1 Introduction and main result
This paper is concerned with the following semilinear elliptic problem:
where are real constants, is a continuous nonlinearity satisfying for some and all , and 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 such that almost everywhere in and
As is well known, problems like (1.1) are models for stationary states of reaction diffusion equations in population dynamics (see, e.g., ). 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  and continued in [16, 9, 24, 25, 5, 15] (see  for a similar cylindrical problem). Currently, the problem of existence and nonexistence of radial solutions is essentially solved in the pure-power case , 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 ): the problem has a radial solution for (see ) and for all the pairs satisfying
(see ), while it has no solution if
(see ) and no radial solution for both
(see  and , respectively). As usual, denotes the critical exponent for the Sobolev embedding in dimension . All these results are portrayed in the picture of the -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 and except for the pair ), whereas white color (of course above the line ) 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 , and , has at least a nonradial solution for every A large enough. This brought Catrina to say, in the introduction of his paper : “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  covered also the case where problem (1.1) has general nonlinearities satisfying the power growth condition for some , 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 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 (where the energy functional of the problem is well defined by the embeddings they proved), but the symmetric criticality type results of  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 , provided that , 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 ).
The main assumptions characterizing our class of nonlinearities are the following, where we define :
for some .
There exists such that for all .
for all .
The function is strictly increasing on .
There exists such that the function is decreasing on .
For , , and , we define
where denotes the ceiling function (i.e., ).
Our main result is the following theorem.
if or , respectively. Then there exists such that for every , 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 and 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 ).
Regarding assumptions (1.5) and (f0), it is worth observing that implies , while implies , 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 for all and , but it is actually more stringent than that, since it requires . We finally observe that (f0) still remains true if one raises and lowers , but this decreases ν (see Figures 2 and 3), and therefore it is convenient to apply Theorem 1.1 with as small as possible and 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 , 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.
denotes the -dimensional measure of the unit sphere of .
is the space of infinitely differentiable real functions with compact support in the open set .
is the usual Sobolev space, which identifies with the completion of with respect to the norm of the gradient.
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 is continuous.
Given any integer K such that , we write every as , and in the space , we consider the following closed subspaces of symmetric functions:
Of course naturally means that for all isometries and of and , respectively. Similarly for . Note that for every K, since . The next lemma clarifies better the relation between the spaces and .
Let . Then .
The proof is essentially an adaptation of the one of [21, Lemma 3.3]. For any , we will denote by its decomposition in , and by its decomposition in . Let , and for every , define
Then we clearly have for every . Let and be such that , i.e., . Suppose that and define by setting
where the first block of zeros has zeros, the second , and the third . Then
which implies , i.e., . If , we repeat the argument with defined by
(with the same lengths of blocks of zeros) and get the same result. Hence, implies , i.e., . ∎
We modify the function f by setting for all and, with a slight abuse of notation, we still denote by f the modified function. Then, by (f0), there exist such that
which, in particular, yields
is of class on and has Fréchet derivative at any given by
This yields that the critical points of 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 readily follows from the results of . By exploiting the compact embeddings of  and the results of  about Nemytskiĭ operators on the sum of Lebesgue spaces, we will show in Section 5 that has a nonzero critical point for every . Thanks to the classical Palais principle of symmetric criticality , 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 for every K, which also implies for , by Lemma 2.1. This will be achieved by showing that the critical levels are lower than all the nonzero critical levels of . The starting points in proving this are the following lemmas.
For every , , there exists such that .
since . On the other hand, condition (2.1) with and the continuous embeddings imply that there exists a constant such that for all , which in turn yields for all small enough. Together with (2.4), this gives the result. ∎
According to Lemma 2.2, define
Assume (f3) and let . If u is a critical point for I, then
As already observed, if u is a critical point for I, then u is nonnegative. We shall now prove that , which obviously yields the result. For , define
As u is a critical point for I, we readily have that is a critical point for g. Indeed, , and thus . We now show that, on the other hand, g has at most one critical point in . We have if and only if , i.e.,
So, if are critical points for g, then one has
where . Since the integrand in (2.5) is nonnegative by assumption (f3), we have that almost everywhere on . Since has positive measure (because ), this implies , again by assumption (f3). As a conclusion, according to Lemma 2.2, we deduce that and the claim ensues. ∎
There exists such that for every , one has
The claim readily follows from (2.1) with and the continuous embeddings , which imply that there exists a constant such that for all . ∎
In Section 4 we will see that takes negative values by choosing a suitable such that . This implies , by (2.6), and therefore the functional has a mountain-pass geometry. In Section 5 we will see that it also satisfies the Palais–Smale condition for , and so it admits a (nonnegative) critical point at the mountain-pass level
3 Estimate of
Let and . Let be a continuous function satisfying (f0)–(f2). This section is devoted to deriving the estimate of 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).
(recall that denotes the -dimensional measure of the unit sphere of ).
Let and let be continuous and such that for almost every . Set
By [6, Lemma 27], we have that for every , whence and
Moreover, for almost every , one has
If , this implies , and therefore
If , then (3.1) gives , and thus we have
as before. Now observe that there exist and such that and . Indeed, if , then for every r smaller than some suitable , one has , and therefore one of the following contradictions ensues:
Similarly, if , then one obtains
We can now prove our estimate for .
Assume , , and let if , or if . Then there exists a constant , independent from A, such that
Let . By Lemma 3.1, we have
since . On the other hand, one has
where denotes the Sobolev constant in dimension N. Then, in either case or , we can argue by interpolation: there exists such that , and by Hölder inequality, we get
where only depends on . Recalling condition (2.1), this implies
where we set for brevity. Hence,
The function attains its maximum in and since
with the definition of being obvious. Since is arbitrary, we conclude
and the proof is complete.∎
If p is as in Proposition 3.2, it is easy to check that
4 Estimate of
In defining , we will use the following construction of positive functions, which is inspired by . Denote by the change to polar coordinates in , namely, for all . Define
and take any such that and . For and , define
in such a way that , where
Then , where .
For future reference, we now compute the relevant integrals of . By means of spherical coordinates in and , one has
where , and by the change of variables
Similarly (recall that ),
where we set and for brevity.
The mapping , , is such that
According to the previous computations, for , we have
In the integration set E, one has , and thus, for small enough (i.e., large enough), we get that and . Hence, there exist two constants such that
Similarly, since in E, all the terms , and are bounded, and bounded away from zero by positive constants independent of (i.e., of ), say and , respectively. Using these bounds in (4.4), we have
The last ratio is positive and independent of A, whence the claim follows. ∎
According to Lemma 4.1, we fix so that
We now distinguish the cases and .
Assume (f4) and . Let and define by setting
Then and the corresponding mountain-pass level (2.7) satisfies
where the constant does not depend on A.
Since , one has . Then an obvious change of variables yields
where the last inequality follows from assumption (f2), since almost everywhere. In order to estimate , consider the straight path , . Clearly, . Thanks to assumption (f4), which implies for all and , we have
where we set and for brevity. The function reaches its maximum in , and so we get
Hence, setting for brevity and recalling that , we obtain
Inserting the definition of λ into (4.6), we get
As in the proof of Lemma 4.1, we take four constants independent of A such that for every , one has and the terms , and are bounded, and bounded away from zero by and , respectively. Hence, we conclude
with the definition of the constant C being obvious. As the last ratio does not depend on A, the conclusion ensues. ∎
Assume (f4) and . Let and define by setting
Then and the corresponding mountain-pass level (2.7) satisfies
where the constant does not depend on A.
The proof is very similar to the one of Proposition 4.2, so we omit here some computational details. As , we have
Recalling the definition of and inserting the one of λ, we get
where 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 . Assume (where is defined by (4.5)) and consider the mountain-pass level defined by (2.7), with given by Lemma 4.2 if , or Lemma 4.3 if . We are going to show that 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  that such a space can be characterized as the set of measurable mappings for which there exists a measurable set such that (see [8, Proposition 2.3]). This is a Banach space with respect to the norm
where is the Hölder conjugate exponent of (see [8, Lemma 2.9]).
is a critical level for the functional .
Thanks to Lemma 2.4 (note that implies ), the claim follows from the mountain pass theorem  if we show that satisfies the Palais–Smale condition. Using the compact embeddings of  and the results of  about Nemytskiĭ operators on , this is a standard proof but we still give some details for the sake of completeness. Let be a sequence in such that is bounded and in the dual space of . Then, recalling (2.2) and (2.3), we have
and so assumption (f1) implies
This yields that is bounded, since . On the other hand, thanks to the fact that , the space is compactly embedded into , since so is the subspace of made up of the mappings with the same symmetries of (see [2, Theorem A.1]). Hence, there exists such that, up to a subsequence, we have in and in . This implies that is bounded in both and , since assumption (f0) ensures that the operator is continuous from into (see [8, Corollary 3.7]). Then, by (5.1), we get
where , since in , and because in the dual space of and is bounded in . This completes the proof. ∎
For every , , we have .
Assume . Since , 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 . Similarly, if , we readily have , and condition
turns out to be equivalent to the second inequality of (1.5). This proves again that . ∎
Proof of Theorem 1.1.
On the one hand, the restriction has a critical point thanks to the results of , since (f0) ensures that one can find such that (cf. (2.1)) and (1.3) holds. On the other hand, according to Lemma 5.2, there are integers K (precisely ) such that
Then, by Lemma 5.1, there exists such that and , where , since and . Both and are also critical points for the functional , by the Palais principle of symmetric criticality . Moreover, it easy to check that they are nonnegative: test with the negative part of and use the fact that for to get ; the same applies for . Therefore, and are weak solutions to problem (1.1). Finally, is not radial, because otherwise Lemma 2.3 would imply , which is false by (5.2). This also implies for , thanks to Lemma 2.1. ∎
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About the article
Published Online: 2017-11-21
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 885–901, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0177.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0