.7) imply
this and the first conclusion in (3.9) yield the second one.
To prove (3.10), let us observe that \(I_{\Omega }(\Phi ^{y})=I(\Phi ^{y})\). Therefore, from (3.2), (2.2), and using Lagrange mean value theorem, we get that there exists a positive constant \(C_{0}\) such that
Then, (3.10) follows from (3.7) and (3.9). Moreover, taking into account (3.2) and (3.4), and using condition (2.2) we get that there exists a positive constant \(C_{1}\) such that
Then the conclusion follows from (3.7). \(\square \)
As already said, the linking sets will be subsets of \(\mathcal N_{\Omega }\), but, as f is asymptotically linear, not every function of \(H^{1}_{0}(\Omega )\) can be projected on \(\mathcal N_{\Omega }\). The following lemma tells us when this projection can be performed.
Lemma 3.5
Let us assume (2.1), (2.3), (2.4), and define the maps
Then, the maps
are well defined and continuous. In addition, for every \(\theta \in \mathbb {R}^{N}{\setminus }\{0\}\), there exists \(R_{0}=R_{0}(|\theta |)\) such that \(\Phi ^{R\theta }\in A_{\Omega }\cap A\) for every \(R\ge R_{0}\) and
Proof
Let us define \(g_{\Omega }\,: [0,+\infty )\times A_{\Omega }\mapsto \mathbb {R}\) by
Observe that \(g_{\Omega }\) is continuous on \(\mathbb {R}^{+}\times A_{\Omega }\); moreover, hypotheses (2.1), (2.3) and (2.4) allow to apply Lebesgue dominated convergence theorem to obtain
So that \(g_{\Omega }\) is continuous on \([0,+\infty )\times A_{\Omega }\) and \(g_{\Omega }(0,u)>0\), while (2.1), (2.3) and (2.4) yield
where the last inequality is implied by the inequality \({\mathcal G}_{\Omega }(u)< 0\). Finally, (2.3) implies that \(g_{\Omega }\) is strictly decreasing with respect to \(\tau \), so that, since it is a continuous function, there exists a unique \(T_{\Omega }(u) > 0\) such that \(g_{\Omega }( T_{\Omega }(u),u) =0\), that is
i.e., \(T_{\Omega }(u) u \in \mathcal {N}_{\Omega }\), showing that \(T_{\Omega }\) is well defined.
In order to show that the map \(T_{\Omega }\) is continuous in \(A_{\Omega }\), let u be in \(A_{\Omega }\), so that we can consider \(\tilde{\Omega }\subset A\) with \(|\tilde{\Omega }|>0\) and \(u^{+}(x)\ne 0\) iff \(x\in \tilde{\Omega }\) and assume that \((u_{n})\in A_{\Omega }\) converges to u. The continuity property of \({\mathcal G}_{\Omega }\) implies that there exists \(n_{0}\) such that \({\mathcal G}_{\Omega }(u_{n})<0\) for every \(n\ge n_{0}\), so that there exists \(T_{\Omega }(u_{n})\) which will be denoted as \(T_{n}\). Assume, by contradiction that, up to a subsequence, \(T_{n}\rightarrow +\infty \), then (2.1), (2.3) and (2.4) allow to apply Lebesgue dominated convergence theorem to obtain that
Then, by definition of \(T_{n}=T_{\Omega }(u_{n})\), and using (2.1), (2.3) and (2.4) it results
implying \({\mathcal G}_{\Omega }(u)=0\), which is a contradiction. Then, up to a subsequence, \(T_{n}\rightarrow T_{0}\); if it were \(T_{0}=0\), then (2.1), (2.3) and (2.4) would imply
again a contradiction, as \({\mathcal G}_{\Omega }(u)<0\). Then, \(T_{n}\rightarrow T_{0}>0\) and passing to the limit in (3.13), we derive that \(T_{0}=T_{\Omega }(u)\), as \(T_{\Omega }(u)\) is unique. In addition, take \(\theta \in \mathbb {R}^{N}{\setminus }\{0\}\) and first observe that \({\mathcal G}_{\Omega }(\Phi ^{R\theta })=\mathcal {G}(\Phi ^{R\theta })\), as \(\Phi ^{R\theta }=0\) for \(|x|<\rho \). Moreover, from Lemma 3.4 it results \(\mathcal {G}(\Phi ^{R\theta })=\mathcal {G}(w)+o(\varepsilon _{R})\), so that there exists \(R_{0}\) such that for every \(R\ge R_{0}\), it results \(\Phi ^{R\theta }\in A\cap A_{\Omega }\). In order to show (3.12), let us consider \(\theta \in \mathbb {R}^{N}{\setminus }\{0\},\) \(R_{n}\rightarrow +\infty \) and set \(T_{n}=T_{\Omega }(\Phi ^{R_{n}\theta })\), which is well defined. By definition, \(T_{n}\) satisfies
Note that (3.2), (3.9), (2.3), (2.4) and (3.7) imply
Moreover, recalling (3.4) and performing a change of variable, we obtain
Now, it is easy to see that \((T_{n})\) is bounded by contradiction, since if it were not the case, then one would obtain that \(\mathcal {G}(w)=0\), contradicting the fact that \(w\in A\). Then \((T_{n})\) has to be bounded, so that we can assume that, up to a subsequence, it converges to \(T\in \mathbb {R}\), and \(T\ne 0\) arguing again by contradiction. Passing to the limit in (3.14), we get
Then \(T=1\) since \(w\in {\mathcal N}\). \(\square \)
Remark 3.6
Lemma 3.5 provides the definition of the continuous projection map
Up to now, we have obtained asymptotic estimates on a function \(\Phi ^{R\theta }\). In proving our existence results, we will use the following convex combination of \(\Phi ^{Ry}\) and \(\Phi ^{Rx_{0}}\),
As a consequence, we will need also asymptotic information on quantities involving the functions \(\Phi ^{Ry}\) and \(\Phi ^{Rx_{0}}\) with \(y\in \partial B_{2}(x_{0})\). Let us start with the following result.
Lemma 3.7
Assume (2.2). Let \(x_{0}\) and y be fixed in (3.15). Then it results
Proof
Notice that
As \(w^{Ry}\) and \(w^{Rx_{0}}\) are solutions of (2.7), we get
Then, the result is proved once one shows that the last two integrals on the right-hand side are \(o(\varepsilon _{R})\). First, we observe that (3.2) imply that there exists a positive constant \(C_{0}\) such that
where the last inequality is implied by (3.7). In addition, exploiting again (3.2) and (3.7), one obtains
yielding the conclusion. \(\square \)
In the following lemma, we prove some crucial estimates for the map \(T_{\Omega }(U_{t}^{R})\).
Lemma 3.8
Assume conditions (2.1), (2.2), (2.3), (2.4), and let \(x_{0},\,y\) and t be given in (3.15). Then, there exists \(R_{1}\) such that the following conclusions hold:
-
(i)
\(U^{R}_{t}\in A_{\Omega } \), for every \(R\ge R_{1}\);
-
(ii)
there exists a positive constant L such that
$$\begin{aligned} |T_{\Omega }(U^{R}_{t})|\le L, \quad \mathrm{for every}\,(t,R)\in [0,1]\times [R_{1},+\infty ). \end{aligned}$$(3.16) -
(iii)
for every \(t_{0}\in (0,1)\), it holds
$$\begin{aligned} \lim _{(t,R)\rightarrow (t_{0},+\infty )}T_{\Omega }(U^{R}_{t})=\dfrac{1}{t_{0}}\quad \text {if and only if}\,t_{0}=\frac{1}{2}. \end{aligned}$$(3.17)
Proof
First of all, let us note that (3.9) and Lemmas 3.2 and 3.7 imply that the following inequality holds for every \(y\in \partial B_{2}(x_{0})\) (recall (3.15))
Then, recalling the definition of \({\mathcal G}_{\Omega }\) stated in Lemma 3.5, using again (3.9), and applying Lemma 3.7, one obtains
Thanks to (3.3), we can apply Lemma 2.1 in [1] with \(\overline{\mu }=\sqrt{\lambda }/2\) to deduce
Then, as \(\mathcal {G}(w)<0\), there exists \(R_{1}\) such that for every \(R>R_{1}\) it holds \({\mathcal G}_{\Omega }(U^{R}_{t})<0\), for every \(y\in \partial B_{2}(x_{0})\), yielding conclusion (i). In order to prove conclusion (ii), first note that \(T_{\Omega }(U^{R}_{t})\) is well defined; then, we argue by contradiction and suppose that there exist sequences \(R_{n}\rightarrow +\infty \), \(t_{n}\in [0,1]\) and \(y_{n}\in \partial B_{2}(x_{0})\) such that
Since \(t_{n}\in [0,1]\), we can suppose, up to a subsequence, that there exists \(t_{0}\in [0,1]\) such that \(t_{n}\rightarrow t_{0}\). Three cases may occur: either \(t_{0}=0\), or \(t_{0}=1\), or \(t_{0}\in (0,1)\).
Suppose first that \(t_{0}=0\), and observe that \(T_{n}U_{n}\in {\mathcal N}_{\Omega }\) so that
Then, taking into account that \(t_{n}\rightarrow 0,\,R_{n}\rightarrow +\infty \) and exploiting (3.18), we obtain
With respect to the right-hand side of (3.20), we observe that the property of the function \(\xi \), joint with assumptions (2.3) and (2.4), gives that there exists a positive constant \(C_{1}\) such that
Moreover, one observes that
and assumptions (2.3), (2.4) and (3.9) yield
In addition, since
almost everywhere in \(\mathbb {R}^{N}\), (2.4) and (2.3) yield
Then (3.21), (3.22), (3.23), (3.25) allow us to pass to the list in (3.20) to obtain \(\mathcal {G}(w)=0\), which is a contradiction as \(w\in A\) (defined in Lemma 3.5).
The case in which \(t_{0}=1\) is similar to the case \(t_{0}=0\) by exchanging the role of \(t_{n}\) with the one of \(1-t_{n}\). Let us handle the third case and suppose that \(t_{0}\in (0,1)\) and come back to (3.18) to observe that (3.9) and Lemma 3.7 imply that
In order to study the right-hand side of (3.20), notice that by (2.3) and (3.2)
and as \(t_{0}\in (0,1)\) it results
Then, taking into account that (3.23) holds also for y in the place of \(x_{0}\), one obtains
Arguing as in (3.24) in the above integrals separately, exploiting (3.2) and (3.19) and taking into account that \( t_{0}\in (0,1)\), one deduces
Therefore, we can use (2.4) and (2.3) to apply Lebesgue dominated convergence theorem and obtain
Finally, passing to the limit in (3.20) and using (3.26), (3.27) it follows
which again contradicts the fact that \(w\in A\), yielding (3.16). In order to prove (3.17), let us first show that
Arguing again by contradiction and supposing that there exist \(\delta >0\) and subsequences \(R_{k}\rightarrow +\infty \), \( t_{k} \rightarrow 1/2\) and \(y_{k}\in \partial B_{2}(x_{0})\) such that the sequence
Estimate (3.16) implies that there exists T such that, up to a subsequence, \(T_{k}\rightarrow T\). Then (3.20) becomes
and (3.26) allows to take the limit on the left-hand side; on the other hand, we can argue in analogous way as in the case \(t_{0}\in (0,1)\) to obtain
As w is a solution of the limit problem (2.7), this implies
yielding (3.29) thanks to (2.3). On the other hand, suppose that there exists \(t_{0}\in (0,1)\) such that the limit in (3.17) holds and use (3.9) and Lemma (3.7) to take the limit in (3.30) and obtain
that is
Finally, (2.3) implies that \(t_{0}=1/2\), yielding the conclusion. \(\square \)
Lemma 3.9
Assume condition (2.2), and let \(x_{0}\), y and t be given in (3.15). Then, it results
where \(T^{R}_{t}=T_{\Omega }(U^{R}_{t})\).
Proof
We use Lemma 2.2 in [1], with \(\alpha :=\min \{(p_1+1)/4,1\}\) and \(p_1\) as in (2.2), and we take into account (3.2) and (3.16) to obtain that
Since \(2\alpha >1\), we have that, denoting with \(\varphi (x)=w^{2\alpha }(x)\), the first hypothesis in Lemma 3.1 is satisfied with \(c=0\). Then Lemma 3.1 implies that
Then, Lemma 3.2 yields the conclusion. \(\square \)
Lemma 3.10
Assume (2.1), (2.2), (2.3), and let \(x_{0}\) and y be given in (3.15). Then, There exist \(C=C(\Vert w\Vert _{\infty })>0\) such that, for all \(\tau _1, \tau _2\in [0,+\infty ) \), it results
Proof
As, performing a change of variable, we have
it is enough to show the first inequality, since the other will follow by a similar argument. In order to do this, taking into account Lemma 3.2, it is sufficient to show that there exists a constant C such that
Note that the positive minimum of w(x) in the ball \(B_{1}(0)\) is achieved for \(|x|=1\); let us denote this minimum value as \(\alpha _{0}>0\). As a consequence of hypothesis (2.3), we have that the function \(g(u)=f(u)/u\) is continuous and monotone increasing in the interval \([\alpha _{0},\Vert w\Vert _{\infty }]\), so that \(g(u)\ge C:=g(\alpha _{0})\). Then we obtain
Note that for every \(x\in B_{1}(0)\), it results for every \(R\ge 1\)
In the following, we will denote with C possibly different positive constants. Condition (3.3) implies that for sufficiently large R, there exists a constant C such that
and (3.32) gives
Using this inequality in (3.31) and applying Lemma 3.2 yield the conclusion. \(\square \)
Lemma 3.11
Assume (2.1), (2.2), (2.3), and let \(x_{0}\) and y be given in (3.15). Then, for every \(\tau \in [0,+\infty )\), it results
Proof
The first inequality is a direct consequence of (2.3), (2.2), (3.2) and (3.7). Indeed, it results
In order to show (3.34), consider the function \( g(\tau ):=\tau f(w^{Rx_{0}}) - f(\tau w^{Rx_{0}})\) and suppose, without loss of generality, that \( \tau >1\). The mean value theorem implies that there exists \(\theta \in (1,\tau )\) such that
Substituting in the integral in (3.34), we obtain
Applying Lemma 3.2, hypothesis (2.2) and taking into account that \(\theta \in (1,\tau )\), one obtains
Applying Lemma 3.1 with \(\varphi _{1}=w\) and \(\varphi _{2}=w^{p_{1}}\) (see also the argument of the proof of Lemma 3.2), we obtain that
so that the conclusion follows. \(\square \)
4 Compactness condition
In this section, we will find the level interval where Palais–Smale condition holds.
Lemma 4.1
Assume (2.1), (2.3),(2.4). There holds
and \(m_{ \Omega }\) is not attained.
Proof
Since any \(u \in H^1_0(\Omega )\) can be extended as zero outside \(\Omega \), we may consider \(H^1_0(\Omega ) \subset H^1(\mathbb {R}^N)\) and so \(m_{\Omega } \ge m\;.\) On the other hand, applying Lemma 3.5 to the sequence \(\phi _n:=\Phi ^{R_{n}\theta }\), with \(R_{n}\rightarrow +\infty \), it follows that there exists \(n_{0}\) such that for every \(n\ge n_{0}\) there exists \(T_n >0\) such that \(T_n (\phi _{n})\phi _{n} \in \mathcal {N}_{\Omega }.\) Then, taking into account (1.5), (3.10) and (3.12) yield
Finally, assume that there exists \(u\in H^{1}_{0}(\Omega )\) such that \(I_{\Omega }(u)=m_{\Omega }\). Then \(\overline{u}\), extension by zero of u outside \(\Omega \), would be a minimizer of the problem at infinity not positive on the whole \(\mathbb {R}^{N}\) contradicting the maximum principle. \(\square \)
The remaining of this section is devoted to recovering compactness properties for \(I_{\Omega }\). The main difficulty in our context is due to the fact that f is asymptotically linear. Moreover, we will need a compactness property just for sequences belonging to the Nehari manifold \({\mathcal N}_{\Omega }\), and then we first prove the following result.
Lemma 4.2
Assume (2.1), (2.2), (2.3), (2.4), (2.5), (2.6). Let \(d\in \mathbb {R}^{+}\) and \((u_n)\) be such that
then \((u_{n})\) is bounded.
Proof
The proof can be started as in Proposition 3.20 in [28], and it can be concluded as in Lemma 5.3 in [27]. We give here a brief summary. First, note that as \(u_{n}\in {\mathcal N}_{\Omega }\), it results
For every \(v\in {\mathcal N}_{\Omega }\), let us consider the one-variable function \(h:\mathbb {R}^{+}\rightarrow \mathbb {R}\)
Notice that (4.1) implies that there exists a positive constant \(C=C(d)\) such that
so that condition (2.3) implies that, for every fixed \(x\in \mathbb {R}^{N}\), \(h_{u_{n}}(p)\) has a unique maximum point for \(p=1\) so that \(I_{\Omega }(pu_{n})\le I_{\Omega }(u_{n})\le C\). Recalling (4.2), one obtains
Then, assuming by contradiction that, up to a subsequence, \(\Vert u_{n}\Vert _{\Omega }\rightarrow +\infty \) and setting \(v_{n}=p_{n}u_{n}\) with \(p_{n}=2\sqrt{C}/\Vert u_{n}\Vert _{\Omega }\), we obtain the uniform lower bound
This lower bound, hypothesis (2.2) and Lions lemma (see [25]) imply that there exist positive numbers r and \(\delta \) and a sequence \((y_{n})\in \mathbb {R}^{N}\) such that
Then, we have to handle two different possible situations: either \(y_{n}\) is bounded, or, up to a subsequence, \(|y_{n}|\rightarrow +\infty \) as \(n\rightarrow +\infty \).
In the first case, we deduce from (4.3) and recalling that \(v_{n}\equiv 0\) outside \(\Omega \), that there exists \(r_{1}>\rho \) such that
and the same lower bound holds for the weak limit v of \(v_{n}\) (up to a subsequence). Then, as \(v_{n}\equiv 0\) in \(\mathbb {R}^{N}{\setminus }\Omega \), there exists a subset \(\Lambda \in B_{r_{1}}(0)\cap \Omega \) with positive measure and such that \(v(x)>0\) in \(\Lambda \), so as \(u_{n}(x)=2\sqrt{C}v_{n}(x)\Vert u_{n}\Vert _{\Omega }\), it follows that \(u_{n}(x)\rightarrow +\infty \) for all \(x\in \Lambda \). This immediately leads to a contradiction using (2.1), (2.5) and (4.1) (for more details, see Lemma 5.3 in [27]).
Then \((y_{n})\) cannot be bounded, and, up to a subsequence, we obtain \(|y_{n}|\rightarrow +\infty \); moreover, we can assume that \(B_{r}(y_{n})\subset \Omega \) for n sufficiently large. Then, it is possible to argue analogously on the sequence \(\tilde{v}_{n}=v_{n}(x+y_{n})\), obtaining, as before, a set \(\Lambda \subseteq B_{r}(0)\) with positive measure such that \(u_{n}(x+y_{n})\rightarrow +\infty \) for every \(x\in \Lambda \). The contradiction follows again by using (2.5) (see Lemma 5.3 in [27]). Indeed, (4.1) yields
and the last integral goes to plus infinity thanks to (2.5). Then, we reach a contradiction, proving the Lemma. \(\square \)
Remark 4.3
Notice that in order to prove that \((u_{n})\) is bounded the classical information \(I_{\Omega }^{\prime }(u_{n})\rightarrow 0\) can be substituted by the information \((u_{n})\subseteq {\mathcal N}_{\Omega }\). Moreover, note that, in order to prove Lemma 4.2, it is sufficient to assume (2.2) for \(k=0\).
In the following lemma, we will show that \((u_{n})\) is a Palais–Smale sequence in the whole space.
Lemma 4.4
Assume (2.1), (2.2), (2.3), (2.4), (2.5), (2.6). Let \((u_{n})\) satisfy (4.1) and be such that
Then \(I_{\Omega }^{\prime }(u_{n})\rightarrow 0\) in \(H^{-1}(\Omega )\).
Proof
First of all, from (4.4), recalling (1.4), we obtain a sequence \((\mu _{n})\subseteq \mathbb {R}\) such that
where \(N_{\Omega }\) is defined in Remark 2.5. Moreover, we can use Lemma 4.2 to obtain that \((u_{n})\) is bounded, so that there exists \(u\in H^{1}_{0}(\Omega )\) such that, up to a subsequence, \(u_{n}\rightharpoonup u\) weakly in \(H^{1}_{0}(\Omega )\) and \(u_{n}\rightarrow u\) almost everywhere. Let us first show that \((\mu _{n})\) is bounded arguing by contradiction, so that we assume that, up to a subsequence, \(|\mu _{n}|\rightarrow +\infty \) and set \(t_{n}=\mu _{n}/|\mu _{n}|\). Since \(|t_{n}|=1\), there exists \(t_{0}\) such that, up to a subsequence, \(|t_{0}|=1\) and \(t_{n}\rightarrow t_{0}\) in \(\mathbb {R}\). Moreover, it results
Since \(|\mu _{n}|\rightarrow +\infty \) and as \((u_{n})\) is bounded, (2.2) implies
because \(t_{n}\rightarrow t_{0}\ne 0\). Then, using (2.1), and taking into account that \(u_{n}\in {\mathcal N}_{\Omega }\) we have
Assume that there exists \(\delta >0\) such that
then there exists \((y_{n})\) such that
arguing as in Lemma 4.2, we observe that two cases may occur, either \(|y_{n}|\) is bounded, or, up to a subsequence, \(|y_{n}|\rightarrow +\infty \). In the first case, there exists \(r_{1}>0\) such that \(\Vert u^{+}_{n}\Vert ^{2}_{L^{2}(B_{r_{1}}(0))}\ge \delta \). Then, since \(u^{+}_{n}\) strongly converges to some v, in \(L^{2}_{\mathrm{loc}}(\mathbb {R}^{N})\), it follows that \(v\ge 0\), and v satisfies \(\Vert v\Vert ^{2}_{L^{2}(B_{r_{1}}(0))}\ge \delta \). Moreover, recalling that \(u^{+}_{n}\equiv 0\) outside \(\Omega \), we deduce that there exists a measurable set \(\Lambda \subseteq \Omega \cap B_{r_{1}}(0)\) with positive measure, such that \(v(x)>0\) for \(x\in \Lambda \). Then, using (2.5), (2.6) and applying Fatou lemma, one obtains
that is a contradiction, so that \(|y_{n}| \) cannot be bounded, and \(|y_{n}|\rightarrow +\infty \), up to a subsequence. Then, we define \(\hat{u}_{n}(\cdot ):=u^{+}_{n}(\cdot +y_{n})\) and observe that there exists \(\hat{u}\ge 0\) almost everywhere in \(\mathbb {R}^{N}\), such that \(\hat{u}_{n}\rightharpoonup \hat{u}\) weakly in \(H^{1}_{0}(\Omega )\), strongly in \(L^{p}_{\mathrm{loc}}(\mathbb {R}^{N})\) and almost everywhere. From the strong convergence in \(L^{2}_\mathrm{loc}(\mathbb {R}^{N})\) and using (4.7), one deduces that \(\Vert \hat{u}\Vert ^{2}_{L^{2}(B_{r_{1}}(0))}\ge \delta \). Then there exists a measurable set \(\Lambda \subseteq B_{r}(0)\) with positive measure, and such that \(\hat{u}(x)> 0\) for every \(x\in \Lambda \). Moreover, as \(|y_{n}|\rightarrow \infty \), we can assume that \(B_{r}(y_{n})\subseteq \Omega \). Then, arguing as in (4.8) we get
so that (2.6) yields again a contradiction, showing that
Then, Lions’ lemma [25] implies
and (2.2) gives
Taking into account that \(u_{n}\in {\mathcal N}_{\Omega }\), by (2.1), one deduces that \(u_{n}\) strongly converges to zero, which is an evident contradiction with (4.1) implying that \((\mu _{n})\) is bounded. As a consequence, there exists \(\mu \in \mathbb {R}\) such that, up to a subsequence, \(\mu _{n}\rightarrow \mu \). Assume, by contradiction, that \(\mu \ne 0\), take \(\varphi =u_{n}\) as test function in (4.5) and, recalling that \(u_{n}\in {\mathcal N}_{\Omega }\), we have
From this point, we can repeat the same argument, starting from (4.6), to get a contradiction. Then \(\mu _{n}\rightarrow 0\) and (4.5) yields the conclusion. \(\square \)
The next lemma studies the asymptotic behavior of a bounded Palais–Smale sequence of \(I_{\Omega }\). In the case \(f(t)=t^{p}\), the proof is given in [9] (see also Chapter 8 in [39]). However, thanks to hypotheses (2.1), (2.2) the proof can be handled as in the polynomial case arguing as in Chapter 8 in [39]. We will include some details for the sake of clearness.
Lemma 4.5
(Splitting) Assume (2.1), (2.2). Let \((u_n)\in H^1_0(\Omega )\) be a bounded sequence such that
Replacing \((u_n)\) by a subsequence, if necessary, there exists a solution \(u_{0}\) of (1.1), such that the following alternative holds: either \(u_{n}\rightarrow u_{0}\) strongly in \(H^{1}(\mathbb {R}^{N})\) or there exists a finite sequence \((u^1,u^2,\ldots ,u^k)\) in \(H^{1}(\mathbb {R}^{N})\) solutions of (2.7) and k sequences of points \((y^{j}_{n})\subset \mathbb {R}^N, 1\le j \le k\), satisfying:
-
(a)
\(|y^{j}_{n}|\rightarrow \infty \) and \(|y^{j}_{n} - y^{i}_{n}|\rightarrow \infty , i\ne j\);
-
(b)
\(\Vert u_n -u_{0}- \displaystyle \sum _{j=1}^{k}u^j(\cdot -y^{j}_{n})\Vert \rightarrow 0\);
-
(c)
\(I_\Omega (u_n)\rightarrow I_\Omega (u_{0}) + \displaystyle \sum _{j=1}^{k}I(u^j)\).
Proof
Since \((u_n)\) is bounded, there exists \(u_0\in H^1_0(\Omega )\) such that, up to a subsequence, \(u_n\rightharpoonup u_0\). Then, thanks to the continuity of f and to (2.2), from (4.9) it follows that
Let \(u_n^1:=u_n-u_0 \in H^1_0(\Omega ) \subset H^1(\mathbb {R}^N)\), and let us show that it results, for \(n\rightarrow \infty ,\)
The proof of (4.11) is standard. To show (4.12), note that the weak convergence of \((u_n)\) to \(u_0\) implies \(u_n^1\rightharpoonup 0\) weakly in \(H^{1}_{0}(\Omega )\). Then, applying Theorem 2 in [12] (see also Lemma 8.1 in [39]), it follows that
where \(o_{n}(1)\) is a quantity tending to zero as n goes to plus infinity. Then, (4.9) yields (4.12). Moreover, (4.13) follows from the following facts: observe that condition (2.2) allows us to use Theorem 2 in [12] or arguing as in Lemma 8.1 in [39], to obtain that \(f(u_{0}+u^{1}_{n})-f(u^{1}_{n})\rightarrow f(u_{0}) \) in \( H^{-1}(\Omega )\). Then, exploiting (4.9), (4.10), it follows, for every \(\varphi \in H^{1}_{0}(\Omega )\)
Hence, (4.13) follows. Let us now consider
If \(\delta =0\), it follows from Lions’ lemma [25] that
On the other hand, as \(\left( u^{1}_{n}\right) \) is bounded in \(H^{1}_{0}(\Omega )\), we can use (2.2) and (4.14) to deduce that \(u^{1}_{n}\rightarrow 0\) strongly in \(H^{1}_{0}(\Omega )\) and since \(u^{1}_{n}\equiv 0\) outside \(\Omega \) it results \(u^{1}_{n}\rightarrow 0\) strongly in \(H^{1}(\mathbb {R}^{N})\). Then the first alternative in the statement of the Lemma holds. While, if \(\delta >0\), there exists a sequence \((y_n^{1})\subset \mathbb {R}^N\) such that
Define a new sequence \((v_n^1)\subset H^1(\mathbb {R}^N)\) by \(v_n^1:=u_n^1(\cdot +y^{1}_n).\) Since \((u_n^1)\) is bounded in \(H^{1}(\mathbb {R}^{N})\), then \((v_n^1)\) is also bounded and there exists \(u^{1}\in H^1(\mathbb {R}^N)\), such that \(v_n^1\rightharpoonup u^1\) in \(H^1(\mathbb {R}^N)\) and \(v_n^1\rightarrow u^1\) almost everywhere in \( \mathbb {R}^N\); then performing a change of variable in (4.15) and applying Fatou lemma, one obtains that
so that \(u^1\not \equiv 0\). Moreover, since \(u_n^1\rightharpoonup 0\) in \(H^1(\mathbb {R}^N)\), it follows that up to a subsequence we can assume that \(|y^{1}_n|\rightarrow \infty \). As shown in [9] (see also Proposition 3.1 in [16] or Chapter 8 in [39]), it results that \(I^{\prime }(u^1)=0\) in \(H^{-1}(\mathbb {R}^{N})\). Indeed, take \(\phi \in C^\infty _c(\mathbb {R}^{N})\) and observe that since \(|y_{n}^{1}|\rightarrow +\infty \), we can find \(n_{0}\) such that \(\phi _{n}:=\phi (x-y_{n}^{1})\in C^{\infty }_{c}(\Omega )\) for every \(n\ge n_{0}\); moreover, \(\Vert \phi _{n}\Vert _{\Omega }\le \Vert \phi \Vert \). As a consequence, (4.13) yields
so that \(u^{1}\) is a solution of (2.7). Defining \(\displaystyle u_n^2(x):=u_n^1(x)-u^1(x-y_n^1),\) it results that \(u_{n}^{2}(\cdot +y^{1}_{n})=v^{1}_{n}-u^{1}\) and repeating for \(u^{2}_{n}\) the argument done for \(u^{1}_{n}\) we deduce that
so that we go on repeating the argument obtaining \((y^{2}_{n})\) satisfying conclusion (a) and \(u^{2}\), another solution of (2.7). From now on, we proceed by iteration. Note that if u is a non-trivial critical point of I and w is the solution of minimum action of (2.7), then
As a consequence, passing from the step l to the step \(l+1\), the action level decreases because in the asymptotic information on the functional \(I(u^{l+1}_{n})\) it appears \(-I(u^{l+1})\). Then, taking into account (4.16), the sum must have a finite number of terms, so that the iteration must be finite and terminate at some index \(k\in \mathbb {N}\), yielding also conclusions (b) and (c). \(\square \)
The following result is a direct consequence of the splitting lemma.
Corollary 4.6
Assume (2.1), (2.2), (2.3), (2.4), (2.5), (2.6) and (U). Let \((u_{n})\) be a sequence satisfying (4.1) and (4.4) with \(d\in (m,2m)\). Then \((u_{n})\) admits a strongly convergent subsequence.
Proof
Applying Lemma 4.2, we deduce that \((u_{n})\) is bounded; moreover, Lemma 4.4 allows us to apply Lemma 4.5 to obtain that there exists \(u_{0}\) solution of (1.1) such that conclusion b) holds. Moreover, conclusion c) and Lemma 4.1 yield
Then, in both cases, \(k<2\), i.e., \(k=0\) or \(k=1\). If \(k=1\) and \(u_{0} \equiv 0\), it follows that \(I_{\Omega }(u_n)\rightarrow I(u_1)=d\), and from hypothesis (2.1), \(u_{1}\) is positive, so that condition (U) yields \(u_{1}=w\) and \(d=I(u_{1})=m\), which contradicts the hypothesis. Also note that the hypothesis \(d<2m\) implies that it is not possible that \(k=1\) and \(u_{0}\not \equiv 0\). Then, the only possibility is that \(k=0\), that is \(d=I(u_{0})=I_{\Omega }(u_{0})\) and Lemma 4.5 implies \(u_{n}\) strongly converges to \(u_{0}\).
\(\square \)
5 The linking argument and proof of the main results
We will prove our existence results by applying the linking theorem on the manifold \({\mathcal N}_{\Omega }\) (see Theorem 8.22 in [5] or Theorem 8.4 in [37] joint with Lemmas 5.14 and 5.15 in [39]). This argument has already been used in [4] to prove existence results for linearly coupled semi-linear non-autonomous equations in \(\mathbb {R}^{N}\). In order to define the linking sets, we will make use of the properties of a barycenter function, already adopted in [9] and then often used when building solution at higher action level than the least one (see, for example, [4, 18] or [8] for an interesting generalization). Here, we will follow the notation in [18] (see also [4]). For every \(u\in H^{1}(\mathbb {R}^{N}){\setminus } \{0\}\), the following maps are well defined
Then, the barycenter of a function \(u\in H^{1}(\mathbb {R}^{N}){\setminus } \{0\}\) defined by
is a continuous function enjoying the following properties
One of the linking set is the following subset of \({\mathcal N}_{\Omega }\)
Remark 5.1
Notice that \(S\ne \emptyset \). Indeed, first note that from the properties of the function w, we can find \(r_{0}>0\) such that \(\{x\in \mathbb {R}^{N}\,:\, \mu (w)>\Vert \mu (w)\Vert _{\infty }/2\}= B_{r_{0}}(0)\). Then, take \(\theta \in \mathbb {R}^{N}\) with \(|\theta |=1\), \(R>4\rho r_{0}\) and the functions \(\eta _{1}\) and \(\eta _{2}\) defined as \(\eta _{1}(x)=1-\xi (4\rho (x-R\theta )/R)\) and \(\eta _{2}(x)=1-\xi (4\rho (x+R\theta )/R)\). Choose the function \(z_{R}=w^{R\theta }\eta _{1}+w^{-R\theta }\eta _{2}\). The properties of the function \(\eta _{1}\) imply that \(\eta _{1}w^{R\theta }<w^{R \theta }\) so that \(\mu (\eta _{1}w^{R\theta }) <\mu (w^{R\theta })\), and it results
where we have taken into account that \(\mu (0)>\mu (x)\) for every \(x\in \mathbb {R}^{N}{\setminus }\{0\}\). As a consequence of the previous facts,
Arguing analogously for \(\eta _{2}w^{-R\theta }\), one obtains \(\widehat{\eta _{1}w}^{R\theta }=\widehat{w}^{R\theta }\) and \(\widehat{\eta _{2}w}^{-R\theta }=\widehat{w}^{-R\theta }\). Moreover, observe that \({\mathcal G}_{\Omega }(z_{R})={\mathcal G}_{\Omega }(w^{R\theta }) +{\mathcal G}_{\Omega }(w^{-R\theta })+ o(R)=2{\mathcal G}_{\Omega }(w)+o(R)\), where o(R) is a quantity tending to zero as R tends to infinity. Then \(z_{R}\in A_{\Omega }\) for R sufficiently large. Therefore, \(v=T_{\Omega }(z_{R})z_{R}\in {\mathcal N}_{\Omega }\) and from (5.2) it results \(\beta (v)=\beta (z_{R})\). In addition, taking into account that \(B_{R/2}(R\theta )\cap B_{R/2}(-R\theta )=\emptyset \), we obtain \(\hat{z}_{R}=\widehat{\eta _{1}w}^{R\theta }+\widehat{\eta _{2}w}^{-R\theta } \). Then it results
showing that \(z_{R}\in S\).
Lemma 5.2
Assume (2.1), (2.2), (2.3), (2.4), (2.5), (2.6). Let \(b = \displaystyle \inf _{S}I_\Omega (u)\), then \(b > m_{\Omega }\).
Proof
It is clear that \(b \ge m_{\Omega }\). To prove the strict inequality, we shall argue by contradiction. Suppose \(b = m_{\Omega }\); then there exists a sequence \((v_n) \subset {\mathcal N}_{\Omega }\) such that \(\beta (v_n)=0\), \(\langle I^{\prime }_\Omega (v_n), v_n\rangle =0\) and \(I_\Omega (v_n) \rightarrow m_{\Omega }\); moreover, \((v_n)\) is not relatively compact because \(m_{\Omega }\) is not attained. By Ekeland variational principle (Theorem 8.5 in [39]) applied to the closed manifold \({\mathcal N}_{\Omega }\), there exists another sequence \((\tilde{v}_n) \subset {\mathcal N}_{\Omega }\) such that:
From Lemma 4.2, \((v_{n})\) and \((\tilde{v}_{n})\) are bounded. Moreover, Lemma 4.4 implies that
By exploiting hypothesis (2.2), we deduce that that \(I^{\prime \prime }_{\Omega }\) maps bounded sets of \(H^{1}_{0}(\Omega )\) in bounded sets, then the mean value theorem implies that the following inequality holds for every \(\phi \in H^{1}_{0}(\Omega )\),
Taking the supremum on \(\phi \) and using (5.5), it follows that also \((v_{n})\) is a Palais–Smale sequence. Therefore, from Lemma 4.5 and since \(m_{\Omega }\) is not attained, we deduce that conclusions (a)–(c) hold. In particular, conclusion (c), (5.4) and Lemma 4.1 imply
where the last inequality is implied by the fact that, either \(u_{0}\equiv 0\) so that \(I_{\Omega }(u_{0})=0\), or \(u_{0}\in {\mathcal N}_{\Omega }\) so that \(I(u_{0})>m_{\Omega }>0\). Then, k has to be equal to one, \(u_{0}\) has to be trivial, and \(u^{1}=w\). This and conclusion (b) of Lemma 4.5 yield \(v_n(\cdot )-w(\cdot -y_n)\rightarrow 0\), strongly in \(H^{1}(\mathbb {R}^{N})\), where \(y_n \in \mathbb {R}^N\), \(|y_n| \rightarrow \infty \). Calculating the barycenter function of \(v_{n}(x)\) and \(w(x-y_{n})\), we have, as \(\beta \) is a continuous function, w is radially symmetric, and using (5.1)
where o(1) a quantity tending to zero as n goes to plus infinity. This immediately gives a contradiction as \(|y_{n}|\rightarrow +\infty \). \(\square \)
In order to define the other linking set, we argue as in [18] (see also Section 7 in [4]) and we take \(x_{0}\) and y as in (3.15) and \(R>\max \{R_{0},R_{1}\}\) where \(R_{0}\) and \(R_{1}\) are introduced in Lemmas 3.5 and 3.8. Let us define the function
Notice that z is a homeomorphism from \(\partial B_2(x_{0}) \times (0,1]\) to \( \overline{B_{2}(x_{0})}{\setminus } \{x_{0}\}\), so that for every point \(\tilde{z}\in \overline{B_{2}(x_{0})}\) there is a unique pair \((y,t)\in \partial B_2(x_{0}) \times (0,1] \), such that \(\tilde{z}=z(y,t) \not = x_{0}\). Therefore, we can define the operator \(\Psi _{R}:\overline{B_2(x_{0})} \mapsto {\mathcal N}_{\Omega }\) by the values that it takes on \(\partial B_{2}(x_{0})\times [0,1]\) as follows
where \(U^{R}_{t}\) is given in (3.15) and \(\Pi _{{\mathcal N}_{\Omega }}\) is defined in (3.6). Notice that \(\Psi _{R}\) is well defined thanks to Lemma 3.8. In order to apply the linking theorem on \({\mathcal N}_{\Omega }\), we recall that S is defined in (5.3) and we set
The linking geometrical structure is shown in the next lemma.
Lemma 5.3
Assume conditions (2.1), (2.2), (2.3), (2.4), (2.5), (2.6). Let Q and \({\mathcal H}\) be defined in (5.7), (5.8). Then, for R sufficiently large, it results
Proof
First of all, notice that \(\Psi _{R}:\overline{B_{2}(x_{0})}\mapsto \Psi _{R}\left( \overline{B_{2}(x_{0})}\right) \) is a continuous bijection defined on a compact set. Indeed, let us first show that \(\Psi _{R}\) is injective in \(\overline{B_{2}(x_{0})}{\setminus } \{x_{0}\}\). In order to do this, let us consider \(z_{1},\,z_{2}\in \overline{B_{2}(x_{0})}{\setminus } \{x_{0}\}\) such that \(\Psi _{R}(z_{1})=\Psi _{R}(z_{2})\). Since z(y, t) is injective, this amounts to consider \((y_{1},t_{1}),\, (y_{2},t_{2})\in \partial B_{2}(x_{0}) \times (0,1]\) such that \(\Psi _{R}[z(y_{1},t_{1})]=\Psi _{R}[z(y_{2},t_{2})]\). Taking into account Remark (3.6), and using the notation
this is equivalent to have
Suppose that \(|y_{1}-y_{2}|=a>0\). Since \(|y_{i}-x_{0}|=2\) for \(i=1,2\), (3.3) implies that \(w(R(y_{2}-y_{1}))=o(R)\) and \(w(R(y_{i}-x_{0}))=o(R)\), where o(R) is a quantity tending to zero as R tends to infinity. Choosing \(x=Ry_{1}\) with \(R> 2\rho \), and recalling (3.8), one obtains \( T_{1}(R)\left( t_{1}w(0)+(1-t_{1})o(R)\right) =T_{2}(R)o(R). \) Then, (3.16) implies that \( T_{1}(R)t_{1}w(0)= o(R).\) Since \(w(0)>0\) and \(t_{1}\in (0,1]\), this implies that \(T_{1}(R)=o(R).\) Arguing as in the proof of Lemma 3.8, one can reach a contradiction, so that \(|y_{1}-y_{2}|=0\), i.e., \(y_{1}=y_{2}\). In order to prove that \(t_{1}=t_{2}\), let us choose \(x=Ry=Ry_{1}=Ry_{2}\), take into account (3.2) and that \(R>2\rho \) to obtain
On the other hand, choosing \(x=Rx_{0}\), we get
Then
Since the function \(h(t)=at+b(1-t)/[bt+a(1-t)]\) is injective, the above equality implies that \(t_{1}=t_{2}\). Then, we have shown that \(\Psi \) is injective in \(\overline{B_{2}(x_{0})}{\setminus }\{x_{0}\}\). Now, take \(z\in \overline{B_{2}(x_{0})}\) such that \(\Psi _{R}(z)=\Psi _{R}(x_{0})\), that is
Choosing \(x=Ry\) and arguing as before, one obtains that \(z=x_{0}\) proving that \(\Psi _{R}\) is injective in \(\overline{B_{2}(x_{0})}\). As a consequence, \(\Psi _{R}\) is an homeomorphism and \(\partial Q=\Psi _{R} (\partial B_{2}(x_{0}))\). In order to show that (5.9) holds, we observe that, since w is radially symmetric, positive and decreasing in \((0,+\infty )\), also \(\mu (w)\) is decreasing with respect to |x|; moreover, as proved in Theorem 2.1 in [8] \(\mu (w)\rightarrow 0\) as \(|x|\rightarrow +\infty \), then, arguing as in Remark 5.1, we obtain that there exists a unique \(r_{0}>0\) such that for every \(|x|=r_{0}\), \(\mu (w)(x)=\Vert \mu (w)\Vert _{\infty }/2\) and by (5.1) the set
is such that
for every \( R\in \mathbb {R}^{+}\). Let us fix R such that \( R>2\rho +1+r_{0}\) and, as \(y\in \partial B_{2}(x_{0})\), it results that \(|x|>2\rho +r_{0}\) for every \(x\in B_{1}(Ry)\). Then, exploiting (3.2), we obtain
Since, \(\Vert \xi \Vert _{\infty }\le 1\), it results
showing that
In addition, for every \(x\in B_{r_{0}}(Ry)\), any \(z\in B_{1}(x)\) satisfies \(|z|>2\rho ,\) showing that \(B_{1}(x)\subseteq \mathbb {R}^{N}{\setminus } B_{2\rho }(0)\), and using again (3.2), we have
Recalling (5.12) we have that, for every \(x\in B_{r_{0}}(Ry)\), \(\mu (w)(x-Ry)>\Vert \mu (w)\Vert _{\infty }/2\), yielding
so that \(\widehat{\Phi }^{Ry}\ne 0\) if \(x\in B_{r_{0}}(Ry)\). If \(x\not \in B_{r_{0}}(Ry)\), recalling (5.12), (5.13) and (5.14), it results
Therefore, \(\widehat{\Phi }^{Ry}\ne 0\) if and only if \(x\in B_{r_{0}}(Ry)\), but, in this set \(\xi \equiv 1\), so that \(\widehat{\Phi }^{Ry}=\widehat{w}(\cdot -Ry)\) and hence
showing (5.9). In order to prove (5.10), for every \(h\in {\mathcal H}\), let us take \(V:\overline{B_{2}(x_{0})}\mapsto \mathbb {R}^{N}\) given by \(V(z)=\left( \beta \circ h\circ \Psi _{R}\right) (z)\). If \(z\in \partial B_{2}(x_{0})\), \(z=z(y,1)\), then it results \(\Psi _{R}(z)\in \partial Q\), so that \(h(\Psi _{R}(z))=\Psi _{R}[y,1]\), then (5.2) and (5.15) yield
Then, by Brouwer theorem, there exists \(z_{0}\in B_{2}(x_{0})\) such that \(V(z_{0})=0\), i.e., \(h(\Psi _{R}(z_{0}))\in S\), and, as by definition \(\Psi _{R}(z_{0})\in Q\), (5.10) follows.
In order to show (5.11), let us first observe that \(\partial Q=\Psi _{R}(\partial B_{2}(x_{0}))=\Psi _{R}[y,1]\); moreover,
From (3.10) and (3.12), it follows that \(I_{\Omega }(\Psi _{R}[y,1])\rightarrow m\) as \(R\rightarrow +\infty \); then, Lemmas 4.1 and 5.2 imply that, fixing R sufficiently large, (5.11) holds. \(\square \)
Lemma 5.4
Assume (2.1), (2.2), (2.3), (2.4). There exists \(R_{2}>\max \{R_{0},R_{1}\}\) such that
Proof
In the proof, we will use the notation introduced in Lemma 3.9
where T(u) is defined in Lemma 3.5. Recalling (5.7), (5.6), and taking into account (3.15), it is sufficient to show that there exists \(R_{2}\) sufficiently large, such that
It results
By adding and subtracting \(I\left( T^{R}_{t}tw^{Ry}\right) +I\left( T^{R}_{t}(1-t)w^{Rx_{0}}\right) \) and taking into account (3.9) and Lemma 3.8, one obtains
Moreover, using (3.11) and (3.16), one obtains
Applying Lemmas 3.7 and 3.9, we deduce
Applying Lemmas 3.8 and 3.11 (see (3.33)) in the last term, one gets
We use (3.34) with the choices \(\tau =T^{R}_{t}t\), \(\tau =T^{R}_{t}(1-t)\) and use (3.16) to get that there exists a positive constant \(C_{1}\) such that
Choosing \((\tau _{1},\tau _{2})=(T^{R}_{t}(1-t),T^{R}_{t}t)\) in the first inequality in Lemma 3.10, and \((\tau _{1},\tau _{2})=(T^{R}_{t}t,T^{R}_{t}(1-t))\) in the second one, we obtain (thanks to (3.16))
with \(C_{2}\) a positive constant. In view of (3.17), there exists \(\delta >0\) such that for every \( t\in \left( \frac{1}{2}-\delta ,\frac{1}{2}+\delta \right) ,\) it results
Moreover, the real-valued function \(h_{v}\) defined as
achieves the unique maximum value for \(p=1\) (see for more details Lemma 4.2 or Proposition 3.20 in [28]); then, as \(w\in {\mathcal N}\), it results
The same conclusion holds for the term \(I\left( T^{R}_{t}(1-t)w^{Rx_{0}}\right) \), so that from (5.18) it follows
which, recalling Lemma 4.1, shows the conclusion if \(t\in \left( \frac{1}{2}-\delta ,\frac{1}{2}+\delta \right) \).
In the other case, namely when \(|t-1/2|>\delta \), we can fix a positive constant \(\sigma _{0}\) such that
Indeed, if (5.21) were not true, we could find positive sequences \((\sigma _{n}),\,(t_{n}),\,(T_{n})\) such that, up to subsequences, \(\sigma _{n}\rightarrow 0\), \(t_{n}\rightarrow t_{0}\in [0,1]\), \(T_{n}\rightarrow T_{0}\in [0,L]\) (thanks to (3.16)), and
Since \(\sigma _{n}\rightarrow 0\), it results \((1-t_{0})T_{0}=1\) and \(t_{0}T_{0}=1\), so that \(T_{0}\ne 0\) and \(t_{0}T_{0}=(1-t_{0})T_{0}\), which implies that \(t_{0}=1/2\), that is not possible since \(|t_{0}-1/2|>\delta \). Then (5.21) is true and we also claim that there exists a positive constant \(\alpha _{0}\) such that
where \(h_{v}(p)\) is defined in (5.19). Indeed, by contradiction, there exists a sequence \(\alpha _{n}\) converging to zero, and a sequence of point \(p_{n}\in [0,L]\) such that
Up to a subsequence, there exists \(p_{0}\in [0,L]\) such that \(p_{n}\rightarrow p_{0}\). Taking limit in (5.23), we find
Since \(\overline{p}=1\) is the unique maximum point of \(h_{w}(p)\), there results
yielding \(p_{0}=\overline{p}=1\) contradicting the first inequality in (5.24), so that (5.22) is true. Assume that the first inequality in (5.21) is true, then we can make the choice \((p,w)=(tT^{R}_{t},w^{Ry})\) in (5.22) yielding
Then, using this information in (5.17), one gets
and an analogous argument gives the same conclusion when the second inequality in (5.21) holds true. This, Lemma 4.1 and (5.20) yield (5.16) giving the conclusion. \(\square \)
Proof of Theorems 2.2 and 1.1
Lemma 5.3 yields all the required geometrical properties to apply a linking theorem. Moreover, from Lemmas 5.2 and 5.4, we deduce that
Indeed, (b) in (5.25) follows directly from Lemma 5.4 as \(h:=id\in {\mathcal H}\), so that
In order to show (a) in (5.25), observe that exploiting property (5.10) we can say that for every \(h\in {\mathcal H}\) there exists \(v\in Q\) such that \(h(v)\in S\), so that
where the last inequality is proved in Lemma 5.2. Corollary 4.6 implies that the (PS) condition is satisfied at level d. Then we can apply the linking theorem (see, e.g., Theorem 8.22 in [5] or Theorem 8.4 in [37] joint with Lemmas 5.14 and 5.15 in [39]) to obtain the existence of a constrained critical point of I in \({\mathcal N}_{\Omega }\). This, taking into account Remark (2.5), gives the conclusion.
Theorem 1.1 follows directly from Theorem 2.2 as f given in (1.2) satisfies all the requested hypotheses. \(\square \)
Remark 5.5
The previous topological argument shows that the set \(\Psi _{R}(\partial {B_R(0)})\) is contractible in the sublevel set \(I_{\Omega }^{\overline{c}}\) with \(\overline{c}\) given by
Because the whole set \(\Psi _{R}(\overline{B_R(0)})\) is contractible in \(I_{\Omega }^{\overline{c}}\), but \(\Psi _{R}(\partial {B_R(0)})\) is not contractible in the sublevel set \(I_{\Omega }^{\underline{c}}\) for \(\underline{c}\) given by
Moreover, we believe that using appropriate topological tools one can show also multiplicity results depending on the topology of \(\Omega \). We leave this topic as an interesting, in our opinion, open problem.
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Acknowledgments
Part of this work has been done while the second author was visiting the University of Brasilia. She wishes to thank all the departamento de Matemática for the warm hospitality. A special thank goes to the Liliane’s family for the hospitality and the friendly atmosphere. The authors thank the anonymous referee whose important comments helped them to improve their work.
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Research of the first author is partially supported by CNPq/Brazil, PROEX/CAPES and FAPDF193.000.939/2015.
Research of the second author is partially supported by MIUR-PRIN-2012-74FYK7 Grant: “Variational and perturbative aspects of nonlinear differential problems,” the Italian Project FIRB 2012: “Dispersive dynamics: Fourier Analysis and Variational Methods,” “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), University Project “Sostegno alla ricerca individuale per il trienno 2015–2017”.
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Maia, L.A., Pellacci, B. Positive solutions for asymptotically linear problems in exterior domains. Annali di Matematica 196, 1399–1430 (2017). https://doi.org/10.1007/s10231-016-0621-4
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DOI: https://doi.org/10.1007/s10231-016-0621-4