A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

In the present paper, we show how to define suitable subgroups of the orthogonal group O(d-m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O}(d-m)$$\end{document} related to the unbounded part of a strip-like domain ω×Rd-m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \times {\mathbb {R}}^{d-m}$$\end{document} with d≥m+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge m+2$$\end{document}, in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of H01(ω×Rd-m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$\end{document} which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document}, as for instance, the ones due to Bartsch and Willem. The techniques used here are new.


Lack of Compactness and Symmetries on Unbounded Domains
Several important problems arising in many research fields such as physics and differential geometry lead to consider semilinear variational elliptic equations defined on unbounded domains of the Euclidean space and a great deal of work has been devoted to their study. From the mathematical point of view, probably the main interest relies on the fact that often the tools of nonlinear functional analysis, based on compactness arguments, cannot be used, at least in a straightforward way, and some new techniques have to be developed.
The seminal paper [15] by Lions has inspired a (nowadays usual) way to overcome the lack of compactness by exploiting symmetry. This approach is fruitful in the study of variational elliptic problems in presence of a suitable continuous action of a topological group on the Sobolev space where the solutions are being sought.
Along this direction, in the present paper, we exploit a group theoretical scheme, raised in the study of problems which are invariant with respect to the action of orthogonal subgroups, to show the existence of multiple solutions distinguished by their different symmetry properties. We emphasize that a wide class of nonlinear problems of this kind can be handled by constructing suitable subspaces, of "partially symmetric" functions, of the ambient Sobolev space, and by applying an appropriate version of the so-called Principle of Symmetric Criticality proved in the seminal paper [21] by Palais. For instance, let R + 0 = [0, ∞), let ψ 1 , ψ 2 : R + 0 → R be two functions that are bounded on bounded sets, with ψ 1 (t) < ψ 2 (t) for every t ∈ R + 0 and consider the strip-like domain of the Heisenberg group H n = C n × R, n ≥ 1, given by ψ := q = (z, t) ∈ C n × R : ψ 1 (|z|) < t < ψ 2 (|z|) .
The existence of weak solutions for subelliptic problems set on ψ has been investigated in [9,16,20] by employing symmetries. The main proofs, crucially based on the Palais Principle, are obtained by developing a suitable algebraic procedure on the unitary group U(n) := U (n) × {id} that fits well with the approach developed along the present paper. This group acts continuously on the Folland-Stein space H W 1,2 0 ( ψ ) by the action : U(n) × H W 1,2 0 ( ψ ) → H W 1,2 0 ( ψ ) pointwise defined by setting, for all τ := τ × id with τ ∈ U (n), ( τ u)(q) = u(τ −1 z, t) for a.e. q = (z, t) ∈ H n .
Here, we are interested on problems settled in strip-like domains of the Euclidean space R d . Without loss of generality, fixed m ∈ N we consider a strip-like domain ω × R d−m , where ω ⊂ R m is an open-bounded Euclidean domain with smooth boundary ∂ω. If d ≥ m +2, we exploit some compact embedding of the space H 1 0,cyl (ω ×R d−m ) of "cylindrically symmetric" functions of the Sobolev space H 1 0 (ω × R d−m ) into the Lebesgue space L ν (ω × R d−m ) for all ν ∈ (2, 2 * ), 2 * := 2d/(d − 2). Subsequently, assuming that d ≥ m + 4, more partial symmetries (in addition to the so-called block radial symmetries) can be used and so more distinct subspaces of H 1 0 (ω × R d−m ) can be introduced on which one can recover compactness (see Proposition 2.2).
The proof of the main compactness result given in Proposition 2.2 somehow follows by [

Fi x H d,m,i (H
are the so-called block-radial subspaces ofH 1 0 (ω × R d−m ) and therefore, by [15,Théorème III.2], are compactly embedded in L ν (ω × R d−m ) for all ν ∈ (2, 2 * ), they are not "mutually disjoint". So, more effort must be done to get the multiplicity result.
To this aim, by adapting the arguments of [13,Theorem 2.2] introduced in the whole Euclidean space, we define on R d−m the involution function η d,m,i (see (2.8) , (see Proposition 2.2) and with the property of being "mutually disjoint", i.e. their mutual intersection reduces to the trivial space, as proved in Proposition 2.3. For the precise statements and the related details see Sect. 2.2.
The new key results, of independent interest, given by Proposition 2.2 and Proposition 2.3 describe a sort of flower-shape geometry in ) introduced above plus the subspace of cylindrically symmetric functions H 1 0,cyl (ω × R d−m ). The advantage of this new type of symmetries in the study of nonlinear Dirichlet problems on strip-like domains has been investigated in Theorem 1.1 and Theorem 1.2 by using variational and topological methods. In particular, while, to get the existence result, we apply the variational argument to the space , to obtain the multiplicity result, we use the same approach in each petal Fi x H d,m, η i (H 1 0 (ω×R d−m )), and so we must require that τ d,m ≥ 1 (and, therefore, that d = m + 4 or d ≥ m + 6) and that the nonlinear term appearing in the equation satisfies suitable symmetry assumptions to assure that the functional associated with the problem is invariant with respect to the action of the group H d,m, η i (according to the statement in (3.34)).

Nonlinear Problems on Strip-Like Domains
In the present paper, we are interested in getting existence and multiplicity results of weak solutions to the following problem where λ is a positive parameter and ω × R d−m is an unbounded strip of R d , being ω an open bounded subset of R m with smooth boundary ∂ω and d, m ∈ N, d ≥ m + 2. Moreover, we assume that α : ω × R d−m → R verifies the following integrability, symmetry and sign conditions α ≥ 0 a.e. in ω × R d−m and there exist r > 0 and α 0 > 0 such that where B(0, r ) is the ball in R d−m centred at 0 with radius r , while on f : R → R we require the next hypotheses where F is the following antiderivative of the function f where 2 * is the critical Sobolev exponent given by 2 * := 2d/(d − 2) . As a model for f we can consider, for fixed q ∈ (2, 2 * ), the function Assumption ( f 3 ) is the well-known Ambrosetti-Rabinowitz condition, which is a superlinear assumption on the term f , namely a superquadratic one on its antiderivative F at infinity.
In this paper, we want to study Problem (P λ ) also under sublinear conditions at infinity on the nonlinearity f . More precisely, we shall also consider the case in which, instead of ( f 3 ), the function f satisfies the following hypotheses where F is given in (1.2). Note that when ( f 5 ) is satisfied then, thanks to ( f 2 ), condition ( f 4 ) is also guaranteed. A prototype for f is given by the odd extension of the function f r defined on R 0 + by setting with fixed q ∈ (2, 2 * ). Problem (P λ ) has a clear variational structure, indeed its solutions can be found as critical points of the following energy functional defined by setting for all u ∈ Since the problem is set on the strip-like domain ω×R d−m , there is no compactness property which can be used with I λ on the whole space. Hence, in order to find a weak solution to Problem (P λ ), we shall choose a suitable subspace of H 1 0 (ω × R d−m ) which allows us, from one side, to recover compactness and to get, by an application of the Mountain Pass Theorem (see [3]), a constrained critical point for the energy functional I λ and, from the other side, to apply the Principle of Symmetric Criticality got by Palais in [21] (see also [27] for some applications) to show that the restriction to that subspace does not play any role.
Finally, when d = m + 4 or d ≥ m + 6 and the nonlinearity f is odd, by exploiting the flower-shape geometric structure in the Sobolev space H 1 0 (ω × R d−m ) described in Sect. 2, we get a multiplicity result for Problem (P λ ), using again variational and topological arguments.
More precisely, our main results for Problem (P λ ) are stated in Theorem 1.1 and Theorem 1.2 below, for the superlinear and, respectively, for the sublinear growth of the nonlinearity at infinity.
In the superlinear framework, our result reads as follows: We would remark that the number s d,m is equal to τ d,m + 1 (with τ d,m given in (1.1)). Actually, the s d,m sequences of weak solutions to Problem (P λ ) found in Theorem 1.1 are characterized by different symmetries since they are found as critical points of the energy functional I λ in s d,m subspaces of H 1 0 (ω × R d−m ) which are "'mutually disjoint" (in the sense that their mutual intersection reduces to the trivial space, see, for a precise statement, Proposition 2.3).
In the sublinear setting, our main result for Problem (P λ ) is stated here below. It is an open problem to establish whether or notλ equals λ * E and if Problem (P λ ) has or not a nontrivial solution for λ =λ or λ = λ * E , as well as to check whether λ * E is exactly λ * M or not. The proof of part (i) in Theorem 1.2 is a rather straightforward consequence of the definition of weak solution to (P λ ) and of the sublinear assumption ( f 5 ) on f . While part (ii) and (iii) in Theorem 1.2 are just a byproduct of a more general existence and multiplicity result for the following problem, which actually can be used to study the stability of Problem (P λ ) with respect to changes of the nonlinearity: where λ and μ are positive parameters, β verifies and g : R → R is a function satisfying g is continuous in R (g 1 ) sup t∈R\{0} |g(t)| |t| + |t| q−1 < +∞ for some q ∈ (2, 2 * ).
(g 2 ) With respect to Problem (P λ,μ ), our main result reads as follows: ). The proof of Theorem 1.3 relies on an abstract critical points result due to Ricceri (see [25,Theorem 2] or Theorem 4.1 below) and again on the flower-shape geometric structure on the Sobolev space H 1 0 (ω×R d−m ) introduced in Sect. 2 and on the Principle of Symmetric Criticality.
We observe that, of course, λ * * E and λ * * M are greater than the constant λ in Theorem 1.2-(i). Moreover, Theorem 1.3 asserts that the existence result of solutions to Problem (P λ ) is stable with respect to small perturbations of the nonlinearity which are of superlinear and subcritical growth type.
The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space R d , as for instance, the ones obtained in the papers [5,6] due to Bartsch and Willem (see also [17]).
The techniques performed in this paper are new. Our abstract approach is in the spirit of the theoretical setting developed by Bartsch and Willem in [5,6], as well as of some recent contributions got in [13], see also the recent book [14] and references therein.
Several research perspective naturally arises exploiting the flower shape geometry constructed along the present paper: for instance, an interesting open problem is to investigate the existence of multiple solutions for nonlocal problems, as in [2], under the action of exterior topological groups (see, for instance, [12] for additional comments and related topics).
The present paper is organized as follows. In Sect. 2, we introduce the abstract setting which allows us to reveal a flower-shape geometry in the Sobolev space . In Sect. 3 we deal with the nonlinear Problem (P λ ) and we prove some existence, non-existence and multiplicity results for it, by using some classical theorems in critical points theory and the geometric construction given in Sect. 2. Section 4 is devoted to Problem (P λ,μ ), which is a nonlinear perturbation of (P λ ) with sublinear growth, and to the proof of Theorem 1.3.

A Flower-Shape Geometry in Sobolev Spaces
In this section, we construct a flower-shape geometry in the Sobolev space a finite number of spaces, "mutually disjoint" and characterized by different symmetries, which are compactly embedded into the classical Lebesgue spaces. These properties will be crucial for getting the existence and multiplicity results for the nonlinear problems (P λ ) and (P λ,μ ).

Preliminaries and Notations
In this subsection, we give some preliminaries and we introduce the notation used along the present paper. Here and in the sequel is the classical Lebesgue space with norm defined as follows Since the embedding endowed with the natural multiplication law which maps any pair Here {id m } denotes the trivial group in R m with the natural product and, from now on, in order to simplify the notation, we shall omit the · symbol.
The map * induces the natural action i.e. in a more involved form, Along the present paper we denote by Fi of cylindrically symmetric functions given by We list below the following properties (see, the celebrated paper [10]) for the embedding of H 1 0,cyl (ω × R d−m ) into Lebesgue spaces: and Now, let either d = m + 4 or d ≥ m + 6, so that the set I d,m defined by (1.1) is nonempty. Then, for any i ∈ I d,m , by "grouping together" the d − m variables of the unbounded part of the strip in blocks of at least two variables, we get τ d, which define the subgroups

The sets
are known in the literature as the subspaces of block-radial (or block-cylindrical) functions of H 1 0 (ω × R d−m ) and they are compactly embedded into L ν (ω × R d−m ) for every ν ∈ (2, 2 * ) (see [15,Théorème III.2]). Unfortunately, these τ d,m subspaces are not "mutually disjoint" and, in term of our problems (P λ ) and (P λ,μ ), this is an obstacle to get a multiplicity result for them.
In order to overcome this difficulty, for any i ∈ I d,m : By (2.2) it is easily seen that for any i ∈ I d,m Finally, for every i ∈ I d,m , we consider the compact subgroup of O(d − m) given by and the action Bearing in mind (2.3) and fixing i ∈ I d,m , the action i can be written, for a.e.
In the next subsection, we prove some interesting properties of this space.

Compactness and Symmetries
In this subsection, we show that each one of the τ d,m spaces Fi for any ν ∈ (2, 2 * ) and we prove some geometric properties for them.

Remark 2.1
We recall that "symmetry" allows to recover compactness when it involves at least two variables. So, any block of variables on which one asks for symmetry should be at least of dimension 2. Thus, the simplest possible setting is the blockradial symmetry in four dimensional Euclidean space with two 2-dimensional blocks. This justifies the requirement d ≥ m + 4 all along this subsection.
With respect to the compactness result, we get that (2.6) and (2.7) hold if we replace Precisely, our result reads as follows: ) be defined as in (2.13). Then, the embedding • is continuous for any ν ∈ [2, 2 * ] • is compact for any ν ∈ (2, 2 * ) .
Proof Let us fix i ∈ I d,m . Since H d,m,i ⊂ H d,m, η i , the first relation of (2.11) (or, equivalently, of (2.12)) and the continuity of the action i imply that Fi [15,Théorème III.2]). Hence, the embedding is also continuous for any ν ∈ [2, 2 * ] and compact for any ν ∈ (2, 2 * ) and this ends the proof of Proposition 2.2.
) are mutually disjoint, as stated here below. A key point along the proof of Proposition 2.3 is the transitive

Proposition 2.3 Let either d
) be defined as in (2.13). Then, the following statements hold: As a consequence of (2.15), the function u is cylindrically symmetric, and we can apply (i) thus obtaining that u is identically zero in ω × R d−m . This concludes the proof of Proposition 2.3.
We suggest the recent monograph [22] as a comprehensive reference for preliminaries and, in particular, for the main properties related to Sobolev spaces.

Dirichlet Problems on Strip-Like Domains
This section is devoted to the study of the nonlinear Problem (P λ ), under either superlinear assumption on the nonlinearity f at infinity (see ( f 3 )) or sublinear condition again at infinity (see ( f 5 )).
As already remarked, since the equation in (P λ ) has a variational nature, its weak solutions can be seen as critical points of the energy functional I λ defined by (1.3). It is standard to see that, thanks to (α 1 ) and ( f 2 ), ( f 4 ) in the superlinear setting or ( f 2 ), ( f 5 ) (note that ( f 2 ) and ( f 5 ) imply ( f 4 )) in the sublinear framework, the functional I λ is well defined on . We shall prove in Sect. 3.1 and in Sect. 3.2, respectively, the existence and multiplicity results stated in Theorem 1.1 and in Theorem 1.2.

Problem (P ) with Superlinear Growth at Infinity
In this subsection, we study the semilinear equation (P λ ), when f satisfies the Ambrosetti-Rabinowitz condition ( f 3 ). The main tools are given by the Mountain Pass Theorem of Ambrosetti and Rabinowitz (see [3,23]), the Principle of Symmetric Criticality of Palais (see [21]) and the flower-shape geometry in the Sobolev space This subsection is devoted to the proof of Theorem 1.1: in particular in Sect. 3.1.1 we prove the existence result (i), while in Sect. 3.1.2 we prove the multiplicity result (ii) of nontrivial weak solutions to Problem (P λ ).

A Mountain Pass Existence Result for Problem (P )
In this subsection, we prove the existence result stated in Theorem 1.1, by applying the Mountain Pass Theorem to the energy functional I λ defined in (1.3).
As it is well known, in order to follow this strategy, it is necessary to have some compactness properties on the functional, and so we shall exploit (2.7) by working, with fixed λ > 0, with the functional J λ defined as the restriction of I λ to the space The main ingredients of our proof are the application of the following results: • the Mountain Pass Theorem by Ambrosetti and Rabinowitz (see [3]) to get a critical point u λ ∈ H 1 0,cyl (ω × R d−m ) for the functional J λ ; • the Principle of Symmetric Criticality by Palais (see [21]) to prove that H 1 0,cyl (ω × R d−m ) is a natural constraint for the functional I λ , i.e. critical points of I λ constrained on H 1 0,cyl (ω × R d−m ) are actually critical points of First of all, let us show that J λ satisfies the geometric Mountain Pass structure. For this, note that by conditions ( f 2 ) and ( f 4 ), it is standard to see that for any ε > 0, there exists δ = δ(ε) > 0 such that for any t ∈ R | f (t)| ≤ ε|t| + δ(ε)|t| q−1 (3.2) and, as a consequence, such that Now, let us proceed by steps.

Claim 3.1.3
The functional J λ satisfies the Palais-Smale condition at any level c ∈ R, that is for any sequence (u k ) k in H 1 0,cyl (ω × R d−m ) such that, as k → +∞, there exists u ∞ ∈ H 1 0,cyl (ω × R d−m ) such that, up to a subsequence, Proof Let (u k ) k be a Palais-Smale sequence for J λ , i.e. a sequence satisfying (3.8) and (3.9) for some fixed c ∈ R. First of all, let us prove that (u k ) k is bounded in . At this purpose, note that, by (3.8) and (3.9), it easily follows that for a suitable positive constant κ, where σ is the constant in ( f 3 ). Moreover, thanks to (α 3 ) and ( f 3 ), we get that, for any k ∈ N, So, by combining (3.11) and (3.12) we get, for a suitable positive constant κ * , that Hence, the sequence (u k ) k is bounded in H 1 0,cyl (ω × R d−m ) and so, by definition of J λ and (3.9), we have that, as k → +∞, is a reflexive space, we also get, up to a subsequence, still denoted by (u k ) k , that there exists (3.14) Moreover, by applying the compact embedding (2.7), we get, again up to a subsequence still denoted by (u k ) k , that, as k → +∞ for any ν ∈ (2, 2 * ) , (3.15) and, as a consequence, that u k → u ∞ a.e. in ω × R d−m as k → +∞ , (3.16) while, by using the continuous embedding (2.6), we deduce that there exist two positive constants κ 2 and κ 2 * such that u k 2 ≤ κ 2 and u k 2 * ≤ κ 2 * for any k ∈ N . (3.17) Now, we claim that, as k → +∞, Indeed, by ( f 1 ) and (3.16), we get that Moreover, since α satisfies condition (α 1 ), it is easy to see (since Now, by (2.1), (3.2) with ε = 1 and set δ := δ(1), by (3.20) and the Hőlder Inequality, we have that, set q := q/(q − 1), the conjugate exponent of q, for any k ∈ N, where C q is the constant given in (2.1) with ν = q. Since (u k ) k is bounded in H 1 0,cyl (ω ×R d−m ), by (3.21) we deduce that the sequence α(·) f (u k (·)) k is bounded in L q (ω × R d−m ), which, together with (3.19), yields that as k → +∞. Then, we get (3.18) by testing this weak convergence with u ∞ . Now, we claim that, as k → +∞, Indeed, by (3.2), the Hőlder Inequality and (3.17), we have that Now, we are in position to conclude our proof. Indeed, as a consequence of (3.13) and (3.25) we deduce that (3.27) as k → +∞. So, by using (3.14) and (3.18) in (3.27), we obtain (3.28) Therefore, (3.26) and (3.28) state that Finally, thanks to (3.14) and (3.29), we have that   and so, as a consequence, we deduce that, for any g ∈ O(d − m), with g = id m × g , g ∈ O(d − m), and for any u ∈ H 1 0 (ω × R d−m ), (3.33) Hence, by (3.31) and (3.33) we obtain, by the Principle of Symmetric Criticality, that u λ is a critical point of I λ . Then, we have shown the existence of a nontrivial weak solution u λ to Problem (P λ ), with cylindrical symmetry, concluding the proof of Theorem 1.1-(i).

A Multiplicity Result for Problem (P )
This subsection is devoted to the proof of the multiplicity result stated in Theorem 1.
Furthermore, as usual when dealing with odd nonlinearities, we apply the Symmetric Mountain Pass Theorem due to Ambrosetti-Rabinowitz (see again [3]) to our functional. Now, we give the following claims, stated for fixed λ > 0 and i ∈ I d,m .
Proof The claim follows verbatim the proof of Claim 3.1.1.
As for the geometry required by the Symmetric Mountain Pass Theorem, we need the next property on J λ,i :

Claim 3.1.5 For any finite dimensional subspace F of Fi x H d,m, η
) and let u ∈ F. By using the same arguments considered in Claim 3.1.2, we have, see (3.7), that and so, by taking into account that in F, all the norms are equivalent and that σ > 2, we get that ) and J λ with J λ,i . By taking into account Proposition 2.2 and using the same arguments considered in the proof of Claim 3.1.3, we easily have that the functional J λ,i satisfies the Palais-Smale (compactness) condition at any level c ∈ R. In addition, it fulfills the geometric conditions stated in Claim 3.1.4 and Claim 3.1.5.
Now, since f is odd, by the Symmetric Mountain Pass Theorem (see [3] and also the version given in [24,Chapter 1]) applied to the functional J λ,i , we obtain the existence of an unbounded sequence (u (i) λ,k ) k of critical points ) and for any k ∈ N . Now, we claim that Fi ) is a natural constraint for I λ , i.e. u (i) λ,k is a critical point of I λ for any k ∈ N. Indeed, not only the action i , defined by (2.11), of the group H d,m, η i on the space H 1 0 (ω × R d−m ) is an isometry, but also I λ is invariant with respect to the action i of the group H d,m, η i . Indeed, since f is odd (and so F is even) and H d,m, η i is a subgroup of the group O(d − m), by (3.32), we have that Then, by applying the Principle of Symmetric Criticality of Palais to I λ , we get that each u ) is a nontrivial weak solution for Problem (P λ ) for any k ∈ N.
Finally, we note that, by Proposition 2.3-(i), for any i ∈ I d,m and any k ∈ N,

Remark 3.1
In order to assure the invariance (see (3.34)) of the functional I λ with respect to the action i of the group H d,m, η i , for any i ∈ I d,m , it is not enough, (as instead happens for the analogous property (3.33)), to assume just the cylindrical symmetry property on the weight α (see condition (α 2 )). Indeed, by (2.12), we have and the presence of the minus sign in the second case makes the evenness of I λ necessary to get the invariance (3.34) and this justifies the oddness requirement on f while getting the multiplicity result. (The same will be true while getting the invariance of the functional I λ,μ associated to Problem (P λ,μ ), and this justifies the oddness requirement on both f and g).

Problem (P ) with Sublinear Growth at Infinity
In this subsection we consider the semilinear Problem (P λ ) in the case in which the term f satisfies sublinear growth assumptions at infinity, namely condition ( f 5 ) (and ( f 6 )) instead of ( f 3 ) and we prove Theorem 1.2. As already remarked, assumptions ( f 2 ) and ( f 5 ) imply ( f 4 ).

Proof of Theorem 1.2 Let us start with assertion (i).
First of all, note that conditions ( f 1 ), ( f 2 ), ( f 5 ) and the Weierstrass Theorem yield that there exists a positive constant κ f , depending on f , such that Now, we argue by contradiction and we assume that there exists a sequence (λ k ) k in R + 0 such that λ k → 0 ask → +∞, (3.37) and such that Problem (P λ ) with λ = λ k admits a nontrivial weak solution u k ∈ H 1 0 (ω × R d−m ) for any k ∈ N. Thus, by taking u k as a test function in the equation and by using (3.36), we get that for any k ∈ N, where C 2 is the constant given in (2.1) with ν = 2. So, unless u k ≡ 0 for all large enough k, we would deduce λ k ≥ (κ f C 2 2 α ∞ ) −1 in contradiction with (3.37) for infinitely many values of k. Hence, the non-existence result stated in (i) is proved.
Finally, for what concerns assertions (ii) and (iii), here we just observe that Problem (P λ ) is a particular case of Problem (P λ,μ ), with μ = 0. So, assertions (ii) and (iii) are a consequence of Theorem 1.3 (whose proof will be provided in Sect. 4). This concludes the proof of Theorem 1.2.

A Nonlinear Perturbation of Problem (P ) with Sublinear Growth
In this section we deal with Problem (P λ,μ ), which can be seen as a nonlinear perturbation of Problem (P λ ). Precisely, here we prove the existence and multiplicity results stated in Theorem 1.3.
Weak solutions to Problem (P λ,μ ) can be found as critical points of the energy functional I λ,μ : H 1 0 (ω × R d−m ) → R naturally associated with it and defined by setting, for any u ∈ H 1 0 (ω × R d−m ), where F is the function defined in (1.2) and G is analogously given by Under the assumptions ( f 1 ), ( f 2 ), ( f 5 ), (g 1 ), (g 2 ), (α 1 ), (β 1 ) and thanks to the embeddings in (2.1), it is standard to check that I λ,μ is well defined on and that I λ,μ ∈ C 1 (H 1 0 (ω × R d−m )) with . For the proof of Theorem 1.3, the main tools are the following ones: • the abstract critical points result stated in [25,Theorem 2] (see also Theorem 4.1 below), which assures the existence of multiple critical points for a suitable functional; • the Principle of Symmetric Criticality due to Palais (see [21]); • the flower-shape geometry in the Sobolev space

Existence of At Least Two Nontrivial Weak Solutions
This subsection is devoted to the proof of the existence of at least two nontrivial weak solutions of Problem (P λ,μ ) in H 1 0 (ω × R d−m ) with cylindrical symmetry. In order to do this, we use the abstract critical points result [25, Theorem 2] due to Ricceri, stated here below for the reader's convenience. : X → R be a coercive, sequentially weakly lower semicontinuous C 1 functional, bounded on each bounded subset of X , whose derivative admits a continuous inverse in the dual of X and such that any sequence (x k ) k ⊂ X such that x k → x weakly in X and lim inf admits a strongly converging subsequence.
has at least three solutions whose norms are less than r .
By looking at the functional I λ,μ , we shall apply Theorem 4.1 by taking X = and so that, since solutions to (4.6) give critical points of I λ,μ constrained on H 1 0,cyl (ω × R d−m ). Then, by using the Principle of Symmetric Criticality by Palais, we get at least three solutions to Problem (P λ,μ ). (1) The space X : as k → +∞. This and the weak convergence of (x k ) k imply that x k → x in X as k → +∞. Therefore, (4. Then, set λ * * E := b −1 , we get that, for any λ > λ * * E , there exists μ λ,E > 0 such that for any μ ∈ [0, μ λ,E ] the functional I λ,μ admits two nontrivial critical points u λ,μ andũ λ,μ constrained on H 1 0,cyl (ω × R d−m ). Finally, thanks to (α 2 ) and (β 2 ), we can apply the Principle of Symmetric Criticality by Palais (arguing as in (3.31) and (3.33)) and deduce that u λ,μ andũ λ,μ are critical points of I λ,μ in H 1 0 (ω × R d−m ), i.e. these critical points are solutions to Problem (P λ,μ ). This ends the proof of Theorem 1.3-(i). Now, the remaining part of this subsection will be devoted to state and prove the lemmas and claims used in the proof of Theorem 1.3-(i).

Proof of Theorem 1.3-(i):
First of all, we start by proving the required compactness property of the functionals J and , as stated in the following lemmas, in which the compactness of the embedding for any ν ∈ (2, 2 * ) (see (2.7)) plays a crucial role.
. Proof The proof of this assertion is quite standard: we repeat it here just for the reader's convenience.

Lemma 4.3
Assume (g 1 ), (g 2 ), (β 1 ) and (β 2 ). Then, the functional ∈ C 1 (H 1 0 (ω × R d−m )) and is compact in Proof In order to get that the functional ∈ C 1 (H 1 0 (ω × R d−m )) we can argue as in the proof of Lemma 4.2, by taking into account assumptions (β 1 ), (g 1 ) and (g 2 ), while the proof of the compactness of is a more delicate question since we can not use the Dominated Convergence Theorem as in the previous lemma (indeed, the function g is not necessarily sublinear and so it does not need to satisfy a relation analogous to (3.36)).
Let For this purpose, note that by (g 2 ) there exists a positive constant C > 0 such that |g(t)| ≤ C(|t| + |t| q−1 ) for all t ∈ R (4.16) so, by (β 1 ), (2.1) (twice applied with ν = 2 and ν = q), (2.6) and the Hőlder Inequality, we have that, for any k ∈ N, . Hence, (4.15) is proved. As a consequence of (4.15), there exists H in the dual space of H 1 0,cyl (ω × R d−m ) such that, as k → +∞, In order to complete the proof, we need to prove that To get this goal, we argue by contradiction and we suppose that there exists δ > 0 and k * ∈ N such that δ − 1 k * > 0, and and (2.7) holds, up to a subsequence, still denoted by (ϕ k ) k , there exists ϕ ∞ ∈ H 1 0,cyl (ω × R d−m ) such that, as k → +∞, and Now, by (β 1 ) (which yields that β ∈ L ν (ω × R d−m ) for any ν ∈ [1, +∞], see the analogous argument (3.20) for the weight α), (g 2 ), the Hőlder Inequality and (4.16), we have that as k → +∞, thanks to (4.22) and to the boundedness of (u k ) k in H 1 0 (ω × R d−m ). Here C is the positive constant in (4.16) and C q is the Sobolev embedding constant given in (2.1) with ν = q and q := q/(q − 1), the conjugate exponent of q.
Moreover, due to (4.21) and, respectively, to (4.18), we have that as k → +∞. Therefore, by (4.23), we have that which contradicts (4.20), since δ − 1 k > δ − 1 k * > 0 for all k > k * . Hence, (4.19) holds and this, as already said, ends the proof of Lemma 4.3. Now, we state the claims concerning the functional and J , used in the proof of Theorem 1.3-(i) in order to get the inequality a < b between the constants a and b defined by (4.4) and (4.5), respectively. Thus, passing to the limsup as u H 1 0 → 0 in the above inequality and by taking into account that ν > 2 and the arbitrariness of ε, we get the assertion, concluding the proof of Claim 4.1.2.
In the next claim, assumptions (α 3 ) and ( f 6 ) are essential.

A Multiplicity Result
In this subsection we provide the multiplicity result stated in Theorem 1.3-(ii). In order to get this goal, the main idea consists in applying, for any i ∈ I d,m (see (1.1)), {λ * E , λ * i }, we get that for any λ > λ * M there exists μ λ,M > 0 such that for any μ ∈ [0, μ λ,M ], the functional I λ,μ admits two nontrivial critical points u λ,μ,i andũ λ,μ,i constrained on Fi x H d,m, η i (H 1 0 (ω × R d−m )). Then, thanks to (α 2 ) and (β 2 ) and the oddness assumption on f and g, we can apply the Principle of Symmetric Criticality by Palais, (see Remark 3.1, (3.31) and (3.33) also with α and f replaced by β and g, respectively) and deduce that these critical points are solutions to Problem (P λ,μ ).
So, we conclude this subsection just by proving the natural counterpart of  In addition, by (4.31) and (4.42), we get that u ε,i ∞ ≤ t 0 |u ε,i (x, y)| = t 0 for a.e. (x, y) ∈ K × S ε,i . α(x, y)F(u ε,i (x, y)) dx dy as ε → 1 − . As a consequence of this, by choosingū i = u ε,i with ε sufficiently close to 1, we obtain (4.37), and so Claim 4.2.1 is proved.