Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Sergio Rolando

doi:10.1515/anona-2017-0177

Artikel Open Access

Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Sergio Rolando

Veröffentlicht/Copyright: 21. November 2017

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Abstract

Many existence and nonexistence results are known for nonnegative radial solutions to the equation

- △ ⁢ u + A | x | α ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , N ≥ 3 , A , α > 0 , u ∈ D 1 , 2 ⁢ ( ℝ N ) ∩ L 2 ⁢ ( ℝ N , | x | - α ⁢ d ⁢ x ) ,

with the nonlinearities satisfying | f ( u ) | ≤ ( const . ) u p - 1 for some p > 2 . The existence of nonradial solutions, by contrast, is known only for N ≥ 4 , α = 2 , f ⁢ ( u ) = u ( N + 2 ) / ( N - 2 ) and A large enough. Here we show that the above equation has multiple nonradial solutions as A → + ∞ for N ≥ 4 , 2 / ( N - 1 ) < α < 2 ⁢ N - 2 and α ≠ 2 , with the nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

Keywords: Semilinear elliptic PDE; singular vanishing potential; symmetry breaking

MSC 2010: 35J60; 35Q55; 35J20

1 Introduction and main result

This paper is concerned with the following semilinear elliptic problem:

(1.1) { - △ ⁢ u + A | x | α ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , N ≥ 3 , u ≥ 0 in ⁢ ℝ N , u ∈ H α 1 , u ≠ 0 .

where A , α > 0 are real constants, f : ℝ → ℝ is a continuous nonlinearity satisfying | f ( s ) | ≤ ( const . ) s p - 1 for some p > 2 and all s ≥ 0 , and H α 1 := D 1 , 2 ⁢ ( ℝ N ) ∩ L 2 ⁢ ( ℝ N , | x | - α ⁢ d ⁢ x ) is the natural energy space related to the equation. We will deal with problem (1.1) in the weak sense, that is, when we speak about solutions to (1.1) we will always mean weak solutions, i.e., functions u ∈ H α 1 ∖ { 0 } such that u ≥ 0 almost everywhere in ℝ N and

(1.2) ∫ ℝ N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x + ∫ ℝ N A | x | α ⁢ u ⁢ v ⁢ 𝑑 x = ∫ ℝ N f ⁢ ( u ) ⁢ v ⁢ 𝑑 x for all ⁢ v ∈ H α 1 .

As is well known, problems like (1.1) are models for stationary states of reaction diffusion equations in population dynamics (see, e.g., [17]). They also arise in many other branches of mathematical physics, such as nonlinear optics, plasma physics, condensed matter physics and cosmology (see, e.g., [11, 27]), where its nonnegative solutions lead to special solutions (solitary waves and solitons) for several nonlinear field theories like nonlinear Schrödinger (or Gross–Pitaevskii) and Klein–Gordon equations. In this context, (1.1) is a prototype for problems exhibiting radial potentials which are singular at the origin and/or vanishing at infinity (sometimes called the zero mass case; see, e.g., [22, 8]).

Although it can be considered as a quite recent investigation, the study of problem (1.1) has already some history, which probably started in [26] and continued in [16, 9, 24, 25, 5, 15] (see [4] for a similar cylindrical problem). Currently, the problem of existence and nonexistence of radial solutions is essentially solved in the pure-power case f ⁢ ( u ) = u p - 1 , where the results obtained rest upon compatibility conditions between α and p. These can be summarized as follows (for a chronological overview of these results see [5]): the problem has a radial solution for ( α , p ) = ( 2 , 2 * ) (see [26]) and for all the pairs ( α , p ) satisfying

(1.3) { 0 < α < 2 , 2 α * < p < 2 * , or { 2 < α < 2 ⁢ N - 2 , 2 * < p < 2 α * , or { α ≥ 2 ⁢ N - 2 , p > 2 * , with 2 α * := 2 2 ⁢ N - 2 + α 2 ⁢ N - 2 - α

(see [25]), while it has no solution if

{ 0 < α < 2 , p ∉ ( 2 α , 2 * ) , or { α = 2 , p ≠ 2 * , or { 2 < α < N , p ∉ ( 2 * , 2 α ) , or { α ≥ N , p ≤ 2 * , with 2 α := 2 ⁢ N N - α

(see [9]) and no radial solution for both

{ 0 < α < 2 , 2 α < p ≤ 2 α * , and { 2 < α < 2 ⁢ N - 2 , 2 α * ≤ p < 2 α

(see [5] and [15], respectively). As usual, 2 * := 2 ⁢ N / ( N - 2 ) denotes the critical exponent for the Sobolev embedding in dimension N ≥ 3 . All these results are portrayed in the picture of the α ⁢ p -plane given in Figure 1, where nonexistence regions are shaded in gray (nonexistence of radial solutions) and light gray (nonexistence of solutions at all, which includes both the lines p = 2 * and p = 2 α except for the pair ( α , p ) = ( 2 , 2 * ) ), whereas white color (of course above the line p = 2 ) means existence of radial solutions. As to nonradial solutions, the only result available is the one contained in [26, Theorem 0.5], where Terracini proves that problem (1.1), with N ≥ 4 , α = 2 and f ⁢ ( u ) = u 2 * - 1 , has at least a nonradial solution for every A large enough. This brought Catrina to say, in the introduction of his paper [15]: “Two questions still remain: whether one can find non-radial solutions in the case when radial solutions do not exist, or in the case when radial solutions exist”.

Figure 1

Regions of nonexistence of solutions (light gray), and existence(white with p > 2 ) and nonexistence (dark gray) of radial solutions.

Su, Wang and Willem [25] covered also the case where problem (1.1) has general nonlinearities satisfying the power growth condition | f ( u ) | ≤ ( const . ) u p - 1 for some p > 2 , and ensured that, under some rather standard additional assumptions on f (precisely (f1) and (f2) below), problem (1.1) has a radial solution for all the pairs ( α , p ) satisfying (1.3). To be precise, they only concerned themselves with radial weak solutions in the sense of the dual space of the radial subspace of H α 1 (where the energy functional of the problem is well defined by the embeddings they proved), but the symmetric criticality type results of [6] actually apply, yielding solutions in the sense of our definition (1.2). No results are known in the literature about nonradial solutions.

This general lack of symmetry breaking results is the motivation of this paper, where we prove that problem (1.1) has multiple nonradial solutions as A → + ∞ , provided that N ≥ 4 , α ∈ ( 2 / ( N - 1 ) , 2 ⁢ N - 2 ) ∖ { 2 } and f belongs to a suitable class of nonlinearities satisfying a power growth condition. We observe straight away that such a class of nonlinearities does not unfortunately contain pure powers (which does not satisfy our assumption (f0), where p 1 ≠ p 2 ).

The main assumptions characterizing our class of nonlinearities are the following, where we define F ⁢ ( s ) := ∫ 0 s f ⁢ ( t ) ⁢ 𝑑 t :

sup s > 0 ⁡ | f ⁢ ( s ) | min ⁡ { s p 1 - 1 , s p 2 - 1 } < + ∞ for some 2 < p 1 < 2 * < p 2 .
There exists θ > 2 such that θ ⁢ F ⁢ ( s ) ≤ f ⁢ ( s ) ⁢ s for all s > 0 .
F ⁢ ( s ) > 0 for all s > 0 .
The function f ⁢ ( s ) s is strictly increasing on ( 0 , + ∞ ) .
There exists μ > 2 such that the function F ⁢ ( s ) s μ is decreasing on ( 0 , + ∞ ) .

For N ≥ 3 , α ∈ ( 0 , 2 ⁢ N - 2 ) , α ≠ 2 and 2 < p 1 < 2 * < p 2 , we define

(1.4) ν := ν N , α , p 1 , p 2 := { ⌈ 2 ⁢ min ⁡ { N - 1 α , N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 } - 2 ⁢ N ⁢ ( 1 α - 1 2 ) ⌉ - 1 if ⁢ 0 < α < 2 , ⌈ 2 ⁢ min ⁡ { N - 1 α , N - 2 α - 2 ⁢ p 2 - 2 * p 2 - 2 } ⌉ - 1 if ⁢ 2 < α < 2 ⁢ N - 2 ,

where ⌈ ⋅ ⌉ denotes the ceiling function (i.e., ⌈ x ⌉ := min ⁡ { n ∈ ℤ : n ≥ x } ).

$Figure 2 ν as a function of p 1 ∈ ( 2 , p 1 * ) {p_{1}\in(2,p_{1}^{*})} for N ≥ 3 {N\geq 3} and α ∈ ( 0 , 2 ) {\alpha\in(0,2)} fixed.$

Figure 2

ν as a function of p 1 ∈ ( 2 , p 1 * ) for N ≥ 3 and α ∈ ( 0 , 2 ) fixed.

$Figure 3 ν as a function of p 2 > p 2 * {p_{2}>p_{2}^{*}} for N ≥ 3 {N\geq 3} and α ∈ ( 2 , 2 ⁢ N - 2 ) {\alpha\in(2,2N-2)} fixed.$

Figure 3

ν as a function of p 2 > p 2 * for N ≥ 3 and α ∈ ( 2 , 2 ⁢ N - 2 ) fixed.

Our main result is the following theorem.

Theorem 1.1.

Let N ≥ 4 and α ∈ ( 2 / ( N - 1 ) , 2 ⁢ N - 2 ) , α ≠ 2 . Let f : R → R be a continuous function satisfying (f1)–(f4). Assume that (f0) holds with

(1.5) p 1 < p 1 * := 2 ⁢ α 2 ⁢ ( N - 1 ) - 2 ⁢ α ⁢ ( N - 1 ) + 4 ⁢ N α 2 ⁢ ( N - 1 ) - 2 ⁢ α ⁢ ( N + 1 ) + 4 ⁢ N 𝑜𝑟 p 2 > p 2 * := 2 ⁢ 2 ⁢ N + 2 - α 2 ⁢ N - 2 - α

if α ∈ ( 2 / ( N - 1 ) , 2 ) or α ∈ ( 2 , 2 ⁢ N - 2 ) , respectively. Then there exists A * > 0 such that for every A > A * , problem (1.1) has both a radial solution and ν different nonradial solutions.

Some comments on Theorem 1.1 are in order. First of all, under the assumptions of the theorem, ν is positive (see Lemma 5.2 below), and so at least one nonradial solution actually exists. On the other hand, it is easy to check that for every fixed N and α, the behavior of ν as a function of p 1 and p 2 is the one portrayed in Figures 2 and 3, respectively, whence one sees that the number ν of nonradial solutions may assume every natural value (as N → ∞ ).

Regarding assumptions (1.5) and (f0), it is worth observing that α ∈ ( 0 , 2 ) implies 2 < p 1 * < 2 * , while 2 < α < 2 ⁢ N - 2 implies p 2 * > 2 * , and so (1.5) and (f0) are consistent with each other. Assumption (f0) is the so-called double-power growth condition and seems to be typical in nonlinear problems with potentials vanishing at infinity (see, e.g., [12, 13, 2, 19, 6, 7, 8, 18, 22, 14, 3]). Such an assumption obviously implies the single-power growth condition | f ( s ) | ≤ ( const . ) s p - 1 for all p ∈ [ p 1 , p 2 ] and s ≥ 0 , but it is actually more stringent than that, since it requires p 1 ≠ p 2 . We finally observe that (f0) still remains true if one raises p 1 and lowers p 2 , but this decreases ν (see Figures 2 and 3), and therefore it is convenient to apply Theorem 1.1 with p 1 as small as possible and p 2 as large as possible (which is also consistent with assumption (1.5)).

The plan of the paper is the following. In Section 2 we define the variational setting and introduce the argument we will use in the proof of Theorem 1.1, which will be given in Section 5. Observe that we cannot use the technique used in [26], where the homogeneity of the nonlinearity is exploited and a nonradial solution is obtained as a global minimizer of the Sobolev type quotient associated to the problem. Our argument, instead, essentially relies on the following two main elements:

The compact embeddings between some suitable functional spaces of symmetric functions, which yield the existence of ν different solutions of mountain-pass type.
The separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.

Sections 3 and 4 are devoted to the estimation of these levels in order to separate them. As a conclusion, we get ν nonradial solutions on which the energy functional of the equation has a lower value than the energy levels of radial solutions.

We end this introductory section by giving some examples of nonlinearities to which Theorem 1.1 applies, and by presenting some notations we use throughout the paper.

Example 1.2.

Let N ≥ 4 and α ∈ ( 2 / ( N - 1 ) , 2 ⁢ N - 2 ) , α ≠ 2 , and let 2 < p 1 < 2 * < p 2 be such that (1.5) holds. The most obvious nonlinearity to which Theorem 1.1 applies is f ⁢ ( s ) = min ⁡ { | s | p 1 - 1 , | s | p 2 - 1 } , which satisfies (f1) and (f4) for θ = p 1 and any μ > p 2 . Other simple examples are

f ⁢ ( s ) = | s | p 2 - 1 1 + | s | p 2 - p 1 , f ⁢ ( s ) = d d ⁢ s ⁢ ( | s | p 2 1 + | s | p 2 - p 1 ) ,

both of which satisfy (f1) with θ = p 1 . In the latter case, (f4) clearly holds for any μ > p 2 . We leave it to the reader to check that (f4) also holds in the former case for μ large enough.

Notations.

σ d denotes the ( d - 1 ) -dimensional measure of the unit sphere of ℝ d .
C c ∞ ⁢ ( Ω ) is the space of infinitely differentiable real functions with compact support in the open set Ω ⊆ ℝ d .
D 1 , 2 ⁢ ( ℝ N ) = { u ∈ L 2 * ⁢ ( ℝ N ) : ∇ ⁡ u ∈ L 2 ⁢ ( ℝ N ) } is the usual Sobolev space, which identifies with the completion of C c ∞ ⁢ ( ℝ N ) with respect to the norm of the gradient.

2 Preliminaries

Let N ≥ 3 and A , α > 0 . Let f : ℝ → ℝ be a continuous function satisfying (f0)–(f2). In this section we define the functional setting and introduce the argument we will use in proving Theorem 1.1.

As already mentioned in the introduction, we define the Hilbert space

H α 1 := { u ∈ D 1 , 2 ⁢ ( ℝ N ) : ∫ ℝ N u 2 | x | α ⁢ 𝑑 x < ∞ } ,

which we endow with the following scalar product and related norm:

( u , v ) A := ∫ ℝ N ( ∇ ⁡ u ⋅ ∇ ⁡ v + A | x | α ⁢ u ⁢ v ) ⁢ 𝑑 x , ∥ u ∥ A 2 := ∫ ℝ N ( | ∇ ⁡ u | 2 + A | x | α ⁢ u 2 ) ⁢ 𝑑 x .

Of course, the embedding H α 1 ↪ D 1 , 2 ⁢ ( ℝ N ) is continuous.

Given any integer K such that 1 ≤ K ≤ N - 1 , we write every x ∈ ℝ N as x = ( y , z ) ∈ ℝ K × ℝ N - K , and in the space H α 1 , we consider the following closed subspaces of symmetric functions:

H r := { u ∈ H : u ⁢ ( x ) = u ⁢ ( | x | ) } and H K := { u ∈ H : u ⁢ ( x ) = u ⁢ ( y , z ) = u ⁢ ( | y | , | z | ) } .

Of course u ⁢ ( y , z ) = u ⁢ ( | y | , | z | ) naturally means that u ⁢ ( y , z ) = u ⁢ ( S 1 ⁢ y , S 2 ⁢ z ) for all isometries S 1 and S 2 of ℝ K and ℝ N - K , respectively. Similarly for u ⁢ ( x ) = u ⁢ ( | x | ) . Note that H r ⊂ H K for every K, since | x | 2 = | y | 2 + | z | 2 . The next lemma clarifies better the relation between the spaces H K and H r .

Lemma 2.1.

Let 1 ≤ K 1 < K 2 ≤ N - 1 . Then H K 1 ∩ H K 2 = H r .

Proof.

The proof is essentially an adaptation of the one of [21, Lemma 3.3]. For any x ∈ ℝ N , we will denote by ( y 1 , z 1 ) its decomposition in ℝ K 1 × ℝ N - K 1 , and by ( y 2 , z 2 ) its decomposition in ℝ K 2 × ℝ N - K 2 . Let u ∈ H K 1 ∩ H K 2 , and for every s , t ≥ 0 , define

u ~ ⁢ ( s , t ) := u ⁢ ( s , 0 , … , 0 , t ) .

Then we clearly have u ⁢ ( x ) = u ~ ⁢ ( | y 1 | , | z 1 | ) = u ~ ⁢ ( | y 2 | , | z 2 | ) for every x = ( y 1 , z 1 ) = ( y 2 , z 2 ) ∈ ℝ N . Let x = ( y 1 , z 1 ) ∈ ℝ K 1 × ℝ N - K 1 and x ′ = ( y 2 ′ , z 2 ′ ) ∈ ℝ K 2 × ℝ N - K 2 be such that | x | = | x ′ | , i.e., | y 1 | 2 + | z 1 | 2 = | y 2 ′ | 2 + | z 2 ′ | 2 . Suppose that | y 1 | ≤ | y 2 ′ | and define x ′′ ∈ ℝ N by setting

x ′′ := ( | y 1 | , 0 , … , 0 , | y 2 ′ | 2 - | y 1 | 2 , 0 , … , 0 , | z 2 ′ | , 0 , … , 0 ) ,

where the first block of zeros has K 1 - 1 zeros, the second K 2 - K 1 - 1 , and the third N - K 2 - 1 . Then

u ⁢ ( x ′′ ) = u ~ ⁢ ( | y 1 ′′ | , | z 1 ′′ | ) = u ~ ⁢ ( | y 1 | , | y 2 ′ | 2 - | y 1 | 2 + | z 2 ′ | 2 ) = u ~ ⁢ ( | y 1 | , | z 1 | )

and

u ⁢ ( x ′′ ) = u ~ ⁢ ( | y 2 ′′ | , | z 2 ′′ | ) = u ~ ⁢ ( | y 1 | 2 + | y 2 ′ | 2 - | y 1 | 2 , | z 2 ′ | ) = u ~ ⁢ ( | y 2 ′ | , | z 2 ′ | ) ,

which implies u ~ ⁢ ( | y 1 | , | z 1 | ) = u ~ ⁢ ( | y 2 ′ | , | z 2 ′ | ) , i.e., u ⁢ ( x ) = u ⁢ ( x ′ ) . If | y 1 | > | y 2 ′ | , we repeat the argument with x ′′ defined by

x ′′ := ( | y 2 ′ | , 0 , … , 0 , | y 1 | 2 - | y 2 ′ | 2 , 0 , … , 0 , | z 1 | , 0 , … , 0 )

(with the same lengths of blocks of zeros) and get the same result. Hence, | x | = | x ′ | implies u ⁢ ( x ) = u ⁢ ( x ′ ) , i.e., u ∈ H r . ∎

We modify the function f by setting f ⁢ ( s ) = 0 for all s < 0 and, with a slight abuse of notation, we still denote by f the modified function. Then, by (f0), there exist M 1 , M 2 > 0 such that

| f ⁢ ( s ) | ≤ M 1 ⁢ min ⁡ { | s | p 1 - 1 , | s | p 2 - 1 } and | F ⁢ ( s ) | ≤ M 2 ⁢ min ⁡ { | s | p 1 , | s | p 2 } for all ⁢ s ∈ ℝ ,

which, in particular, yields

(2.1) | f ⁢ ( s ) | ≤ M 1 ⁢ | s | p - 1 and | F ⁢ ( s ) | ≤ M 2 ⁢ | s | p for all ⁢ p ∈ [ p 1 , p 2 ] ⁢ and ⁢ s ∈ ℝ .

By the continuous embeddings H α 1 ↪ D 1 , 2 ⁢ ( ℝ N ) ↪ L 2 * ⁢ ( ℝ N ) , one can check (see, for example, [20]) that condition (2.1) with p = 2 * implies that the energy functional associated to the equation of (1.1), i.e.,

(2.2) I ⁢ ( u ) := 1 2 ⁢ ∥ u ∥ A 2 - ∫ ℝ N F ⁢ ( u ) ⁢ 𝑑 x ,

is of class C 1 on H α 1 and has Fréchet derivative I ′ ⁢ ( u ) at any u ∈ H α 1 given by

(2.3) I ′ ⁢ ( u ) ⁢ v = ( u , v ) A - ∫ ℝ N f ⁢ ( u ) ⁢ v ⁢ 𝑑 x for all ⁢ v ∈ H α 1 .

This yields that the critical points of I : H α 1 → ℝ satisfy (1.2). A standard argument shows that such critical points are nonnegative (see the proof of Theorem 1.1 in Section 5), and therefore we conclude that the nonzero critical points of I are weak solutions to problem (1.1).

Accordingly, our argument in proving Theorem 1.1 will be essentially the following. The existence of a critical point for the restriction I | H r readily follows from the results of [25]. By exploiting the compact embeddings of [2] and the results of [8] about Nemytskiĭ operators on the sum of Lebesgue spaces, we will show in Section 5 that I | H K has a nonzero critical point u K for every 2 ≤ K ≤ N - 2 . Thanks to the classical Palais principle of symmetric criticality [23], all these critical points are also critical points of I, and thus weak solutions to (1.1). Hence, Theorem 1.1 is proved if we show that u K ∉ H r for every K, which also implies u K 1 ≠ u K 2 for K 1 ≠ K 2 , by Lemma 2.1. This will be achieved by showing that the critical levels I ⁢ ( u K ) are lower than all the nonzero critical levels of I | H r . The starting points in proving this are the following lemmas.

Lemma 2.2.

For every u ∈ H r ∖ { 0 } , u ≥ 0 , there exists t u > 0 such that I ⁢ ( t u ⁢ u ) = max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) .

Proof.

Since u ≥ 0 and u ≠ 0 , we can fix δ > 0 such that the set { x ∈ ℝ N : u ≥ δ } has positive measure. From assumptions (f1) and (f2), we deduce that there exists a constant C > 0 such that F ⁢ ( s ) ≥ C ⁢ s θ for all s ≥ δ . Then, for every t > 1 , one has

∫ ℝ N F ⁢ ( t ⁢ u ) ⁢ 𝑑 x = ∫ { x ∈ ℝ N : t ⁢ u ≥ δ } F ⁢ ( t ⁢ u ) ⁢ 𝑑 x + ∫ { x ∈ ℝ N : t ⁢ u < δ } F ⁢ ( t ⁢ u ) ⁢ 𝑑 x

≥ ∫ { x ∈ ℝ N : t ⁢ u ≥ δ } F ⁢ ( t ⁢ u ) ⁢ 𝑑 x

≥ C ⁢ t θ ⁢ ∫ { x ∈ ℝ N : t ⁢ u ≥ δ } u θ ⁢ 𝑑 x

≥ C ⁢ t θ ⁢ ∫ { x ∈ ℝ N : u ≥ δ } u θ ⁢ 𝑑 x ,

and therefore

(2.4) I ⁢ ( t ⁢ u ) ≤ 1 2 ⁢ t 2 ⁢ ∥ u ∥ A 2 - C ⁢ t θ ⁢ ∫ { x ∈ ℝ N : u ≥ δ } u θ ⁢ 𝑑 x → - ∞ as ⁢ t → + ∞ ,

since θ > 2 . On the other hand, condition (2.1) with p = 2 * and the continuous embeddings H r ↪ H α 1 ↪ D 1 , 2 ⁢ ( ℝ N ) ↪ L 2 * ⁢ ( ℝ N ) imply that there exists a constant C > 0 such that I ⁢ ( t ⁢ u ) ≥ ∥ u ∥ A 2 ⁢ t 2 2 - C ⁢ ∥ u ∥ A 2 * ⁢ t 2 * for all t ≥ 0 , which in turn yields I ⁢ ( t ⁢ u ) > 0 for all t > 0 small enough. Together with (2.4), this gives the result. ∎

According to Lemma 2.2, define

m A := inf u ∈ H r ∖ { 0 } , u ≥ 0 ⁡ max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) .

Lemma 2.3.

Assume (f3) and let u ∈ H r ∖ { 0 } . If u is a critical point for I, then I ⁢ ( u ) ≥ m A .

Proof.

As already observed, if u is a critical point for I, then u is nonnegative. We shall now prove that I ⁢ ( u ) = max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) , which obviously yields the result. For t ≥ 0 , define

g ⁢ ( t ) := I ⁢ ( t ⁢ u ) = 1 2 ⁢ t 2 ⁢ ∥ u ∥ A 2 - ∫ ℝ N F ⁢ ( t ⁢ u ) ⁢ 𝑑 x .

As u is a critical point for I, we readily have that t = 1 is a critical point for g. Indeed, g ′ ⁢ ( t ) = I ′ ⁢ ( t ⁢ u ) ⁢ u , and thus g ′ ⁢ ( 1 ) = I ′ ⁢ ( u ) ⁢ u = 0 . We now show that, on the other hand, g has at most one critical point in ( 0 , + ∞ ) . We have g ′ ⁢ ( t ) = 0 if and only if I ′ ⁢ ( t ⁢ u ) ⁢ u = 0 , i.e.,

I ′ ⁢ ( t ⁢ u ) ⁢ u = t ⁢ ∥ u ∥ A 2 - ∫ ℝ N f ⁢ ( t ⁢ u ) ⁢ u ⁢ 𝑑 x = 0 .

So, if 0 < t 1 < t 2 are critical points for g, then one has

∥ u ∥ A 2 = 1 t 1 ⁢ ∫ ℝ N f ⁢ ( t 1 ⁢ u ) ⁢ u ⁢ 𝑑 x = 1 t 2 ⁢ ∫ ℝ N f ⁢ ( t 2 ⁢ u ) ⁢ u ⁢ 𝑑 x ,

which implies

(2.5) ∫ E u ( f ⁢ ( t 2 ⁢ u ) t 2 ⁢ u - f ⁢ ( t 1 ⁢ u ) t 1 ⁢ u ) ⁢ u 2 ⁢ 𝑑 x = 0 ,

where E u := { x ∈ ℝ N : u > 0 } . Since the integrand in (2.5) is nonnegative by assumption (f3), we have that f ⁢ ( t 2 ⁢ u ) t 2 ⁢ u - f ⁢ ( t 1 ⁢ u ) t 1 ⁢ u = 0 almost everywhere on E u . Since E u has positive measure (because 0 ≠ u ≥ 0 ), this implies t 1 = t 2 , again by assumption (f3). As a conclusion, according to Lemma 2.2, we deduce that t u = 1 and the claim ensues. ∎

Lemma 2.4.

There exists R > 0 such that for every 1 ≤ K ≤ N - 1 , one has

(2.6) inf u ∈ H K , ∥ u ∥ A ≤ R ⁡ I ⁢ ( u ) = 0 𝑎𝑛𝑑 inf u ∈ H K , ∥ u ∥ A = R ⁡ I ⁢ ( u ) > 0 .

Proof.

The claim readily follows from (2.1) with p = 2 * and the continuous embeddings H K ↪ H α 1 ↪ D 1 , 2 ⁢ ( ℝ N ) ↪ L 2 * ⁢ ( ℝ N ) , which imply that there exists a constant C > 0 such that I ⁢ ( u ) ≥ ∥ u ∥ A 2 / 2 - C ⁢ ∥ u ∥ A 2 * for all u ∈ H K . ∎

In Section 4 we will see that I | H K takes negative values by choosing a suitable u ¯ K ∈ H K such that I ⁢ ( u ¯ K ) < 0 . This implies ∥ u ¯ K ∥ A > R , by (2.6), and therefore the functional I | H K has a mountain-pass geometry. In Section 5 we will see that it also satisfies the Palais–Smale condition for 2 ≤ K ≤ N - 2 , and so it admits a (nonnegative) critical point u K at the mountain-pass level

(2.7) c A , K := inf γ ∈ Γ ⁡ max t ∈ [ 0 , 1 ] ⁡ I ⁢ ( γ ⁢ ( t ) ) > 0 , where ⁢ Γ := { γ ∈ C ⁢ ( [ 0 , 1 ] ; H K ) : γ ⁢ ( 0 ) = 0 , γ ⁢ ( 1 ) = u ¯ K } .

With a view to obtaining the separation inequality c A , K < m A , Sections 3 and 4 are devoted to estimating m A and c A , K .

3 Estimate of m A

Let N ≥ 3 and α , A > 0 . Let f : ℝ → ℝ be a continuous function satisfying (f0)–(f2). This section is devoted to deriving the estimate of m A given in Proposition 3.2 below, which relies on the following radial lemma (see also [6, Appendix] and [25, Lemmas 4 and 5] for similar results).

Lemma 3.1.

Every u ∈ H r satisfies

| u ⁢ ( x ) | ≤ 2 / σ N A 1 / 4 ⁢ ∥ u ∥ A | x | ( 2 ⁢ N - 2 - α ) / 4 almost everywhere in ⁢ ℝ N

(recall that σ N denotes the ( N - 1 ) -dimensional measure of the unit sphere of R N ).

Proof.

Let u ∈ H r and let u ~ : ( 0 , + ∞ ) → ℝ be continuous and such that u ⁢ ( x ) = u ~ ⁢ ( | x | ) for almost every x ∈ ℝ N . Set

v ⁢ ( r ) := r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) 2 for all ⁢ r > 0 .

By [6, Lemma 27], we have that u ~ ∈ W 1 , 1 ⁢ ( ( a , b ) ) for every 0 < a < b < + ∞ , whence v ∈ W 1 , 1 ⁢ ( ( a , b ) ) and

v ⁢ ( b ) - v ⁢ ( a ) = ∫ a b v ′ ⁢ ( r ) ⁢ 𝑑 r .

Moreover, for almost every r ∈ ( a , b ) , one has

(3.1) v ′ ⁢ ( r ) = ( N - 1 - α 2 ) ⁢ r N - 2 - α / 2 ⁢ u ~ ⁢ ( r ) 2 + 2 ⁢ r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) ⁢ u ~ ′ ⁢ ( r ) .

If α < 2 ⁢ N - 2 , this implies v ′ ⁢ ( r ) ≥ 2 ⁢ r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) ⁢ u ~ ′ ⁢ ( r ) , and therefore

v ⁢ ( a ) ≤ v ⁢ ( b ) - ∫ a b v ′ ⁢ ( r ) ⁢ 𝑑 r

≤ v ⁢ ( b ) - ∫ a b 2 ⁢ r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) ⁢ u ~ ′ ⁢ ( r ) ⁢ 𝑑 r

≤ v ⁢ ( b ) + 2 ⁢ ∫ a b r N - 1 - α / 2 ⁢ | u ~ ⁢ ( r ) | ⁢ | u ~ ′ ⁢ ( r ) | ⁢ 𝑑 r

= v ⁢ ( b ) + 2 ⁢ ∫ a b r ( N - 1 ) / 2 ⁢ | u ~ ′ ⁢ ( r ) | ⁢ r ( N - 1 ) / 2 - α / 2 ⁢ | u ~ ⁢ ( r ) | ⁢ 𝑑 r

≤ v ⁢ ( b ) + 2 ⁢ ( ∫ 0 + ∞ r N - 1 ⁢ | u ~ ′ ⁢ ( r ) | 2 ⁢ 𝑑 r ) 1 / 2 ⁢ ( ∫ 0 + ∞ r N - 1 - α ⁢ | u ~ ⁢ ( r ) | 2 ⁢ 𝑑 r ) 1 / 2

≤ v ( b ) + 2 σ N ⁢ A ( ∫ ℝ N | ∇ u | 2 d x ) 1 / 2 ( ∫ ℝ N A ⁢ u 2 | x | α d x ) 1 / 2

(3.2) ≤ v ⁢ ( b ) + 2 σ N ⁢ A ⁢ ∥ u ∥ A 2 .

If α ≥ 2 ⁢ N - 2 , then (3.1) gives v ′ ⁢ ( r ) ≤ 2 ⁢ r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) ⁢ u ~ ′ ⁢ ( r ) , and thus we have

(3.3) v ⁢ ( b ) ≤ v ⁢ ( a ) + ∫ a b 2 ⁢ r N - 1 - α / 2 ⁢ u ~ ⁢ ( r ) ⁢ u ~ ′ ⁢ ( r ) ⁢ 𝑑 r ≤ v ⁢ ( a ) + 2 σ N ⁢ A ⁢ ∥ u ∥ A 2 ,

as before. Now observe that there exist 0 < a n → 0 and b n → + ∞ such that v ⁢ ( a n ) → 0 and v ⁢ ( b n ) → 0 . Indeed, if l := lim inf r → 0 + ⁡ v ⁢ ( r ) > 0 , then for every r smaller than some suitable r 0 > 0 , one has | u ~ ⁢ ( r ) | ≥ l / 2 ⁢ r - ( N - 1 - α / 2 ) / 2 , and therefore one of the following contradictions ensues:

∫ ℝ N u 2 | x | α ⁢ 𝑑 x ≥ l 2 ⁢ ∫ B r 0 1 | x | N - 1 + α / 2 ⁢ 𝑑 x = + ∞ if ⁢ α ≥ 2

∫ ℝ N | u | 2 * d x ≥ ( l 2 ) 2 * / 2 ∫ B r 0 1 | x | N - 1 - α / 2 N - 2 ⁢ N d x = + ∞ if α ≤ 2 .

Similarly, if lim inf r → + ∞ ⁡ v ⁢ ( r ) > 0 , then one obtains

∫ ℝ N u 2 | x | α d x = + ∞ if α ≤ 2 and ∫ ℝ N | u | 2 * d x = + ∞ if α ≥ 2 .

Therefore, the claim follows by letting n → ∞ in (3.2) with a = r and b = b n , and in (3.3) with a = a n and b = r . ∎

We can now prove our estimate for m A .

Proposition 3.2.

Assume 0 < α < 2 ⁢ N - 2 , α ≠ 2 , and let p = max ⁡ { 2 α * , p 1 } if 0 < α < 2 , or p = min ⁡ { 2 α * , p 2 } if 2 < α < 2 ⁢ N - 2 . Then there exists a constant C 0 > 0 , independent from A, such that

m A ≥ C 0 ⁢ A N - 2 α - 2 ⁢ p - 2 * p - 2 .

Proof.

Let u ∈ H r ∖ { 0 } . By Lemma 3.1, we have

∫ ℝ N | u | 2 α * d x = ∫ ℝ N | u | 2 α * - 2 u 2 d x

≤ ( 2 σ N ) ( 2 α * - 2 ) / 2 A ( 2 α * - 2 ) / 4 ⁢ ∥ u ∥ A 2 α * - 2 ⁢ ∫ ℝ N u 2 | x | [ ( 2 ⁢ N - 2 - α ) / 4 ] ⁢ ( 2 α * - 2 ) ⁢ 𝑑 x

= ( 2 σ N ) 2 ⁢ α / ( 2 ⁢ N - 2 - α ) A α / ( 2 ⁢ N - 2 - α ) ⁢ ∥ u ∥ A 2 α * - 2 A ⁢ ∫ ℝ N A ⁢ u 2 | x | α ⁢ 𝑑 x

≤ ( 2 σ N ) 2 ⁢ α / ( 2 ⁢ N - 2 - α ) A ( 2 ⁢ N - 2 ) / ( 2 ⁢ N - 2 - α ) ⁢ ∥ u ∥ A 2 α * ,

since 2 α * - 2 = 4 ⁢ α / ( 2 ⁢ N - 2 - α ) . On the other hand, one has

∫ ℝ N | u | 2 * d x ≤ S N 2 * ( ∫ ℝ N | ∇ u | 2 d x ) 2 * / 2 ≤ S N 2 * ∥ u ∥ A 2 * ,

where S N denotes the Sobolev constant in dimension N. Then, in either case p = max ⁡ { 2 α * , p 1 } < 2 * or p = min ⁡ { 2 α * , p 2 } > 2 * , we can argue by interpolation: there exists λ ∈ [ 0 , 1 ) such that p = λ ⁢ 2 * + ( 1 - λ ) ⁢ 2 α * , and by Hölder inequality, we get

∫ ℝ N | u | p d x = ∫ ℝ N | u | λ ⁢ 2 * + ( 1 - λ ) ⁢ 2 α * d x ≤ ( ∫ ℝ N | u | 2 * d x ) λ ( ∫ ℝ N | u | 2 α * d x ) 1 - λ ≤ C ∥ u ∥ A p A ( 2 ⁢ N - 2 ) ⁢ ( 1 - λ ) / ( 2 ⁢ N - 2 - α ) ,

where C := S N λ ⁢ 2 * ⁢ ( 2 σ N ) 2 ⁢ α ⁢ ( 1 - λ ) / ( 2 ⁢ N - 2 - α ) only depends on N , α , p . Recalling condition (2.1), this implies

| ∫ ℝ N F ( u ) d x | ≤ M 2 ∫ ℝ N | u | p d x ≤ M 2 C ∥ u ∥ A p A ( 2 ⁢ N - 2 ) ⁢ ( 1 - λ ) / ( 2 ⁢ N - 2 - α ) ,

and therefore

I ⁢ ( u ) ≥ 1 2 ⁢ ∥ u ∥ A 2 - a ⁢ ∥ u ∥ A p ,

where we set a = M 2 ⁢ C ⁢ A - ( 2 ⁢ N - 2 ) ⁢ ( 1 - λ ) / ( 2 ⁢ N - 2 - α ) for brevity. Hence,

I ( t u ) ≥ 1 2 t 2 ∥ u ∥ A 2 - a t p ∥ u ∥ A p = : g u ( t ) for every t ≥ 0 .

The function g u : [ 0 , + ∞ ) → ℝ attains its maximum in t u := ( a ⁢ p ) - 1 / ( p - 2 ) ∥ u ∥ A and since

1 - λ = p - 2 * 2 α * - 2 * = ( p - 2 * ) ⁢ ( 2 ⁢ N - 2 - α ) ⁢ ( N - 2 ) 4 ⁢ ( α - 2 ) ⁢ ( N - 1 ) ,

one computes

g u ⁢ ( t u ) = ( 1 a ⁢ p ) 2 / ( p - 2 ) ⁢ ( 1 2 - 1 p )

= p - 2 2 ⁢ p p / ( p - 2 ) ⁢ ( 1 a ) 2 / ( p - 2 )

= p - 2 2 ⁢ p p / ( p - 2 ) ⁢ ( A ( 2 ⁢ N - 2 ) ⁢ ( 1 - λ ) / ( 2 ⁢ N - 2 - α ) M 2 ⁢ C ) 2 / ( p - 2 )

= p - 2 2 ⁢ p p / ( p - 2 ) ⁢ A N - 2 α - 2 ⁢ p - 2 * p - 2 ( M 2 ⁢ C ) 2 / ( p - 2 ) .

Hence,

max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) ≥ max t ≥ 0 ⁡ g u ⁢ ( t ) = g u ⁢ ( t u ) = C 0 ⁢ A N - 2 α - 2 ⁢ p - 2 * p - 2 ,

with the definition of C 0 being obvious. Since u ∈ H r ∖ { 0 } is arbitrary, we conclude

m A = inf u ∈ H r ∖ { 0 } , u ≥ 0 ⁡ max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) ≥ inf u ∈ H r ∖ { 0 } ⁡ max t ≥ 0 ⁡ I ⁢ ( t ⁢ u ) ≥ C 0 ⁢ A N - 2 α - 2 ⁢ p - 2 * p - 2 ,

and the proof is complete.∎

Remark 3.3.

If p is as in Proposition 3.2, it is easy to check that

N - 2 α - 2 ⁢ p - 2 * p - 2 = { min ⁡ { N - 1 α , N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 } if ⁢ 0 < α < 2 , min ⁡ { N - 1 α , N - 2 α - 2 ⁢ p 2 - 2 * p 2 - 2 } if ⁢ 2 < α < 2 ⁢ N - 2 .

4 Estimate of c A , K

Let N ≥ 3 , 2 ≤ K ≤ N - 2 and α > 0 , α ≠ 2 . Let f : ℝ → ℝ be a continuous function satisfying (f0)–(f2). In this section we define a suitable u ¯ K ∈ H K such that I ⁢ ( u ¯ K ) < 0 and estimate the corresponding mountain-pass level (2.7).

In defining u ¯ K , we will use the following construction of positive H K functions, which is inspired by [10]. Denote by ϕ : D → ℝ 2 ∖ { 0 } the change to polar coordinates in ℝ 2 ∖ { 0 } , namely, ϕ ⁢ ( ρ , θ ) = ( ρ ⁢ cos ⁡ θ , ρ ⁢ sin ⁡ θ ) for all ( ρ , θ ) ∈ D := ( 0 , + ∞ ) × [ 0 , 2 ⁢ π ) . Define

E := ( 1 4 , 3 4 ) × ( π 6 , π 3 )

and take any ψ : ℝ 2 → ℝ such that ψ ∈ C c ∞ ⁢ ( E ) and ψ > 0 . For 0 < ε < 1 and ( ρ , θ ) ∈ ℝ 2 , define

ψ ε ⁢ ( ρ , θ ) := ψ ⁢ ( ρ 1 / ε , θ ε ) ,

in such a way that ψ ε ∈ C c ∞ ⁢ ( E ε ) , where

E ε := { ( ρ , θ ) ∈ ℝ 2 : ( ρ 1 / ε , θ ε ) ∈ E } = { ( ρ , θ ) ∈ ℝ 2 : ( 1 4 ) ε < ρ < ( 3 4 ) ε , π ⁢ ε 6 < θ < π ⁢ ε 3 } .

Finally, define

v ε ⁢ ( y , z ) := ψ ε ⁢ ( ϕ - 1 ⁢ ( | y | , | z | ) ) for ⁢ x = ( y , z ) ∈ ( ℝ K × ℝ N - K ) ∖ { 0 } , v ε ⁢ ( 0 ) := 0 .

Then v ε ∈ C c ∞ ⁢ ( Ω ε ) ∩ H K , where Ω ε := { ( y , z ) ∈ ℝ K × ℝ N - K : ( | y | , | z | ) ∈ ϕ ⁢ ( E ε ) } .

For future reference, we now compute the relevant integrals of v ε . By means of spherical coordinates in ℝ K and ℝ N - K , one has

∫ ℝ N v ε 2 | x | α ⁢ 𝑑 x = ∫ Ω ε ψ ε ⁢ ( ϕ - 1 ⁢ ( | y | , | z | ) ) 2 | x | α ⁢ 𝑑 x

= σ K ⁢ σ N - K ⁢ ∫ ϕ ⁢ ( E ε ) ψ ε ⁢ ( ϕ - 1 ⁢ ( s , t ) ) 2 ( s 2 + t 2 ) α / 2 ⁢ s K - 1 ⁢ t N - K - 1 ⁢ 𝑑 s ⁢ 𝑑 t

= σ K ⁢ σ N - K ⁢ ∫ E ε ψ ε ⁢ ( ρ , θ ) 2 ρ α - N + 1 ⁢ H ⁢ ( θ ) ⁢ 𝑑 ρ ⁢ 𝑑 θ

= σ K ⁢ σ N - K ⁢ ∫ E ε ψ ⁢ ( ρ 1 / ε , θ / ε ) 2 ρ α - N + 1 ⁢ H ⁢ ( θ ) ⁢ 𝑑 ρ ⁢ 𝑑 θ ,

where H ⁢ ( θ ) := ( cos ⁡ θ ) K - 1 ⁢ ( sin ⁡ θ ) N - K - 1 , and by the change of variables

r = ρ 1 / ε , φ = θ ε ,

one obtains

∫ ℝ N v ε 2 | x | α ⁢ 𝑑 x = σ K ⁢ σ N - K ⁢ ∫ E ψ ⁢ ( r , φ ) 2 r ( α - N + 1 ) ⁢ ε ⁢ H ⁢ ( ε ⁢ φ ) ⁢ ε 2 ⁢ r ε - 1 ⁢ 𝑑 r ⁢ 𝑑 φ

(4.1) = σ K ⁢ σ N - K ⁢ ε 2 ⁢ ∫ E ψ ⁢ ( r , φ ) 2 r ( α - N ) ⁢ ε + 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ .

Similarly (recall that F ⁢ ( 0 ) = 0 ),

∫ ℝ N F ⁢ ( v ε ) ⁢ 𝑑 x = ∫ Ω ε F ⁢ ( ψ ε ⁢ ( ϕ - 1 ⁢ ( | y | , | z | ) ) ) ⁢ 𝑑 x

= σ K ⁢ σ N - K ⁢ ∫ ϕ ⁢ ( E ε ) F ⁢ ( ψ ε ⁢ ( ϕ - 1 ⁢ ( s , t ) ) ) ⁢ s K - 1 ⁢ t N - K - 1 ⁢ 𝑑 s ⁢ 𝑑 t

= σ K ⁢ σ N - K ⁢ ∫ E ε F ⁢ ( ψ ε ⁢ ( ρ , θ ) ) ⁢ ρ N - 1 ⁢ H ⁢ ( θ ) ⁢ 𝑑 ρ ⁢ 𝑑 θ

(4.2) = σ K ⁢ σ N - K ⁢ ε 2 ⁢ ∫ E F ⁢ ( ψ ⁢ ( r , φ ) ) ⁢ r N ⁢ ε - 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ

and

∫ ℝ N | ∇ v ε | 2 d x = ∫ Ω ε | ∇ ψ ε ( ϕ - 1 ( | y | , | z | ) ) ⋅ J ϕ - 1 ( | y | , | z | ) | 2 d x

= σ K σ N - K ∫ ϕ ⁢ ( E ε ) | ∇ ψ ε ( ϕ - 1 ( s , t ) ) ⋅ J ϕ - 1 ( s , t ) | 2 s K - 1 t N - K - 1 d s d t

= σ K σ N - K ∫ E ε | ∇ ψ ε ( ρ , θ ) ⋅ J ϕ - 1 ( ρ , θ ) | 2 ρ N - 1 H ( θ ) d ρ d θ

= σ K ⁢ σ N - K ⁢ ∫ E ε ( ∂ ⁡ ψ ε ∂ ⁡ ρ ⁢ ( ρ , θ ) 2 + 1 ρ 2 ⁢ ∂ ⁡ ψ ε ∂ ⁡ θ ⁢ ( ρ , θ ) 2 ) ⁢ ρ N - 1 ⁢ H ⁢ ( θ ) ⁢ 𝑑 ρ ⁢ 𝑑 θ

= σ K ⁢ σ N - K ⁢ ∫ E ε 1 ε 2 ⁢ ( ρ 2 / ε ⁢ ∂ ⁡ ψ ∂ ⁡ r ⁢ ( ρ 1 / ε , θ ε ) 2 + ∂ ⁡ ψ ∂ ⁡ φ ⁢ ( ρ 1 / ε , θ ε ) 2 ) ⁢ ρ N - 3 ⁢ H ⁢ ( θ ) ⁢ 𝑑 ρ ⁢ 𝑑 θ

(4.3) = σ K ⁢ σ N - K ⁢ ∫ E ( ψ r ⁢ ( r , φ ) 2 + 1 r 2 ⁢ ψ φ ⁢ ( r , φ ) 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ,

where we set ψ r = ∂ ⁡ ψ ∂ ⁡ r and ψ φ = ∂ ⁡ ψ ∂ ⁡ φ for brevity.

Lemma 4.1.

The mapping w A := v A - 1 / 2 ∈ H K , A > 1 , is such that

lim A → + ∞ ⁡ ∥ w A ∥ A 2 ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x = + ∞ .

Proof.

According to the previous computations, for ε = A - 1 / 2 < 1 , we have

∥ v ε ∥ A 2 ∫ ℝ N F ⁢ ( v ε ) ⁢ 𝑑 x = ∫ ℝ N | ∇ v ε | 2 d x + A ∫ ℝ N v ε 2 | x | α d x ∫ ℝ N F ⁢ ( v ε ) ⁢ 𝑑 x

= ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 + A ⁢ ε 2 ⁢ ψ 2 ⁢ r ( N - α ) ⁢ ε - 1 ) ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ε 2 ⁢ ∫ E F ⁢ ( ψ ) ⁢ r N ⁢ ε - 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ

(4.4) = A ⁢ ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 + ψ 2 ⁢ r ( N - α ) ⁢ ε - 1 ) ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ∫ E F ⁢ ( ψ ) ⁢ r N ⁢ ε - 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ .

In the integration set E, one has ε ⁢ π 6 < ε ⁢ φ < ε ⁢ π 3 , and thus, for ε > 0 small enough (i.e., A > 1 large enough), we get that ε ⁢ φ 2 < sin ⁡ ε ⁢ φ < ε ⁢ φ and 1 2 < cos ⁡ ε ⁢ φ < 1 . Hence, there exist two constants C ¯ 1 , C ¯ 2 > 0 such that

C ¯ 1 ⁢ ε N - K - 1 < H ⁢ ( ε ⁢ φ ) < C ¯ 2 ⁢ ε N - K - 1 .

Similarly, since 1 4 < r < 3 4 in E, all the terms r ( N - 2 ) ⁢ ε + 1 , r ( N - α ) ⁢ ε - 1 and r N ⁢ ε - 1 are bounded, and bounded away from zero by positive constants independent of ε ∈ ( 0 , 1 ) (i.e., of A > 1 ), say C ¯ 3 and C ¯ 4 , respectively. Using these bounds in (4.4), we have

∥ v ε ∥ A 2 ∫ ℝ N F ⁢ ( v ε ) ⁢ 𝑑 x ≥ A ⁢ C ¯ 1 ⁢ ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 + ψ 2 ⁢ r ( N - α ) ⁢ ε - 1 ) ⁢ 𝑑 r ⁢ 𝑑 φ C ¯ 2 ⁢ ∫ E F ⁢ r N ⁢ ε - 1 ⁢ 𝑑 r ⁢ 𝑑 φ

≥ A ⁢ C ¯ 1 ⁢ C ¯ 4 ⁢ ∫ E ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 + ψ 2 ) ⁢ 𝑑 r ⁢ 𝑑 φ C ¯ 2 ⁢ C ¯ 3 ⁢ ∫ E F ⁢ ( ψ ) ⁢ 𝑑 r ⁢ 𝑑 φ .

The last ratio is positive and independent of A, whence the claim follows. ∎

According to Lemma 4.1, we fix A 0 > 1 so that

(4.5) ∥ w A ∥ A 2 ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x > 1 for every ⁢ A > A 0 .

We now distinguish the cases 0 < α < 2 and α > 2 .

Proposition 4.2.

Assume (f4) and 0 < α < 2 . Let A > A 0 and define u ¯ K ∈ H K by setting

u ¯ K ⁢ ( x ) := w A ⁢ ( x λ ) , with ⁢ λ := ∥ w A ∥ A 2 / α ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 1 / α .

Then I ⁢ ( u ¯ K ) < 0 and the corresponding mountain-pass level (2.7) satisfies

c A , K ≤ C 1 ⁢ A ( K - 1 ) / 2 + N ⁢ ( 1 / α - 1 / 2 ) ,

where the constant C 1 > 0 does not depend on A.

Proof.

Since A > A 0 , one has λ > 1 . Then an obvious change of variables yields

I ( u ¯ K ) = λ N - 2 2 ∫ ℝ N | ∇ w A | 2 d x + λ N - α 2 ∫ ℝ N A | x | α w A 2 d x - λ N ∫ ℝ N F ( w A ) d x

≤ λ N - α 2 ( ∫ ℝ N | ∇ w A | 2 d x + ∫ ℝ N A | x | α w A 2 d x ) - λ N ∫ ℝ N F ( w A ) d x

= λ N 2 ⁢ ( λ - α ⁢ ∥ w A ∥ A 2 - 2 ⁢ ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x )

= - λ N 2 ⁢ ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x < 0 ,

where the last inequality follows from assumption (f2), since w A > 0 almost everywhere. In order to estimate c A , K , consider the straight path γ ⁢ ( t ) := t ⁢ u ¯ K , t ∈ [ 0 , 1 ] . Clearly, c A , K ≤ max t ∈ [ 0 , 1 ] ⁡ I ⁢ ( γ ⁢ ( t ) ) . Thanks to assumption (f4), which implies F ⁢ ( t ⁢ s ) ≥ t μ ⁢ F ⁢ ( s ) for all s > 0 and t ∈ [ 0 , 1 ] , we have

I ⁢ ( γ ⁢ ( t ) ) = 1 2 ⁢ t 2 ⁢ ∥ u ¯ K ∥ A 2 - ∫ ℝ N F ⁢ ( t ⁢ u ¯ K ) ⁢ 𝑑 x

≤ 1 2 ⁢ t 2 ⁢ ∥ u ¯ K ∥ A 2 - t μ ⁢ ∫ ℝ N F ⁢ ( u ¯ K ) ⁢ 𝑑 x

= 1 2 ⁢ t 2 ⁢ a - t μ ⁢ b ,

where we set a := ∥ u ¯ K ∥ A 2 and b := ∫ ℝ N F ⁢ ( u ¯ K ) ⁢ 𝑑 x for brevity. The function g ⁢ ( t ) := 1 2 ⁢ t 2 ⁢ a - t μ ⁢ b reaches its maximum in t = ( a b ⁢ μ ) 1 / ( μ - 2 ) , and so we get

I ⁢ ( γ ⁢ ( t ) ) ≤ g ⁢ ( ( a b ⁢ μ ) 1 / ( μ - 2 ) ) = a ⁢ ( a b ⁢ μ ) 2 / ( μ - 2 ) ⁢ ( 1 2 - 1 μ ) .

Hence, setting m := ( 1 μ ) 2 / ( μ - 2 ) ⁢ ( 1 2 - 1 μ ) for brevity and recalling that λ > 1 , we obtain

c A , K ≤ m ⁢ ∥ u ¯ K ∥ A 2 ⁢ μ / ( μ - 2 ) ( ∫ ℝ N F ⁢ ( u ¯ K ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

= m ⁢ ( λ N - 2 ∫ ℝ N | ∇ w A | 2 d x + λ N - α ∫ ℝ N A | x | - α w A 2 d x ) μ / ( μ - 2 ) ( λ N ⁢ ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

≤ m ⁢ λ μ ⁢ ( N - α ) / ( μ - 2 ) ( ∫ ℝ N | ∇ w A | 2 d x + ∫ ℝ N A | x | - α w A 2 d x ) μ / ( μ - 2 ) λ 2 ⁢ N / ( μ - 2 ) ⁢ ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

(4.6) = m ⁢ λ μ ⁢ ( N - α ) - 2 ⁢ N μ - 2 ⁢ ∥ w A ∥ A 2 ⁢ μ / ( μ - 2 ) ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 ) .

Inserting the definition of λ into (4.6), we get

c A , K ≤ m ⁢ ∥ w A ∥ A 2 ⁢ μ μ - 2 + 2 α ⁢ μ ⁢ ( N - α ) - 2 ⁢ N μ - 2 ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 ( μ - 2 ) + 1 α ⁢ μ ⁢ ( N - α ) - 2 ⁢ N μ - 2 = m ⁢ ∥ w A ∥ A 2 ⁢ N / α ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) ( N - α ) / α

and therefore, using computations (4.1)–(4.3) with ε = A - 1 / 2 , we have

C ¯ 1 ⁢ ε N - K - 1 < H ⁢ ( ε ⁢ φ ) < C ¯ 2 ⁢ ε N - K - 1

and

c A , K ≤ m ⁢ σ K ⁢ σ N - K ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 + ψ 2 ⁢ r ( N - α ) ⁢ ε - 1 ) ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) N / α ε 2 ⁢ ( N - α ) / α ⁢ ( ∫ E F ⁢ ( ψ ) ⁢ r N ⁢ ε - 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - α ) / α .

As in the proof of Lemma 4.1, we take four constants C ¯ 1 , … , C ¯ 4 > 0 independent of A such that for every ( r , φ ) ∈ E , one has C ¯ 1 ⁢ ε N - K - 1 < H ⁢ ( ε ⁢ φ ) < C ¯ 2 ⁢ ε N - K - 1 and the terms r ( N - 2 ) ⁢ ε + 1 , r ( N - α ) ⁢ ε - 1 and r N ⁢ ε - 1 are bounded, and bounded away from zero by C ¯ 3 and C ¯ 4 , respectively. Hence, we conclude

c A , K ≤ m ⁢ σ K ⁢ σ N - K ⁢ ( C ¯ 2 ⁢ C ¯ 3 ⁢ ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) + ψ 2 ⁢ r ) ⁢ ε N - K - 1 ⁢ 𝑑 r ⁢ 𝑑 φ ) N / α ε 2 ⁢ ( N - α ) / α ⁢ ( C ¯ 1 ⁢ C ¯ 4 ⁢ ∫ E F ⁢ ( ψ ) ⁢ ε N - K - 1 ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - α ) / α

= C ⁢ ε ( N - K - 1 ) ⁢ N / α ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) + ψ 2 ⁢ r ) ⁢ 𝑑 r ⁢ 𝑑 φ ) N / α ε ( N - K + 1 ) ⁢ ( N - α ) / α ⁢ ( ∫ E F ⁢ ( ψ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - α ) / α

= C ⁢ A ( K - 1 ) / 2 + N ⁢ ( 1 / α - 1 / 2 ) ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) + ψ 2 ⁢ r ) ⁢ 𝑑 r ⁢ 𝑑 φ ) N / α ( ∫ E F ⁢ ( ψ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - α ) / α ,

with the definition of the constant C being obvious. As the last ratio does not depend on A, the conclusion ensues. ∎

Proposition 4.3.

Assume (f4) and α > 2 . Let A > A 0 and define u ¯ ∈ H K by setting

u ¯ K ⁢ ( x ) := w A ⁢ ( x λ ) , with ⁢ λ := ∥ w A ∥ A ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 1 / 2 .

Then I ⁢ ( u ¯ K ) < 0 and the corresponding mountain-pass level (2.7) satisfies

c A , K ≤ C 2 ⁢ A ( K - 1 ) / 2 ,

where the constant C 2 > 0 does not depend on A.

Proof.

The proof is very similar to the one of Proposition 4.2, so we omit here some computational details. As α > 2 , we have

I ( u ¯ K ) ≤ λ N - 2 2 ( ∫ ℝ N | ∇ w A | 2 d x + ∫ ℝ N A | x | α w A 2 d x ) - λ N ∫ ℝ N F ( w A ) d x = - λ N 2 ∫ ℝ N F ( w A ) d x < 0

and

max t ∈ [ 0 , 1 ] ⁡ I ⁢ ( t ⁢ u ¯ K ) ≤ m ⁢ ∥ u ¯ K ∥ A 2 ⁢ μ / ( μ - 2 ) ( ∫ ℝ N F ⁢ ( u ¯ K ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

= m ⁢ ( λ N - 2 ∫ ℝ N | ∇ w A | 2 d x + λ N - α ∫ ℝ N A | x | - α w A 2 d x ) μ / ( μ - 2 ) ( λ N ⁢ ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

≤ m ⁢ λ μ ⁢ ( N - 2 ) / ( μ - 2 ) ( ∫ ℝ N | ∇ w A | 2 d x + ∫ ℝ N A | x | - α w A 2 d x ) μ / ( μ - 2 ) λ 2 ⁢ N / ( μ - 2 ) ⁢ ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 )

= m ⁢ λ μ ⁢ ( N - 2 ) - 2 ⁢ N μ - 2 ⁢ ∥ w A ∥ A 2 ⁢ μ / ( μ - 2 ) ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 / ( μ - 2 ) .

Recalling the definition of c A , K and inserting the one of λ, we get

c A , K ≤ m ⁢ ∥ w A ∥ A 2 ⁢ μ μ - 2 + μ ⁢ ( N - 2 ) - 2 ⁢ N μ - 2 ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) 2 μ - 2 + 1 2 ⁢ μ ⁢ ( N - 2 ) - 2 ⁢ N μ - 2 = m ⁢ ∥ w A ∥ A N ( ∫ ℝ N F ⁢ ( w A ) ⁢ 𝑑 x ) ( N - 2 ) / 2 ,

and therefore, using computations (4.1)–(4.3) with ε = A - 1 / 2 , we have

c A , K ≤ m ⁢ σ K ⁢ σ N - K ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) ⁢ r ( N - 2 ) ⁢ ε + 1 + ψ 2 ⁢ r ( N - α ) ⁢ ε - 1 ) ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) N / 2 ε N - 2 ⁢ ( ∫ E F ⁢ ( ψ ) ⁢ r N ⁢ ε - 1 ⁢ H ⁢ ( ε ⁢ φ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - 2 ) / 2

≤ C ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) + ψ 2 ⁢ r ) ⁢ ε N - K - 1 ⁢ 𝑑 r ⁢ 𝑑 φ ) N / 2 ε N - 2 ⁢ ( ∫ E F ⁢ ( ψ ) ⁢ ε N - K - 1 ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - 2 ) / 2

= C ⁢ A ( K - 1 ) / 2 ⁢ ( ∫ E ( ( ψ r 2 + 1 r 2 ⁢ ψ φ 2 ) + ψ 2 ⁢ r ) ⁢ 𝑑 r ⁢ 𝑑 φ ) N / 2 ( ∫ E F ⁢ ( ψ ) ⁢ 𝑑 r ⁢ 𝑑 φ ) ( N - 2 ) / 2 ,

where C > 0 is a suitable constant independent of A. This concludes the proof. ∎

5 Proof of Theorem 1.1

This section is entirely devoted to the proof of Theorem 1.1, so we assume all the hypotheses of the theorem. The proof will be achieved through some lemmas.

Let K be any integer such that 2 ≤ K ≤ N - 2 . Assume A > A K (where A K is defined by (4.5)) and consider the mountain-pass level c A , K defined by (2.7), with u ¯ K ∈ H K given by Lemma 4.2 if α ∈ ( 2 N - 1 , 2 ) , or Lemma 4.3 if α ∈ ( 2 , 2 ⁢ N - 2 ) . We are going to show that c A , K is a critical level for the energy functional I defined in (2.2). To this end, we will make use of the sum space

L p 1 + L p 2 := { u 1 + u 2 : u 1 ∈ L p 1 ⁢ ( ℝ N ) , u 2 ∈ L p 2 ⁢ ( ℝ N ) } .

We recall from [8] that such a space can be characterized as the set of measurable mappings u : ℝ N → ℝ for which there exists a measurable set E ⊆ ℝ N such that u ∈ L p 1 ⁢ ( E ) ∩ L p 2 ⁢ ( ℝ N ∖ E ) (see [8, Proposition 2.3]). This is a Banach space with respect to the norm

∥ u ∥ L p 1 + L p 2 := inf u 1 + u 2 = u ⁡ max ⁡ { ∥ u 1 ∥ L p 1 ⁢ ( ℝ N ) , ∥ u 2 ∥ L p 2 ⁢ ( ℝ N ) }

(see [8, Corollary 2.11]), and the continuous embedding L p ⁢ ( ℝ N ) ↪ L p 1 + L p 2 holds for all p ∈ [ p 1 , p 2 ] (see [8, Proposition 2.17]), in particular, for p = 2 * . Moreover, for every u ∈ L p 1 + L p 2 and every φ ∈ L p 1 ′ ⁢ ( ℝ N ) ∩ L p 2 ′ ⁢ ( ℝ N ) , one has

(5.1) ∫ ℝ N | u φ | d x ≤ ∥ u ∥ L p 1 + L p 2 ( ∥ φ ∥ L p 1 ′ ⁢ ( ℝ N ) + ∥ φ ∥ L p 2 ′ ⁢ ( ℝ N ) ) ,

where p i ′ = p i / ( p i - 1 ) is the Hölder conjugate exponent of p i (see [8, Lemma 2.9]).

Lemma 5.1.

c A , K is a critical level for the functional I | H K .

Proof.

Thanks to Lemma 2.4 (note that I ⁢ ( u ¯ K ) < 0 implies ∥ u ¯ K ∥ A > R ), the claim follows from the mountain pass theorem [1] if we show that I | H K satisfies the Palais–Smale condition. Using the compact embeddings of [2] and the results of [8] about Nemytskiĭ operators on L p 1 + L p 2 , this is a standard proof but we still give some details for the sake of completeness. Let { u n } be a sequence in H K such that { I ⁢ ( u n ) } is bounded and I ′ ⁢ ( u n ) → 0 in the dual space of H K . Then, recalling (2.2) and (2.3), we have

1 2 ⁢ ∥ u n ∥ A 2 - ∫ ℝ N F ⁢ ( u n ) ⁢ 𝑑 x = O ⁢ ( 1 ) and ∥ u n ∥ A 2 - ∫ ℝ N f ⁢ ( u n ) ⁢ u n ⁢ 𝑑 x = o ⁢ ( 1 ) ⁢ ∥ u n ∥ ,

and so assumption (f1) implies

1 2 ⁢ ∥ u n ∥ A 2 + O ⁢ ( 1 ) = ∫ ℝ N F ⁢ ( u n ) ⁢ 𝑑 x ≤ 1 θ ⁢ ∫ ℝ N f ⁢ ( u n ) ⁢ u n ⁢ 𝑑 x = 1 θ ⁢ ∥ u n ∥ A 2 + o ⁢ ( 1 ) ⁢ ∥ u n ∥ .

This yields that { ∥ u n ∥ A } is bounded, since θ > 2 . On the other hand, thanks to the fact that p 1 < 2 * < p 2 , the space H K is compactly embedded into L p 1 + L p 2 , since so is the subspace of D 1 , 2 ⁢ ( ℝ N ) made up of the mappings with the same symmetries of H K (see [2, Theorem A.1]). Hence, there exists u ∈ H K such that, up to a subsequence, we have u n ⇀ u in H K and u n → u in L p 1 + L p 2 . This implies that { f ⁢ ( u n ) } is bounded in both L p 1 ′ ⁢ ( ℝ N ) and L p 2 ′ ⁢ ( ℝ N ) , since assumption (f0) ensures that the operator v ↦ f ⁢ ( v ) is continuous from L p 1 + L p 2 into L p 1 ′ ⁢ ( ℝ N ) ∩ L p 2 ′ ⁢ ( ℝ N ) (see [8, Corollary 3.7]). Then, by (5.1), we get

| ∫ ℝ N f ( u n ) ( u n - u ) d x | ≤ ∫ ℝ N | f ( u n ) | | u n - u | d x

≤ ∥ u n - u ∥ L p 1 + L p 2 ⁢ ( ∥ f ⁢ ( u n ) ∥ L p 1 ′ ⁢ ( ℝ N ) + ∥ f ⁢ ( u n ) ∥ L p 2 ′ ⁢ ( ℝ N ) )

≤ ( const . ) ∥ u n - u ∥ L p 1 + L p 2 = o ( 1 ) ,

and therefore

∥ u n - u ∥ A 2 = ( u n , u n - u ) A - ( u , u n - u ) A

= I ′ ⁢ ( u n ) ⁢ ( u n - u ) + ∫ ℝ N f ⁢ ( u n ) ⁢ ( u n - u ) ⁢ 𝑑 x - ( u , u n - u ) A = o ⁢ ( 1 ) ,

where ( u , u n - u ) A = o ⁢ ( 1 ) , since u n ⇀ u in H K , and I ′ ⁢ ( u n ) ⁢ ( u n - u ) = o ⁢ ( 1 ) because I ′ ⁢ ( u n ) → 0 in the dual space of H K and { u n - u } is bounded in H K . This completes the proof. ∎

The next lemma clarifies why our separation of c A , K and m A needs assumption (1.5) and the lower bound α > 2 N - 1 . Recall the definition (1.4) of ν = ν N , α , p 1 , p 2 .

Lemma 5.2.

For every α ∈ ( 2 N - 1 , 2 ⁢ N - 2 ) , α ≠ 2 , we have ν ≥ 1 .

Proof.

Assume 2 N - 1 < α < 2 . Since α > 2 N - 1 , we have

2 ⁢ N - 1 α - 2 ⁢ N ⁢ ( 1 α - 1 2 ) = N - 2 α > 1 .

On the other hand, by easy computations, condition

2 ⁢ N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 - 2 ⁢ N ⁢ ( 1 α - 1 2 ) > 1

turns out to be equivalent to the first inequality of assumption (1.5). This proves that

2 ⁢ min ⁡ { N - 1 α , N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 } - 2 ⁢ N ⁢ ( 1 α - 1 2 ) > 1 ,

which means

⌈ 2 ⁢ min ⁡ { N - 1 α , N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 } - 2 ⁢ N ⁢ ( 1 α - 1 2 ) ⌉ ≥ 2 ,

and thus ν ≥ 1 . Similarly, if 2 < α < 2 ⁢ N - 2 , we readily have 2 ⁢ ( N - 1 ) α > 1 , and condition

2 ⁢ N - 2 α - 2 ⁢ p 2 - 2 * p 2 - 2 > 1

turns out to be equivalent to the second inequality of (1.5). This proves again that ν ≥ 1 . ∎

Proof of Theorem 1.1.

On the one hand, the restriction I | H r has a critical point u r ≠ 0 thanks to the results of [25], since (f0) ensures that one can find p ∈ [ p 1 , p 2 ] such that | f ( u ) | ≤ ( const . ) u p - 1 (cf. (2.1)) and (1.3) holds. On the other hand, according to Lemma 5.2, there are ν ≥ 1 integers K (precisely K = 2 , … , ν + 1 ) such that

K - 1 2 + N ⁢ ( 1 α - 1 2 ) < min ⁡ { N - 1 α , N - 2 2 - α ⁢ 2 * - p 1 p 1 - 2 } if ⁢ 2 N - 1 < α < 2

and

K - 1 2 < min ⁡ { N - 1 α , N - 2 α - 2 ⁢ p 2 - 2 * p 2 - 2 } if ⁢ 2 < α < 2 ⁢ N - 2 .

Let K be any of such integers. By Remark 3.3 and Propositions 3.2, 4.2 and 4.3, there exists A * > A K such that

(5.2) c A , K < m A for every ⁢ A > A * .

Then, by Lemma 5.1, there exists u K ∈ H K such that I ⁢ ( u K ) = c A , K and I | H K ′ ⁢ ( u K ) = 0 , where u K ≠ 0 , since c A , K > 0 and I ⁢ ( 0 ) = 0 . Both u r and u K are also critical points for the functional I : H α 1 → ℝ , by the Palais principle of symmetric criticality [23]. Moreover, it easy to check that they are nonnegative: test I ′ ⁢ ( u K ) with the negative part u K - ∈ H α 1 of u K and use the fact that f ⁢ ( s ) = 0 for s < 0 to get I ′ ⁢ ( u K ) ⁢ u K - = - ∥ u K - ∥ A 2 = 0 ; the same applies for u r . Therefore, u r and u K are weak solutions to problem (1.1). Finally, u K is not radial, because otherwise Lemma 2.3 would imply c A , K = I ⁢ ( u K ) ≥ m A , which is false by (5.2). This also implies u K 1 ≠ u K 2 for K 1 ≠ K 2 , thanks to Lemma 2.1. ∎

References

[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Suche in Google Scholar

[2] A. Azzollini and A. Pomponio, Compactness results and applications to some “zero-mass” elliptic problems, Nonlinear Anal. 69 (2008), 602–617. 10.1016/j.na.2007.09.041Suche in Google Scholar

[3] M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc. (JEMS) 9 (2007), 355–381. 10.4171/JEMS/83Suche in Google Scholar

[4] M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Differential Equations 12 (2007), 1321–1362. 10.57262/ade/1355867405Suche in Google Scholar

[5] M. Badiale, M. Guida and S. Rolando, A nonexistence result for a nonlinear elliptic equation with singular and decaying potential, Commun. Contemp. Math. 17 (2015), Article ID 1450024. 10.1142/S0219199714500242Suche in Google Scholar

[6] M. Badiale, M. Guida and S. Rolando, Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence, NoDEA Nonlinear Differential Equations Appl. 23 (2016), Article ID 67. 10.1007/s00030-016-0411-0Suche in Google Scholar

[7] M. Badiale, M. Guida and S. Rolando, Compactness and existence results for the p-Laplace equations, J. Math. Anal. Appl. 451 (2017), 345–370. 10.1016/j.jmaa.2017.02.011Suche in Google Scholar

[8] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 18 (2011), 369–405. 10.1007/s00030-011-0100-ySuche in Google Scholar

[9] M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 16 (2006), 1–13. 10.4171/RLM/450Suche in Google Scholar

[10] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud. 4 (2004), 453–467. 10.1515/ans-2004-0406Suche in Google Scholar

[11] V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations. Solitary Waves, Hylomorphic Solitons and Vortices, Springer Monogr. Math., Springer, Cham, 2014. 10.1007/978-3-319-06914-2Suche in Google Scholar

[12] V. Benci, C. R. Grisanti and A. M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrödinger equation with V ⁢ ( ∞ ) = 0 , Topol. Methods Nonlinear Anal. 26 (2005), 203–220. 10.12775/TMNA.2005.031Suche in Google Scholar

[13] V. Benci, C. R. Grisanti and A. M. Micheletti, Existence of Solutions for the Nonlinear Schrödinger Equation with V ⁢ ( ∞ ) = 0 , Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel, 2006. 10.1007/3-7643-7401-2_4Suche in Google Scholar

[14] V. Benci and A. M. Micheletti, Solutions in exterior domains of null mass nonlinear scalar field equations, Adv. Nonlinear Stud. 6 (2006), 171–198. 10.1515/ans-2006-0204Suche in Google Scholar

[15] F. Catrina, Nonexistence of positive radial solutions for a problem with singular potential, Adv. Nonlinear Anal. 3 (2014), 1–13. 10.1515/anona-2013-0023Suche in Google Scholar

[16] M. Conti, S. Crotti and D. Pardo, On the existence of positive solutions for a class of singular elliptic equations, Adv. Differential Equations 3 (1998), 111–132. 10.1080/10236199708808110Suche in Google Scholar

[17] P. C. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc. 84 (1978), 693–728. 10.1090/S0002-9904-1978-14502-9Suche in Google Scholar

[18] M. Ghimenti and A. M. Micheletti, Existence of minimal nodal solutions for the nonlinear Schrödinger equations with V ⁢ ( ∞ ) = 0 , Adv. Differential Equations 11 (2006), 1375–1396. 10.57262/ade/1355867589Suche in Google Scholar

[19] M. Guida and S. Rolando, Nonlinear Schrödinger equations without compatibility conditions on the potentials, J. Math. Anal. Appl. 439 (2016), 347–363. 10.1016/j.jmaa.2016.02.061Suche in Google Scholar

[20] I. Kuzin and S. Pohožaev, Entire Solutions of Semilinear Elliptic Equations, Progr. Nonlinear Differential Equations Appl. 33, Birkhäuser, Basel, 1997. Suche in Google Scholar

[21] Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations 83 (1990), 348–367. 10.1016/0022-0396(90)90062-TSuche in Google Scholar

[22] J. Mederski, Ground states of time-harmonic semilinear Maxwell equations in ℝ 3 with vanishing permittivity, Arch. Ration. Mech. Anal. 218 (2015), 825–861. 10.1007/s00205-015-0870-1Suche in Google Scholar

[23] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19–30. 10.1007/BF01941322Suche in Google Scholar

[24] J. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying potentials, Commun. Contemp. Math. 9 (2007), 571–583. 10.1142/S021919970700254XSuche in Google Scholar

[25] J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 201–219. 10.1016/j.jde.2007.03.018Suche in Google Scholar

[26] S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), 241–264. 10.57262/ade/1366896239Suche in Google Scholar

[27] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, 2001. 10.1007/978-1-4757-6548-9Suche in Google Scholar

Received: 2017-08-03

Revised: 2017-09-01

Accepted: 2017-09-24

Published Online: 2017-11-21

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Semilinear elliptic PDE; singular vanishing potential; symmetry breaking

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