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

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha }}u=f\left( u\right) \quad \textrm{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha>0, \] with nonlinearites satisfying $\left| f\left( u\right) \right| \leq \left(\mathrm{const.}\right) u^{p-1}$ for some $p>2$. Existence of nonradial solutions, by contrast, is known only for $N\geq 4$, $\alpha =2$, $f\left( u\right) =u^{(N+2)/(N-2)}$ and $A$ large enough. Here we show that the equation has multiple nonradial solutions as $A\rightarrow +\infty$ for $N\geq 4$, $2/(N-1)<\alpha<2N-2$, $\alpha\neq 2$, and 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.


Introduction and main result
This paper is concerned with the following semilinear elliptic problem: (1.1) 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 1 α := D 1,2 (ℝ N ) ∩ L 2 (ℝ N , |x| −α dx) 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 1 α \ {0} such that u ≥ 0 almost everywhere in ℝ N and 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., [8,22]).
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 [5,9,15,16,24,25] (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 (see [25]), while it has no solution if p ≤ 2 * , with 2 α := 2N N − α (see [9]) and no radial solution for both (see [5] and [15], respectively). As usual, 2 * := 2N/(N − 2) denotes the critical exponent for the Sobolev embedding in dimension N ≥ 3. All these results are portrayed in the picture of the αp-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". 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 1 α (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.
Our main result is the following theorem.
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 → ∞).
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: (i) The compact embeddings between some suitable functional spaces of symmetric functions, which yield the existence of ν different solutions of mountain-pass type. (ii) 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.
Other simple examples are 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 .
with respect to the norm of the gradient.

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 which we endow with the following scalar product and related norm: Of course, the embedding H 1 α → 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 1 α , we consider the following closed subspaces of symmetric functions: 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 .
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 , .
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 which, in particular, yields , 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., is of class C 1 on H 1 α and has Fréchet derivative I (u) at any u ∈ H 1 α given by This yields that the critical points of I : H 1 α → 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 Nemytskiȋ 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.
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) ≥ Cs θ for all s ≥ δ. Then, for every t > 1, one has and therefore On the other hand, condition (2.1) with p = 2 * and the continuous embeddings 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(tu), which obviously yields the result. For t ≥ 0, define As u is a critical point for I, we readily have that t = 1 is a critical point for g. Indeed, g (t) = I (tu)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 (tu)u = 0, i.e., So, if 0 < t 1 < t 2 are critical points for g, then one has where E u := {x ∈ ℝ N : u > 0}. Since the integrand in (2.5) is nonnegative by assumption (f3), we have that 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. Proof. The claim readily follows from (2.1) with p = 2 * and the continuous embeddings 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 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 .

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).  1 ((a, b)) for every 0 < a < b < +∞, whence v ∈ W 1,1 ((a, b)) and Moreover, for almost every r ∈ (a, b), one has 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 |ũ (r)| ≥ √l/2r −(N−1−α/2)/2 , and therefore one of the following contradictions ensues: Similarly, if lim inf r→+∞ v(r) > 0, then one obtains 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 .
Then there exists a constant C 0 > 0, independent from A, such that On the other hand, one has 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 = λ2 * + (1 − λ)2 * α , and by Hölder inequality, we get and therefore I(u) ≥ with the definition of C 0 being obvious. Since u ∈ H r \ {0} is arbitrary, we conclude and the proof is complete.
According to Lemma 4.1, we fix A 0 > 1 so that 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 Then I(u K ) < 0 and the corresponding mountain-pass level (2.7) satisfies where the constant C 1 > 0 does not depend on A.
Then I(u K ) < 0 and the corresponding mountain-pass level (2.7) satisfies 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 Recalling the definition of c A,K and inserting the one of λ, we get and therefore, using computations (4.1)-(4.3) with ε = A −1/2 , we have where C > 0 is a suitable constant independent of A. This concludes the proof.

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, 2N − 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 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 [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 ∫ 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 Nemytskiȋ 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 and so assumption (f1) implies 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 Proof. Assume 2 N−1 < α < 2. Since α > 2 N−1 , we have On the other hand, by easy computations, condition turns out to be equivalent to the first inequality of assumption (1.5). This proves that and thus ν ≥ 1. Similarly, if 2 < α < 2N − 2 , we readily have 2(N−1) α > 1, and condition 2 N − 2 α − 2 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 Let K be any of such integers. 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 1 α → 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 1 α of u K and use the fact that f(s) = 0 for s < 0 to get I (u K )u − K = −‖u − K ‖ 2 A = 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.