Bounds for eigenfunctions of the Neumann p -Laplacian on noncompact Riemannian manifolds

: Eigenvalue problems for the p -Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q - or L ∞ -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.


Introduction and main results
Let Ω be a connected open set in an n-dimensional Riemannian manifold , which will be assumed to be without boundary throughout.Suppose that n ≥ 2 and where H n denotes the n-dimensional Hausdorff measure on , i.e. the volume measure on  induced by its Riemannian metric.In particular, if  = ℝ n equipped with the Euclidean metric, then H n agrees with the Lebesgue measure.The choice Ω =  is also admissible, provided that H n () < ∞.
We are concerned with eigenfunctions of the p-Laplace operator in Ω, subject to homogeneous Neumann boundary conditions on ∂Ω, if Ω ̸ = .Namely, we deal with solutions to the equation for some γ ∈ ℝ, satisfying the condition ∂u ∂n = 0 on ∂Ω, if ∂Ω ̸ = 0. Here, p > 1, and n stands for the normal unit vector on ∂Ω.
A unified definition of an eigenfunction u of the problems under consideration amounts to requiring that u ∈ W 1,p (Ω) and satisfies the equation for some γ ∈ ℝ and every test function ϕ ∈ W 1,p (Ω), where the dot " ⋅ " denotes the scalar product associated with the Riemannian structure on , and |∇u| denotes the norm of the gradient ∇u defined via this scalar product.
Classical variational methods ensure that the eigenvalue problems in question do admit non-trivial (i.e.non-constant) eigenfunctions under suitable assumptions on Ω; see, e.g., [37].This is the case if Ω has a compact closure and a regular boundary -a Lipschitz domain, for instance.The same conclusion holds if Ω =  and the latter is compact.The regularity of Ω also guarantees that any eigenfunction of problem (1.2) does not merely belong to L p (Ω), but is in fact globally essentially bounded in Ω.On the other hand, membership of eigenfunctions in L ∞ (Ω), and even in L q (Ω) for q > p, is not guaranteed if Ω is an arbitrary open set with The present paper is aimed at offering minimal assumptions on the geometry of Ω for any eigenfunction of problem (1.2) to belong to L ∞ (Ω), and to admit a corresponding bound of the form ‖u‖ L ∞ (Ω) ≤ c‖u‖ L p (Ω) (1.3) for some constant c depending on Ω, γ and p. Estimates in L q (Ω) for every q < ∞, namely inequalities of the type ‖u‖ L q (Ω) ≤ c‖u‖ L p (Ω) (1.4) for some constant c depending on Ω, γ, p and q, are also established under slightly weaker conditions on Ω.The description of the geometry of Ω adopted in our results calls into play certain functions defined in terms of inequalities of geometric-functional nature for subsets of Ω.They are the isoperimetric function and the p-isocapacitary function of Ω.
The isoperimetric function λ Ω is the largest non-decreasing function in [0, H n (Ω) 2 ] such that λ Ω (H n (E)) ≤ P(E; Ω) (1.5) for every measurable set E ⊂ Ω with H n (E) ≤ H n (Ω) 2 .Here, P(E; Ω) denotes the perimeter of E relative to Ω. Let us emphasize that, in view of our applications, only the asymptotic behavior of the isoperimetric function λ Ω near 0 is relevant.If, for instance, Ω has a Lipschitz continuous boundary and Ω is compact, or Ω =  and the latter is compact, then λ Ω (s) = O(s n−1 n ) near 0. (1.6) Here, the relation "O near zero" between two functions means that they are bounded by each other, up to positive multiplicative constants, for small values of their argument.
The behavior near 0 is also the only piece of information about the p-isocapacitary function which is needed for our purposes.This function is denoted by ν Ω,p , and is defined in analogy with λ Ω , save that the perimeter of a set E ⊂ Ω is replaced by its condenser capacity cap p (E, G) relative to any set G such that E ⊂ G ⊂ Ω.Therefore, ν Ω,p is the largest non-decreasing function in [0, H n (Ω)  2 ] which renders the inequality If Ω is sufficiently regular, as specified in connection with equation (1.6) for instance, then near 0.
More details can be found in the next section.Let us just mention here that the regularity of a domain Ω affects the decay of the functions λ Ω and ν Ω,p near 0. A more irregular geometry of the domain Ω is reflected in a faster decay to 0 when their argument approaches 0. In particular, neither λ Ω (s) nor ν Ω,p (s) can decay more slowly to 0 as s → 0 than the functions appearing on the right-hand sides of equations (1.6) and (1.8), respectively, whatever Ω is.
In particular, the special choice p = 2 in (1.2) reproduces the linear eigenvalue problem for the Neumann Laplacian.The analysis of spectral problems for this classical operator has been the center of numerous investigations, especially in the case when Ω agrees with a compact manifold.The vast bibliography on this topic includes the monographs [4,9] and the papers [5, 7, 11, 12, 23-26, 29, 30, 51, 57, 58, 60].Results for the Laplace operator in the noncompact case, in the same vein as those established here, can be found in [18].The papers [16,17] deal with related topics.Our approach in the nonlinear framework at hand has a start reminiscent of that of [18].However, different techniques have to be exploited in fundamental steps of the proofs of estimates (1.3) and (1.4).For instance, certain customary fixed point theorems, which are well suited for the linear case, do not fit the nonlinear setting.
All criteria that will be proposed are invariant under the replacement of ν Ω,p or λ Ω with equivalent functions near 0, in the sense of the relation "≈" defined as follows.Given two functions f, g : The conditions in terms of the isoperimetric function λ Ω ensuring bounds in L q (Ω) or L ∞ (Ω) for eigenfunctions are presented in our first result.Interestingly enough, the condition for L q -estimates is independent of p and q.The dependence on these exponents only enters the constant involved in the estimates.By contrast, the dependence on p is crucial in the condition for L ∞ -estimates.
Theorem 1.1 (Bounds for eigenfunctions via λ Ω ).Assume that n ≥ 2 and p > 1.Let Ω be a connected open subset of an n-dimensional Riemannian manifold  such that H n (Ω) < ∞.Let u be any eigenfunction of problem (1.2) associated with any eigenvalue γ.
Assumptions (1.9) and (1.11) are optimal, in a sense specified in the next theorem, for (1.10) and (1.12), respectively, to hold in classes of sets Ω with a prescribed decay of λ Ω at 0. In particular, the gap between condition (1.11), ensuring L ∞ (Ω) bounds for eigenfunctions, and condition (1.9), just yielding L q (Ω) bounds for any q < ∞, cannot be essentially filled.This can be demonstrated, for instance, via manifolds  of revolution as in Figure 1, whose isoperimetric function is equivalent to a function λ such that ≈ a non-decreasing function near 0.
(1.13)Such an assumption is consistent with the fact that, as noticed above, λ Ω decays as in (1.6) in the best possible case.
(i) Given any q ∈ (p, ∞), there exists an n-dimensional Riemannian manifold , with and the p-Laplacian on  has an eigenfunction u ∉ L q ().(ii) Let λ : [0, ∞) → [0, ∞) be any non-decreasing function, vanishing only at 0, such that Assume in addition that condition (1.13) is in force.Then there exists an n-dimensional Riemannian manifold , with H n () < ∞, fulfilling λ  (s) ≈ λ(s) near 0, and such that the p-Laplacian on  has an unbounded eigenfunction.
Although the isocapacitary function has a less transparent geometric meaning than the isoperimetric function, and its behavior can be more difficult to detect, it is in a sense more appropriate in the framework at hand.Its use provides yet finer conditions for L q -and L ∞ -estimates of eigenfunctions, which are exhibited in Theorem 1.3 below.Indeed, not only are these conditions optimal in classes of sets Ω whose isocapacitary function ν Ω,p has an assigned decay at 0 (see the subsequent Theorem 1.4), but there also exist specific sets and entire manifolds where the criteria of Theorem 1.3 apply, whereas those of Theorem 1.1 fail.Typically, this may happen in the presence of complicated geometric configurations.Instances of this kind of manifolds are those depicted in Figure 2 and discussed in Section 5.5.
The sharpness of condition (1.18) will be exhibited via manifolds  whose isocapacitary function is equivalent to a function ν such that either and such that the p-Laplacian on  has an unbounded eigenfunction.
Although the existence of eigenfunctions is not a main focus of this paper, we conclude this section by pointing out that it is ensured under the assumptions of Theorem 1.1 (i), and even under those of Theorem 1.3 (i).
Let Ω be a connected open subset of an n-dimensional Riemannian manifold  such that H n (Ω) < ∞.Assume that the p-isocapacitary function of Ω fulfills condition (1.17).Then there exists γ > 0 such that problem (1.2) admits an eigenfunction u.In particular, the same conclusion holds if the isoperimetric function of Ω fulfills condition (1.9).
Theorem 1.5 follows from an application of the Ljusternik-Schnirelman variational principle as in [37], thanks to the compactness of the embedding W 1,p (Ω) → L p (Ω), which holds under assumption (1.17) or (1.9).The compactness of this embedding is proved in [16,Theorem 2.4] when p = 2.The proof in the general case is analogous and is omitted.

Background
Let Ω be an open set in an n-dimensional Riemannian manifold , possibly Ω = , and let E be a measurable subset of .The perimeter P(E; Ω) of E relative to Ω can be defined by where ∂ * E stands for the essential boundary of E in the sense of geometric measure theory, and H n−1 denotes the (n − 1)-dimensional Hausdorff measure on  induced by its Riemannian metric.Recall that ∂ * E agrees with the topological boundary ∂E of E when the latter is sufficiently regular -a Lipschitz domain with compact closure, for instance.
Assume that H n (Ω) < ∞.The isoperimetric function of Ω is defined by Obviously, the function λ Ω is non-decreasing.The isoperimetric inequality relative to Ω is a straightforward consequence of the definition of λ Ω and has the form (1.5).
In particular, if Ω is connected, then as shown via an analogous argument as in [46,Lemma 3.2.4].The Sobolev space W 1,p (Ω) is defined, for p ∈ [1, ∞], by and is endowed with the norm Here, ∇ stands for the gradient operator, that is, covariant differentiation on .We denote by W 1,p 0 (Ω) the closure in W 1,p (Ω) of the set of continuously differentiable compactly supported functions in Ω.
The homogeneous Sobolev space V 1,p (Ω) is defined by If the set Ω is connected, and ω is an open set such that ω is compact and ω ⊂ Ω, then V 1,p (Ω) is a Banach space equipped with the norm Note that replacing ω by another set with the same properties results in an equivalent norm.
The isocapacitary function ν Ω,p of Ω is defined in analogy with (2.1), provided that the perimeter of a set E relative to Ω is replaced by its condenser capacity.Specifically, recall that the standard p-capacity of a set E ⊂  can be defined, for p ≥ 1, by 1,p (Ω) has a representative ũ, called the precise representative, enjoying the property that for every ε > 0 there exists a set E ⊂ Ω, with C p (E) < ε, such that ũ restricted to Ω \ E is continuous.The function ũ is unique, up to subsets of p-capacity zero.A pointwise property which holds up to sets of p-capacity zero is said to hold p-quasi everywhere.
In view of a classical result in potential theory (see, e.g., [39, Corollary 2.25]), we adopt the following definition of capacity of a condenser.Let E ⊂ G ⊂ Ω.Then we set The function ν Ω,p is clearly non-decreasing.The isocapacitary inequality (1.7) on Ω is a consequence of the very definition (2.2).If p > 1, then the function λ Ω is related to ν Ω,p via the inequality as shown by an analogous argument to the one in [46, Lemma 2.2.5].
For any measurable function u on Ω, we define the distribution function Note that here μ u is defined in terms of u, instead of |u| as customary.The signed decreasing rearrangement The median of u is defined by ).
Since u and u ∘ are equimeasurable functions, one has that where the representative of u appearing on the right-hand side is the one which renders the coarea formula true.Via (a version on manifolds of) [46, Lemma 2.2.2/1], one can deduce that, if 2 ). (2.6)

Proofs of Theorems 1.1 and 1.3
Our main task in the present section is to establish Theorem (1.3).Theorem 1.1 will then easily follow, thanks to inequality (2.3).The proof of Theorem (1.3) (ii) relies upon an analysis of an integral equation fulfilled by the signed rearrangement of any eigenfunction of problem (1.2).This integral equation is derived after obtaining an equation involving integrals of the eigenfunction over its level sets.Assumption (1.18) is the piece of information which, through inequality (2.6), enables us to deduce the existence and uniqueness of solutions to the integral equation in suitable spaces.
The proof of Theorem (1.3) (i) makes use of an iteration argument, which in turn rests on the Sobolev-type inequality contained in the following lemma.The inequality in question is standard in regular domains.The objective of the lemma is to show that it is also supported by domains which merely satisfy assumption (1.17).Lemma 3.1.Let  be an n-dimensional Riemannian manifold and let Ω be a connected open set in  such that H n (Ω) < ∞.Assume that condition (1.17) holds for some p > 1.Then for every ε > 0 there exists a constant c such that for every u ∈ W 1,p (Ω).
Proof.Fix any s ∈ (0, H n (Ω)/2), and let E be any compact set in Ω such that H n (Ω \ E) < s (such a set E certainly exists since Ω is, in particular, a locally compact, separable topological space with a countable basis).Let ξ be any continuously differentiable compactly supported function on Ω such that 0 ≤ ξ ≤ 1 and ξ = 1 in E. Denote by U the support of ξ .Consider the precise representative u of any function in W 1,p (Ω).
We have that Clearly, v ∈ W 1,p (Ω), and the support of v is contained in Ω \ E. Thus, 2 for every t > 0. Hence, by inequality (1.7), Owing to the monotonicity of capacity, Hence, by the very definition of the capacity of a condenser, From inequalities (3.3)-(3.5),one can infer that there exists a constant c such that Consequently, and, trivially, Inequalities (3.2), (3.6) and (3.7) imply that for some constant c.Now, let Ω  be a Lipschitz domain such that Ω  is compact, Ω  ⊂ Ω and U ⊂ Ω  .Our assumptions on Ω  ensure that a version of the standard Sobolev inequality holds, which tells us that for some constant c and for every u ∈ W 1,p (Ω  ).This follows, for instance, via an argument analogous to the one in [46, proof of Theorem 1.4.6/1].Inequality (3.1) follows from inequalities (3.8) and (3.9).
We are now in a position to prove our criteria for bounds of eigenfunctions to problem (1.2).
Proof of Theorem 1.3.(i) Let u be an eigenfunction of problem (1.2) and let t, α > 0. Define the function T t : ℝ → [0.∞) by T t (s) = min{|s|, t} for s ∈ ℝ. Choose the test function ϕ = T t (u) α u in equation (1.2).Note that this choice is admissible, since ϕ ∈ W 1,p (Ω), by classical results on truncations of Sobolev functions.One obtains that ∫ On the other hand, Thus, for some constant c 1 = c 1 (α, p).From inequality (3.11), via inequality (3.1) applied to the function T t (u) α/p u, one deduces that In order to estimate the right-hand side of equation (3.10), we observe that Multiplying through inequality (3.10) by εc 1 and making use of equation (3.12) tell us that Hence, owing to equation (3.13), Choose ε in a such a way that for some constant c = c(Ω, γ, α, p).Now, apply inequality (3.15) with α = p 2 − p.This results in This shows that u ∈ L p 2 (Ω).Next, choose α = p 3 − p in (3.15).Hence, for some constant c 2 = c 2 (Ω, γ, p).Passing to the limit as t → ∞ implies that Thus, u ∈ L p 3 (Ω).Iterating this argument, with α = p k − p for k ∈ ℕ, shows that u ∈ L p k (Ω) for every k ∈ ℕ.Hence, u ∈ L q (Ω) and inequality (1.4) holds for all q > p.
(ii) Assume that u is an eigenfunction of problem (1.2), and choose its representative which supports the coarea formula for Sobolev functions.Given s ∈ (0, H n (Ω)) and h > 0, let ϕ be the test function in equation (1.2) given by for x ∈ Ω.Notice that ϕ ∈ W 1,p (Ω) by standard results on truncations of Sobolev functions.One obtains that As recalled in equation (2.4), the function u ∘ is locally absolutely continuous in (0, is locally absolutely continuous, since, thanks to the coarea formula, Being the composition of monotone locally absolutely continuous functions, the function V is also locally absolutely continuous, and Here, and in what follows, the superscript "  " denotes differentiation.Therefore, dividing by h in (3.16) and passing to the limit as h → 0 + tell us that for a.e.s ∈ (0, H n (Ω)).On the other hand, inasmuch as the functions u and u ∘ are equimeasurable, From equations (3.17) and (3.18), one infers that where ψ u is the function defined by (2.5).Hence, Integrating both sides of equation (3.19) over the interval (s, ε) yields and observe that med(w) = 0, w ∘ = u ∘ − med(u) and By setting, for simplicity, Let us choose ε ∈ (0, H n (Ω)/2] so small that u ∘ (r) > 0 in (0, ε].Define the operator Our aim is now to prove that the equation has a solution f ∈ L ∞ (0, ε).In order to establish this fact, define the sequence of functions {f k } via iteration by We preliminarily observe that, by Fubini's theorem and inequality (2.6), Hence, owing to assumption (1.18), given δ ∈ (0, 1), there exists ε as above such that, in addition, We claim that for k ∈ ℕ ∪ {0}, whence f k ∈ L ∞ (0, ε).Inequality (3.24) can be verified by induction.Clearly, Assume now that inequality (3.24) holds for some k ∈ ℕ ∪ {0}.From inequality (3.23), one then deduces that namely inequality (3.24) with k replaced by k + 1.Hence, our claim follows.In particular, we have that We next distinguish the cases when p ≥ 2 or 1 < p < 2. Assume first that p ≥ 2. Under this assumption, one has that for k ∈ ℕ. Inequality (3.26) can be shown by induction again.Inequality (3.23) guarantees that that is, inequality (3.26) with k replaced by k + 1. Assume now that 1 < p < 2. We shall make use of the inequalities ) for some constants c 1 = c 1 (p) and c 2 = c 2 (p).
In this case, one has that for k ∈ ℕ. Inequality (3.30) holds with k = 1 thanks to (3.27), which is still valid even if 1 < p < 2. We argue again by induction and assume now that inequality (3.30) holds for some k ∈ ℕ.Then )ϖ(r) dr where the second inequality holds by (3.28), the third by (3.25), the fourth by (3.29), the fifth by the fact that f k (ρ) ≥ u ∘ (ε), the sixth by (3.30), and the last one by (3.23).Inequality (3.26) when p ≥ 2 and inequality (3.30) when 1 < p < 2 ensure that, if ε is sufficiently small, then the sequence {f k } converges in L ∞ (0, ε) to some function f , which solves equation (3.22).Now, we already know that equation (3.22) admits a solution f = u ∘ ∈ L p (0, ε).Our next goal is to show that such a function is the unique solution in L p (0, ε).Assume, by contradiction, that f and g are distinct functions in L p (0, ε) satisfying equation (3.22).Let us again distinguish the cases when p ≥ 2 or 1 < p < 2.
Next, observe that The following chain holds: where the first inequality is due to (3.28), the second to Hölder's inequality, the third to (3.29) and the fourth to (3.32).Combining inequalities (3.37) and (3.38), and an application of Minkowski's integral inequality as in (3.31), enable one to deduce that Inequality (3.39) yields a contradiction, provided that δ is chosen small enough.

Sharpness
The sharpness of the results from Theorems 1.1 and 1.3 will be demonstrated in our proofs of Theorems 1.2 and 1.4 via model "manifolds of revolution", patterned as in Figure 1 of Section 1, and defined as follows. Let ) Given n ≥ 2, we call the "n-dimensional manifold of revolution  built upon φ" the space ℝ n parametrized, in polar coordinates, as {(r, ω) : r ∈ [0, ∞), ω ∈  n−1 } and equipped with the Riemannian metric Here, dω 2 denotes the standard metric on  n−1 .Our assumptions on φ ensure that the metric (4.3) is of class C 1 ().Observe that for any integrable function u :  → ℝ.In particular, H n () < ∞ if and only if We shall make use of functions u :  → ℝ depending only on r, which, with some abuse of notation, will simply be denoted by u = u(r).For functions of this kind, one has that Moreover, the p-Laplace operator takes the form Thus, equation (1.1) on  reduces to the ordinary differential equation The membership of u in W 1,p () reads Thus, u is an eigenfunction of problem (1.2) if it satisfies condition (4.5) and for every locally absolutely continuous function ϕ : It will be convenient to perform a change of variables, in order to get rid of the coefficient φ n−1 in the differential operator in (4.4).To this end, define the function ψ : (0, ∞) → ℝ by where r 0 is any number in (0, ∞) if p ≤ n, and r 0 = 0 if p > n.Note that where we have set ψ(0) = lim r→0 + ψ(r) when 1 < p ≤ n.Under the change of variables where we have set ψ(∞) = lim s→∞ ψ(s).Moreover, condition (4.5) reads A locally absolutely continuous function v : (ψ(0), ψ(∞)) → ℝ is a solution to problem (4.8) if it satisfies condition (4.9) and for every locally absolutely continuous function ϕ : We introduce now a few notations to be employed in what follows.Let I be an interval of the form I = (a, ∞), where either a ∈ ℝ or a = −∞, and let η : I → [0, ∞) be a function such that η ∈ L 1 (I).We define, for p ∈ [1, ∞], the weighted Lebesgue space to be compact are given in the following proposition.
Proof.Assume that a = −∞, namely I = ℝ, the proof when a ∈ ℝ being analogous.Fix ε > 0. By assumptions (4.12) and (4.13), there exists ℓ > 0 such that sup ℓ<t<∞ Pick a compactly supported continuously differentiable function ξ : By setting A ℓ = (−ℓ − 1, ℓ + 1) and B ℓ = ℝ \ (−ℓ, ℓ), we have that Thus, as a consequence of standard weighted Hardy-type inequalities (see, e.g., [41, Now, consider any bounded sequence ≤ C for some constant C > 0 and every k ∈ ℕ.By inequality (4.16), applied with u replaced by u k − u m in the definition of v, one has that for some constants c, c  , c  .On the other hand, owing to the compactness of the embedding W 1,p (A ℓ ) → L p (A ℓ ), the sequence {u k }, restricted to A ℓ , admits a Cauchy subsequence, still denoted by {u k }, in L p (A ℓ ).Our assumptions on the function η entail that ess sup A ℓ η < ∞, a property which guarantees that {u k χ A ℓ } is also a Cauchy sequence in L p (ℝ, η).This piece of information, combined with inequality (4.17), tells us that {u k } is a Cauchy sequence in the Banach space L p (ℝ, η), and hence converges to some function u ∈ L p (ℝ, η).
The following theorem extends the results of [18, Theorem 4.1 and Corollary 4.2] to the case when p ̸ = 2, with an analogous proof.The details are omitted for brevity.
there exists L 0 > 0 such that φ is decreasing and convex in (L 0 , ∞), and let be the function defined by and such that (i) The metric of the n-dimensional manifold of revolution  built upon φ is of class C = 0 for any r 0 ∈ (0, ∞).
(iii) The following conditions are equivalent: The construction of the manifolds of revolution provided by the following proposition relies on Theorem 4.2.Owing to the monotonicity of the function ϑ, we have that Hence, via equations (4.23) and (4.24) and the ∆ 2 -condition near 0 for ν, we deduce that there exist constants c 3 and c 4 such that Define now the function λ : (0, ∞) → (0, ∞) by Moreover, v is convex in a left neighborhood of 1 and in a right neighborhood of −1, whereas it is concave in a right neighborhood of 1 and in a left neighborhood of −1.
Finally, in a neighborhood I 0 of 0, define for k > 0. Then the function v, given by is a convex solution to (4.8) in I 0 .
One can verify that, if β is sufficiently large, s 2 and −s 1 are sufficiently large depending on β, and I 1 , I −1 and I 0 are sufficiently small, then v can be continued to the whole of ℝ in such a way that , for some positive constant C.
Thus, ψ(0) = −∞ and ψ(∞) = ∞.Next, define the function φ : [0, ∞) → [0, ∞) by One has that lim r→0 + φ(r) = 0 and lim r→∞ φ(r) = 0. Furthermore, The existence of an eigenfunction of problem (4.8) could hence be established via the general Ljusternik-Schnirelman principle, as hinted at the end of Section 1.However, we also give a direct, more elementary proof, exploiting the one-dimensional nature of the problem at hand.Let J be the functional given by for v ∈ W 1,p (I).We claim that J achieves its minimum among all (not identically vanishing) functions v ∈ W It remains to show that the function v fulfills the Euler-Lagrange equation (4.10) for all test functions ϕ ∈ W 1,p (I, η).To verify this assertion, we make use of an argument reminiscent of that of [22,Lemma 2.4].Observe that the function v also minimizes the functional G defined by Consider, for the time being, test functions ϕ ∈ W 1,p (I) ∩ L ∞ (I).Given any h ∈ (0, 1), there exists β h ∈ ℝ such that the function v + hϕ + β h fulfills constraint (4.48).This is due to the fact that, fixing h, the function is convex and tends to ∞ as β → ±∞.Hence, it admits a minimum point β h , at which i.e. condition (4.48) is actually satisfied with v replaced by v + hϕ + β h .Next, there exists x h ∈ ℝ such that ϕ(x h ) + β h h = 0. Indeed, if hϕ(s) + β h were positive (resp.negative) for every s ∈ I, then, by equation (4.48) and the monotonicity of the function |t| t−2 t, the integral in equation (4.49) would be positive (resp.negative).
Since we are assuming that the function ϕ is bounded, there exist a sequence {h k } and a number c ∈ ℝ such that lim k→∞ ϕ(x h k ) = c, whence As a consequence of the minimizing property of the function v, one can thus infer that Equation (4.10) hence follows under the assumption ϕ ∈ W 1,p (I) ∩ L ∞ (I).Next, we claim that To verify this assertion, we first show that the space |v| p η ds) .
Inasmuch as v ∈ W 1,p (I, η), the rightmost side of this chain of inequalities tends to 0 as k → ∞.Hence, v k → v in W 1,p (I).The density of the space W 1,p (I) ∩ L ∞ (I) in W 1,p (I, η) ∩ L ∞ (I) is thus established.On the other hand, the space W 1,p (I, η) ∩ L ∞ (I) is in turn dense in W 1,p (I, η), as can be shown by approximating any function in the latter space by its truncations.Altogether, property (4.50) follows.
It remains to show that any eigenfunction v of problem (4.8) is unbounded.By [33, Theorem 3], condition (4.12) entails that equation (4.8) is nonoscillatory at infinity, and hence that every solution has constant sign at infinity.Thus, we may assume that v(s) > 0 for large s.Consequently, (p − 1)|v  | p−2 v  = (|v  | p−2 v  )  < 0 for large s, and hence v is concave near ∞.Now, assume by contradiction that v is bounded.Then lim s→∞ v(s) exists and, by denoting by v(∞) this limit, one has that v(∞) ∈ (0, ∞).Moreover, v is increasing for large s and [14, Theorem 1] for an earlier direct proof of (5.2) when n = 2. Hence, the boundedness of the eigenfunctions of problem (1.2) in Ω can also be deduced via Theorem 1.1.

Cusp-shaped domains
Here we deal with cusp-shaped sets of the form (see Figure 3) An application of Theorem 1.3 (i) ensures, via (5.3), that all eigenfunctions of problem (1.2), with Ω = , belong to L q (), provided that α > 1.
On the other hand, from Theorem 1.3 (ii) and equation (5.3) one can infer that the relevant eigenfunctions are bounded under the more stringent assumption that α > p.
The same conclusions can be derived via Theorem 1.1.Thus, like for any other manifold of revolution of the kind considered in Theorem 4.2, isoperimetric and isocapacitary methods lead to equivalent results for this family of noncompact manifolds.
In both cases, the existence of eigenfunctions is guaranteed by Theorem 1.5.

A family of manifolds with clustering submanifolds
Here, we are concerned with a class of noncompact surfaces  in ℝ 3 , which are shaped as in Figure 2 of Section 1, and are patterned on an example appearing in [21], dealing with a planar domain.Their main feature is the presence of a sequence of mushroom-shaped submanifolds {N k } clustering at some point.Let us emphasize that the submanifolds {N k } are not obtained just by dilation of each other.Roughly speaking, the diameter of the head and the length of the neck of N k decay to 0 as 2 −k when k → ∞, whereas the width of the neck of N k decays to 0 as σ(2 −k ), where σ is a function such that lim s→0 σ(s) s = 0.
The isoperimetric and isocapacitary functions of  depend on the behavior of σ at 0 in a way described in the next result (Proposition 5.1).Qualitatively, a faster decay to 0 of the function σ(s) as s → 0 results in a faster decay to 0 of λ  (s) and ν ,p (s), and hence in a manifold  with a more irregular geometry.The proof of Proposition 5.1 can be found in [18, Propositions 7.1 and 7.2], to which we also refer for a more precise definition of the manifold .Compliance with ethical standards.Conflict of interest: The authors declare that they have no conflict of interest.

Figure 2 :
Figure 2: A manifold with a family of clustering submanifolds.