Oscillating solutions for nonlinear Helmholtz Equations
Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schrödinger equations with large frequencies.
Key words and phrases:Nonlinear Helmholtz equations, standing waves, oscillating solutions
1991 Mathematics Subject Classification:35J05, 35J20, 35Q55.
The main aim of this paper is to give existence results for the following class of nonlinear equations
with and assuming that the nonlinearity is such that
|(1.5)||on and negative on .|
There is a huge literature concerning (1.1) and nonautonomous variants of it under the assumption . Two seminal papers in this context are the contributions by Berestycki-Lions and Strauss [BL83, Str77] who proved the existence of smooth radially symmetric and exponentially decaying solutions for a large class of nonlinearities with this property. We refer to the monographs [AM07, Wil96] for more results in this context. One of the main interests in finding solutions of (1.1) is motivated by the fact that a solution of (1.1) gives rise to a standing wave, i.e. a solution of the form , of the nonlinear time-dependent Klein-Gordon equation
with . Therefore, the assumption (1.4) amounts to look for standing waves having low frequencies and numerous existence results for -solutions under this assumption can be found in the references mentioned above. In this paper we deal with nonlinearities satisfying , which gives rise to standing waves with large frequencies . Looking at the form of the linearized operator , one realizes that lies in its essential spectrum and we are actually dealing with a class of nonlinear Helmholtz equations. Furthermore, as explained in subsection 2.2 in [BL83], the hypothesis (1.4) has the striking consequence that radially symmetric solutions of (1.1) can not exist, and usual variational methods fail. On the other hand, (1.4) is naturally linked to (1.5); in particular, if decreases in , then (1.4) turns out to be necessary in order to have solutions. Actually, the relevant solutions naturally lie outside this functional space. This fact can also be illustrated by an examining the behaviour of the minimal energy solutions on a sequence of large bounded domains. Namely, in Theorem 4.4 we will show that if one takes a sequence of bounded domains invading , then (1.4) guarantees the existence of a sequence of global minimizers of the associated action functional over for sufficiently large . But, it results that converges in to the constant solution .
Therefore, under the assumption (1.4), one has to look for solutions in a broader class of functions. Our focus will be on oscillating and localized ones which we define as follows.
A distributional solution of (1.1) is called oscillating if it has an unbounded sequence of zeros. It is called localized when it converges to zero at infinity.
Let us notice that, the Strong Maximum Principle implies that oscillating solutions of (1.1) change sign at each of their zeros; so that we are going to find solutions that change sign infinitely many times.
In our study, we will pay particular attention to the following model cases
Our interest in these examples has various motivations. The nonlinearity is related to the study of the propagation of lights beams in a photorefractive crystals (see [CCMS97, Yan04]) when a saturation effect is taken into account. Differently from the more frequently studied model
see e.g. [CBWM04], describes a transition from the linear propagation and the saturated one. This difference has important consequences, for instance for there are solutions of (1.1) (e.g. see Theorem 3.6 in [SZ99]), whereas, as we have already observed, this is not the case if due to . Notice that, as , equation (1.1) for can be rewritten in the following form
which allows to settle the problem in in every dimension and that shows that also this saturable model is included in the class of the nonlinear Helmholtz equations. The principal difference between (1.7) and (1.6) is that the formers are superlinear and homogeneous nonlinearities, while the latter is not homogeneous and it is asymptotically linear. However, all of them satisfy our general assumptions, with for and , and for .
Up to now nonlinear Helmholtz equations (1.1) have been mainly investigated for the model nonlinearity or more general superlinear nonlinearities, even not autonomous. In a series of papers [EW14, EW15, Eve15, EW] Evéquoz and Weth proved the existence of radial and nonradial real, localized solutions of this equation under various different assumptions on the nonlinearity. Let us mention that some of the tools used in [EW15] had already appeared in a paper by Gutiérrez [Gut04] where the existence of complex-valued solutions was proved for space dimensions . Let us first focus our attention on radially symmetric solutions and state our first result, which provides a complete description of the radially symmetric solutions of (1.1).
Moreover, for all these solutions are periodic; whereas, for they are localized and satisfy the following asymptotic behavior:
There are positive numbers such that
Here a continuum in is a connected subset of with respect to the uniform convergence of the zeroth, first and second derivatives. The continuum found in Theorem 1.2 is even maximal in the sense that there are no further radially symmetric localized solutions as we will see in section 2. Moreover, conclusion states that the property is equivalent to , and this happens if and only if . Notice that this implies that , showing again that the solutions, as expected, live outside the commonly used energy space.
Furthermore, let us stress that the behaviour of the nonlinearity beyond is completely irrelevant, in particular, the negativity of on is actually not needed. This is the reason why we do not need to assume any subcritical growth condition on the exponent in the model nonlinearities . Let us recall that, in the autonomous setting, Theorem 4 in [EW14] yields nontrivial radially symmetric solutions of (1.1) for superlinear nonlinearities, so that their results hold for the nonlinearity , but not for .
Theorem 1.2 admits generalizations to some nonautonomous radially symmetric nonlinearities. In particular we can prove a nonautonomous version of this result that applies to the nonlinearities
The existence of non-radially symmetric solutions is clearly a more difficult topic and here we can give a partial positive answer in this direction, by exploiting the argument developed in [EW14, EW15, Eve15, EW], where the authors study the equation
where is a super-linear nonlinearity satisfying suitable hypotheses, that include, for example, a non-autonomous generalization of our model nonlinearity , i.e.
Our contribution to this issue is that the positivity assumption on may be replaced by a negativity assumption in order to make the dual variational approach work, so that, using Fourier transform we show that the main ideas from [EW15] may be modified in such a way that their main results remain true for negative . Our results read as follows.
Let and let be periodic and negative almost everywhere. Then the equation (1.11) has a nontrivial localized oscillating strong solution in for all .
Let and let be negative almost everywhere with as . Then the equation (1.11) has a sequence of pairs of nontrivial localized oscillating strong solutions in for all such that
Since the above results together with those from [EW15] provide some existence results for the Nonlinear Helmholtz equation associated with the nonlinearity from (1.10), one is lead to wonder whether similar results hold true for asymptotically linear nonlinearities like in (1.9). Here, the dual variational framework does not seem to be convenient since even the choice of the appropriate function spaces is not clear. A thorough discussion of such nonlinear Helmholtz equations leading to existence results for nonradial solutions still remains to be done.
Let us observe that there is a gap in the admissible range of exponent between Theorem 1.2 and Theorem 1.3. Reading Theorem 1.2 one is naturally lead to the conjecture that nontrivial nonradial solutions in may be found regardless of any sign condition on and for all exponents . On the contrary, Theorem 1.3 only holds for exponents , so that it is still an open question whether or not nonradial -solutions exist for .
The paper is organized as follows: In section 2 we present the proof of Theorem 1.2 as well as a generalization to the radial nonautonomous case (see Theorem 2.10 and Corollaries 2.11, 2.13). In section 3 we present the proofs of Theorem 1.3 and Theorem 1.4. In section 4 we will discuss in detail the attempt to obtain a solution by approximating by bounded domains.
2. Radial solutions
2.1. The autonomous case
for and . Notice that our assumptions on require that there exists a such that
Such a positivity region is in fact almost necessary as the following result shows.
Assume that satisfies for all . Then there is no nontrivial localized solution and there is no nontrivial oscillating solution of (1.1).
Assume that is a nontrivial localized or oscillating solution. Then it attains a positive local maximum or a negative local minimum in some point . Hence we obtain
In view of elliptic regularity theory the above result is also true for weak solutions since these solutions coincide almost everywhere with classical solutions and decay to zero at infinity by Theorem C.3 in [Sim82]. Notice that in case we can deduce the non-existence of solutions from the fact that in violates the necessary condition (1.3) in [BL83], see section 2.2 in that paper. In the case the same follows from Remarque 1 in [BGK83].
First we briefly address the one-dimensional initial value problem
In view of the oddness of it suffices to discuss the inital value problem for . The uniquely determined solution of the initial value problem will be denoted by with maximal existence interval for .
Let . Then the following holds:
If then and if then .
If then strictly increases to on .
If then is periodic and oscillating with .
Conclusion (i) immediately follows from (1.5). Then we only have to prove (ii) and (iii). For notational convenience we write instead of . In the situation of (ii) we set . From and (2.2) we get and thus . We even have , because otherwise
In order to show (iii) we notice that (1.3) implies that solutions are symmetric about critical points and antisymmetric about zeros. Therefore, it suffices to show that decreases until it attains a zero. By the choice of we have so that decreases on a right neighbourhood of . Exploiting (1.5) and (2.2) we deduce that is negative whenever . As a consequence, we obtain that decreases as long as it remains positive. Moreover, it cannot be positive on since this would imply, thanks and the assumptions (1.4),(1.5),
Hence, Sturm’s comparison theorem (p.2 in [Swa68]) ensures that vanishes somewhere, so that it cannot be positive in , a contradiction. Hence, attains a zero and the proof is finished.
Next, we consider the initial value problem (2.1) in the higher dimensional case . Again, we may restrict our attention to the case and we will denote with the primitive of the function , such that . The following result furnishes the study of the solution set which are needed in the proof of Again, the uniquely determined solution of the initial value problem (2.1) will be denoted by with maximal existence interval .
There are many contributions concerning (1.1) in dimension , mainly related to some resonance phenomena. In this context, some “Landesman-Lazer” type conditions, joint with suitable hypotheses on the nonlinearity , are assumed in oder to obtain existence of bounded, periodic or oscillating solution, eventually with arbitrarily large norm, by taking advantage of the presence on a forcing term in the equation (see [SV14, Ver03] and the references therein). Here the situation is different, as we do not need any monotonicity assumption on , nor the knowledge of the asymptotic behavior at infinity of is important, as it is in [SV14, Ver03]. Moreover, our solutions satisfy a uniform bound, so that the phenomenon we are dealing with is actually different from the resonant one.
Let . Then the following holds:
If then and if then .
If then strictly increases to on .
If then is oscillating, localized and satisfies
as well as
for some depending on the solution but not on .
The existence and uniqueness of a twice continuously differentiable solution can be deduced from Theorem 1 and Theorem 2 in [RW97]. We write again in place of . The proof of (i) is direct and assertion (ii) follows similar to the one-dimensional case. Indeed, note that because of
Then, letting , it results . Assuming by contradiction that and using that , from (1.5) we obtain
Step 1: decreases to a first zero. For all such that on we have
showing that decreases as long as it remains positive, as in the one-dimensional case. Moreover, the function can not remain positive on because otherwise would be a positive solution of
As in the proof of Proposition 2.3 we observe for sufficiently large so that Sturm’s comparison theorem tells us that vanishes somewhere. This is a contradiction to the positivity of and thus attains a first zero.
Step 2: oscillates and satisfies (2.3). Let us first show that there are such that all are local maximizers, all are local minimizers and all are zeros of . Moreover, we will find that all zeros or critical points of are elements of this sequence and
In order to prove this we consider the function
and we observe that decreases as
The existence of a first zero of has been shown in Step 1 and the strict monotonicity of implies . Concerning the behaviour of on there are now three alternatives:
decreases until it attains
decreases on to some value
decreases until it attains a critical point at some with .
Let us show that the cases (a) and (b) do not occur. Indeed, if there exists such that , then, by (2.7) we deduce that
which again contradicts (2.8). So the case (c) occurs and there must be a critical point with
Hence, is a local minimizer. Using that is decreasing we can now repeat the argument to get a zero , a local maximizer , a zero and so on. By the strict monotonicity of one obtains (2.6) and thus (2.3). Notice that this reasoning also shows that there are no further zeros or critical points.
Step 3: is localized. First we show as . Our proof is similar to the one of Lemma 4.1 in [GZ08] and it will be presented for the convenience of the reader. Take the sequence of maximizers and assume by contradiction that . Then (2.1) and Ascoli-Arzelà Theorem imply that converges locally uniformly to the unique solution of (2.2) with . Proposition 2.3, (iii) implies that this solution is -periodic with two zeroes at . As a consequence, there exists such that on . Hence, for sufficiently large we have for
From this we deduce for
Let us fix such that for . Then (2.10) implies
Choosing now sufficiently large we obtain that
because the harmonic series diverges, but (2.3)implies that , yielding a contradiciton.
As a consequence, converges to zero as and
analogously we deduce that also . In the end, we
obtain as .
Since is decreasing and nonnegative it follows that as . Hence, by (2.7), also has a limit at infinity which must be zero because converges to 0. Finally, from the differential equation we deduce that as , i.e.
As in Lemma 4.2 in [GZ08] we get that for any there exists such that
This yields for and some positive integrable function . Dividing this inequality by the positive function and integrating the resulting inequality over shows that is bounded from below and from above by a positive number. From this we obtain the lower and upper bounds (2.4) and the proof is finished.
We are now ready to give the proof of Theorem 1.2.
Proof of Theorem 1.2 Let us define the set
where denotes the unique solution of the initial value problem (2.1). The set is a subset of , and it is a continuum thanks to the Ascoli-Arzelà Theorem. From Lemma 2.5 we obtain that all elements of are oscillating localized solutions satisfying (2.3) and (2.4).
Let us mention that an analogous result to Theorem 1.2 in Theorem 1 [GLZ09] and it is applied to a more restrictive class of nonlinearities. Moreover, the above theorem is related to Theorem 4 in [EW14] but we do not need their assumption . Actually, this hypothesis is not satisfied in our model cases or .
The arguments from the proof of Theorem 1.2 also show the existence of oscillating localized solutions to initial value problems which are not of nonlinear Helmholtz type. For instance, one can treat concave-convex problems such as
for with , see for instance [ABC94] or [BEP95] for corresponding results on a bounded domain with homogeneous Dirichlet boundary conditions. The existence of solutions is provided by Theorem 1 in [RW97] so that the steps 1,2,3 are proven in the same way as above and we obtain infinitely many radially symmetric, oscillating, localized, solutions of (2.15).
Using nonlinear oscillation theorems instead of Sturm’s comparison theorem we can even extend the above observation towards superlinear nonlinearities satisfying for
Indeed, in the first case the function from (2.5) satisfies the estimate so that Atkinson’s oscillation criterion applies, see the first line and third column of the table on p.153 in [Swa79]. In the second case Noussair’s oscillation criterion result can be used in order finish step 1, see the third line and third column of the table on p.153 in [Swa79].
If is decreasing, then one can show that the first zero of is smaller than the first zero of whenever . Indeed, we set . Then the interval
is open, connected and nonempty and thus for some . On its right boundary we either have or ; so it remains to exclude the first possibility. Using on and (2.1) we have
Integrating (2.16) from to the assumption leads to
On the other hand on and implies , a contradiction. Thus so that the first zero of comes before the first zero of .
2.2. The nonautonomous case
In this section we generalize Theorem 1.2 to a nonautonomous setting. Our aim is to identify mild assumptions on a nonautonomous nonlinearity that ensure the existence of a continuum of oscillating localized solutions of the initial value problems
that behave like at infinity in the sense of (2.4). Before formulating such assumptions and stating the corresponding existence result let us mention that our result applies to the nonlinearities (1.9),(1.10) under suitable conditions on the coefficient functions. This will be seen in Corollary 2.11 and Corollary 2.13 at the end of this section. Our existence results for (2.17) will be proven assuming that
|(2.18)||is continuously differentiable w.r.t. .|
Moreover, we suppose that there exist positive numbers and a locally Lipschitz continuous function such that
These assumptions will allow us to prove the mere existence of an oscillating localized solution. In order to show the desired asymptotic behaviour we need some extra condition ”at infinity” where is large and the solution itself is small: We will assume that there exist and some integrable function such that
These assumptions are rather technical but can be verified easily in concrete situations as we show in the proof of Corollary 2.11. Let us remark that our assumptions (1.2),(1.3),(1.4)(1.5) from the autonomous case (for any choice ) are more restrictive than the assumptions used above. In particular, the following theorem generalizes our autonomous result.
The proof of our result follows the same argument of the proof of Theorem 1.2, so we only mention the main differences. For simplicity we only treat the case with (2.24). The existence of a maximally extended solution of (2.17) follows from a Peano type existence theorem for singular initial value problems, see Theorem 1 in [RW97].
Step 1 is proven as in the autonomous case where the function from (2.5) has to be replaced by . Assumption (2.21) ensures that is bounded from below by a positive constant as long as so that has to attain a first zero. In step 2 one shows that is nondecreasing due to for , see (2.20). Arguing as in the autonomous case we find that decreases until it attains a local minimum at some with . More precisely one finds a sequence such that all are critical points and all are zeros of with the additional property (the co