Oscillating solutions for prescribed mean curvature equations: Euclidean and Lorentz-Minkowski cases

This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.


INTRODUCTION
Starting from the milestones papers [4,5,27], the literature is plentiful of results concerning the following class of nonlinear equations under the assumption g ′ (0) < 0. In particular, for a large class of nonlinearities, the existence of ground state solutions, namely radially symmetric positive solutions decaying at infinity, has been proved. Non-existence results of ground state solutions are present in [23]. The so called zero mass case, that is when g ′ (0) = 0, instead, has been object of study for example in [4,24]. The arguments in both the situations rely on variational techniques or on an ODE approach, such as the shooting method. Of course, we cannot mention all the contributes on this topic here. Later on, different authors studied the existence and the non-existence of ground state solutions for a larger class of equations, replacing the Laplacian with different differential operators, under the assumption g ′ (0) 0. In particular, some results for the prescribed mean curvature equation in the Euclidean case (1.2) − div ∇u and for some generalizations can be found, among others, in [3,10,12,18,19,21,25,26]. More recently, a lot of attention has been paid on the prescribed mean curvature equation in the Lorentz-Minkowski case Existence and non-existence results of ground state solutions and of sign changing solutions, for the cases g ′ (0) < 0 and g ′ (0) = 0 are contained in [1,2,6]. We remark that this kind of differential operator appears naturally also in the contest of Born-Infeld electro-magnetic theory, see [7,9,8,17]. The case g ′ (0) > 0, conversely, is completely different, indeed, as well explained in [4], a direct consequence of this hypothesis is that radially symmetric H 1 (R N ) solutions of (1.1) can not exist, and usual variational methods fail. Nevertheless this case is very important since it is related to the study of the propagation of lights beams in a photorefractive crystals when a saturation effect is taken into account (see [22] for a more precise description about these phenomena). Under this condition we can find, for example, the well known nonlinear Helmholtz equations. Contrary to the other cases, this one has not been studied intensively: some results for (1.1), under several types of assumptions, can be found in [13,14,15,16]. We mention in particular the recent paper [22], where the authors prove the existence of oscillating solutions (which are actually periodic for N = 1) for (1.1) with N 1 and assuming that the nonlinearity g is odd, with g ′ (0) > 0 and such that there exists α ∈ (0, +∞] such that g is positive on (0, α) and negative on (α, +∞).
Up to our knowledge, very little is known for problems (1.2) and (1.3) under these kind of assumptions on g and, in particular, there is no existence result of oscillating solutions. Aim of this paper, therefore, is to extend the results of [22] to the prescribed mean curvature equations (1.2) and (1.3), namely both in the Euclidean case and in the Lorentz-Minkowski case, for N 1.
In the following, we will refer to a solution of each equation in (1.2) and (1.3) as a classical solution. More precisely, in the Euclidean case, u is a solution of (1.2) if u ∈ C 2 (R N ) and satisfies the equation pointwise in R N ; in the Lorentz-Minkowski case, instead, u is a solution of (1.3) if u ∈ C 2 (R N ), |∇u(x)| < 1 for all x ∈ R N , and u satisfies the equation pointwise in R N . Moreover, we need the following Definition 1.1. A solution u of (1.2) or of (1.3) is called oscillating if it has an unbounded sequence of zeros. It is called localized when it converges to zero at infinity together with its partial derivatives up to order 2.
Moreover, in the following we will denote by G(t) = t 0 g(s) ds. For what concerns the Euclidean case, our main result is the following Theorem 1.2. Assume (g1)-(g3) and, if N 2, also (g4). Then, for any |ξ| < α such that G(ξ) < 1, there exists an oscillating solution In the Lorentz-Minkowski case, instead, our main result is the following Theorem 1.3. Assume (g1)-(g3) and, if N 2, also (g4). Then, for any |ξ| < α, there exists an If one deals with these two cases using variational techniques, the approaches and the functional settings are quite different. In the Lorentz-Minkowski case, for example, a variational approach to the problem can not be performed in the usual functional spaces. In particular, the quantity 1/ 1 − |∇u(x)| 2 makes sense when x ∈ R N is such that |∇u(x)| < 1, being this inequality a necessary constraint to be considered in the functional setting (see [7]). However, since we are interested in radially symmetric solutions, we will see that our approach, based on ODE techniques, works well in both cases with only some suitable modifications. Our arguments are inspired by [5,20,22] but are more involved and require some additional effort. For instance, if u is a solution of (1.2), we don not know, in general, if its gradient is uniformly bounded, while in the Lorentz-Minkowski case, this uniform bound is obtained for free, since, for any x ∈ R N , we have to require that |∇u(x)| < 1, but we have to be sure that |∇u| remains uniformly far away from 1. For this reason the assumptions of the two theorems are similar but not equal. In the Euclidean case, indeed, it is well known, at least for N = 1, that problem (1.2) could have no classic solutions, but only bounded variation solutions, (see [11] and the references therein). Therefore we have to add an additional assumption in order to avoid this particular situation.
The paper is organized as follows. In Section 2 we deal with the Euclidean case and Theorem 1.2 will be an immediate consequence of Theorems 2.1 and 2.2. In Section 3, instead, we treat the Lorentz-Minkowski case and Theorem 1.3 will follow easily from Theorems 3.1 and 3.2. Since the arguments are similar in both cases, in the last section we will skip some details underlying only the necessary differences. We think that the theorems present in Sections 2 and 3 themselves could be of interest.

THE EUCLIDEAN CASE
This section will be devoted to the Euclidean case. Let us start with the one-dimensional case, where we can simply consider the following Cauchy problem The following result is partially already known, see for example [11] and the references therein, but we present it for sake of completeness and because it is a crucial step for the study of the multi-dimensional case.
Proof. By standard arguments, (see for example [11, Section 1.2]), there exists a local solution In the following we simply write u, R instead of u ξ , R ξ , respectively, for brevity. Moreover, being g an odd function, by (g2), we can reduce ourselves to consider only the case ξ 0.
By the assumptions on g, (i) follows immediately.
Let us prove (ii), supposing that α ∈ R. We first prove that u is strictly increasing in [0, R).
Let us prove (iii) and (iv). We start observing that multiplying equation of (2.1) by u ′ and integrating over [0, r], we obtain the following equality for any r ∈ (0, R) 3) is even with respect to u, by (g2), and with respect to u ′ , it is standard to prove that u is symmetric about critical points and antisymmetric about zeros. Therefore, it suffices to show that u decreases until it attains a zero in order to prove that u is periodic and u L ∞ = ξ. By (2.1) and (g3), for all r > 0 such that 0 < u < α on [0, r] we have and so u is strictly decreasing as long as it remains positive. Suppose that u(r) > 0, for all r ∈ [0, R), then by (2.2) and since g(u) > 0, we have that u ′′ (r) < 0 for all r ∈ [0, R). Being u strictly positive, decreasing and concave, the possibilities are two: either |u ′ | blows up at R − or |u ′ | is uniformly bounded in [0, R). In the former case, which happens, as observed in [11, Section 1.2], whenever G(ξ) 1, we deduce that R ∈ R. In the latter case, which occurs, at contrary, if and only if G(ξ) < 1, we have that R = +∞ reaching a contradiction: u vanishes at some r > 0 and, therefore, u is periodic and oscillating. In both cases, we see that the L ∞ -norm of u is ξ, as desired.
We pass now to consider the case N 2. Since we look for radial solutions, we can reduce equation (1.2) to the following Cauchy problem We have the following 1, then R ξ = +∞ and u ξ is oscillating and localized with u ξ L ∞ (R + ) = |ξ|.
Proof. Also in this case, by [24], there exists a local solution u ξ of the Cauchy problem (2.4). Now let R ξ > 0 be such that [0, R ξ ) is the maximal interval where the function u ξ is defined. We have u ξ ∈ C 2 ([0, R ξ )). In the following we simply write u, R instead of u ξ , R ξ , for brevity. Moreover, being g an odd function, by (g2), we can reduce ourselves to consider only the case ξ 0.
By the assumptions on g, (i) follows immediately.
Let us prove (ii), in the case of α ∈ R. We first prove that u is strictly increasing in [0, R). Since observing that u satisfies in (0, R) We have thatr = R, indeed, otherwise, ifr < R, multiplying the equation in (2.4) by r N −1 , integrating over (0,r), and by (g3), we would havē reaching a contradiction. Being u strictly increasing in [0, R), there exists L = lim r→R − u(r). If R ∈ R, then, by the maximality of R, we conclude that either L = +∞ or lim r→R − u ′ (r) = +∞ and we reach the conclusion. Let us consider, therefore, the case R = +∞. Then, since 0 u ′ (r)/ 1 + (u ′ (r)) 2 1, for all r 0, we have Hence, by (2.5) and by (g3), we infer that u is strictly asymptotically convex and so L = +∞, as claimed.
Let us prove (iii). Since the proof is quite long, we divide it into intermediate steps.
Step 2: u is oscillating and u L ∞ (R + ) = ξ. Let us first show that there exist 0 = r 0 < r 1 < r 2 < r 3 < · · · such that all r 4j are local maximizers, all r 4j+2 are local minimizers and all r 2j+1 are zeros of u. The existence of a first zero r 1 > 0 = r 0 of u has been shown in Step 1 and the strict monotonicity of Z + in [r 0 , r 1 ] implies Z + (r 1 ) < Z + (r 0 ). Concerning the behaviour of u on [r 1 , +∞), there are now three possibilities: (a) u decreases until it attains −ξ at somer > r 1 ; (b) u decreases on [r 1 , +∞) to some value u ∞ ∈ [−ξ, 0); (c) u decreases until it attains a critical point at some r 2 > r 1 with −ξ < u(r 2 ) < 0. First of all, let us observe that, arguing as before, by (2.6), u ′ remains bounded in all these three possibilities. Let us show that the cases (a) and (b) do not occur. Indeed, if there existedr > 0 such that u(r) = −ξ, then we would deduce that Z + (r) G(u(r)) = G(ξ) = G(u(0)) = Z + (0) which is in contradiction with the strictly decreasing monotonicity of Z + in any interval whose extreme points are consecutive stationary points of u. Hence, the case (a) is impossible. Let us now suppose that (b) holds. Arguing as in Step 1, we can prove that lim r→+∞ u ′ (r) = 0 and so, by (2.5), we have lim r→+∞ u ′′ (r) = −g(u ∞ ). This implies that g(u ∞ ) must be equal to 0. Being u ∞ ∈ [−ξ, 0), we deduce that u ∞ = −ξ. Hence reaching again a contradiction. Therefore, we can say that u decreases until it attains a critical point at some r 2 > r 1 with −ξ < u(r 2 ) < 0. Moreover, by (2.5), (g2) and (g3), we have u ′′ (r 2 ) = −g(u(r 2 )) > 0.
Hence, r 2 is a local minimizer. By (g2) and (g3), we can now repeat the argument to get a zero r 3 > r 2 , a local maximizer r 4 > r 3 , a zero r 5 > r 4 and so on, such that ξ = u(r 0 ) > −u(r 2 ) > u(r 4 ) > · · · . Notice that this reasoning also shows that there are no further zeros or critical points. Moreover we conclude, also, that u L ∞ (R + ) = ξ.
So, for all j j 0 , (u ′ (r)) 2 Finally, by (2.8), for k j 0 and r r 4j + τ + δ, we have Now the arguments proceed as in [20,22], but we give some details for the sake of completeness. Let us fix c(δ ′ ) such that log(1 + x) c(δ ′ )x for all 0 x 2δ/(r 4j + τ − δ). Then by (2.10), we deduce that Since the harmonic series diverges, choosing k and r sufficiently large, we obtain that Z + (r) → −∞ reaching a contradiction with the non-negativity of Z + . As a consequence u(r 4j ) → 0 and analogously we deduce that also u(r 4j+2 ) → 0. This implies that u(r) → 0, as → +∞.
Being u L ∞ (R + ) = ξ, with ξ ∈ (0, α), by (g3), Z + is non-negative; moreover, since Z + is decreasing, it follows that it admits a finite and non-negative limit at infinity. Hence, by (2.7), also |u ′ | has a limit at infinity which must be zero because u converges to 0. Finally, from (2.5) we deduce that also u ′′ converges to zero at infinity.

Remark 2.3.
It is notable to observe that, if 0 < |ξ| < α and G(ξ) = 1, we obtain a different behavior for N = 1 and for N 2. Moreover, it would be interesting to understand what happens in the case if 0 < |ξ| < α and G(ξ) > 1 for the multi-dimensional case.

THE LORENTZ-MINKOWSKI CASE
This section will be devoted to the Lorentz-Minkowski case. Let us start with the one-dimensional case, where we can simply consider the following Cauchy problem , in (0, +∞), As in the Euclidean case, the following result is partially already known, see for example [11] and the references therein, but we present it for sake of completeness and because it is a crucial step for the study of the multi-dimensional case. Theorem 3.1. Assume (g1)-(g3). For any ξ ∈ R there exists a solution u ξ ∈ C 2 ([0, R ξ )) of the Cauchy problem (3.1), where R ξ ∈ (0, +∞] is such that [0, R ξ ) is the maximal interval where the function u ξ is defined. Moreover, we have (i) if |ξ| = α ∈ R or ξ = 0, then u ξ ≡ ξ; (ii) if α ∈ R and |ξ| > α, then |u ξ | strictly increases to +∞ on [0, R ξ ); (iii) if 0 < |ξ| < α, then R ξ = +∞ and u ξ is oscillating and periodic with u ξ L ∞ (R + ) = |ξ|.
Proof. By standard arguments (see for example [11,Section 3.2]), there exists a local solution u ξ of the (3.1). Now let R ξ > 0 be such that [0, R ξ ) is the maximal interval where the function u ξ is defined. We have u ξ ∈ C 2 ([0, R ξ )). In the following we simply write u, R instead of u ξ , R ξ , respectively, for brevity. Moreover, being g an odd function, by (g2), we can reduce ourselves to consider only the case ξ 0. We start observing that multiplying equation of (3.1) by u ′ and integrating over (0, r), we obtain the following equality for any r ∈ (0, R) By (3.2), we infer that G(ξ) − G(u(r)) 0, for any r ∈ [0, R), and that H − (u ′ (r)) is bounded if the right hand side is bounded: in particular, we have that By the assumptions on g, (i) follows immediately.
Let us prove (ii), assuming that α ∈ R. We first prove that u is strictly increasing in [0, R).
Let us prove (iii). Since (3.2) is even with respect to u, by (g2), and to u ′ , then u is symmetric about critical points and antisymmetric about zeros and so it is periodic. Therefore, it suffices to show that u decreases until it attains a zero. By (3.1) and (g3), for all r > 0 such that 0 < u < α on [0, r] we have and so u is decreasing as long as it remains positive. Suppose by contradiction that u(r) > 0, for all r > 0, then, by (3.4) and (g3), we have that u ′′ (r) < 0 for all r > 0. Being u strictly positive, decreasing and concave, we reach immediately a contradiction. Hence, u attains a zero and the proof is finished.
Proof. Also in this case, by [24], there exists a local solution u ξ of the Cauchy problem (3.5). Now let R ξ > 0 be such that [0, R ξ ) is the maximal interval where the function u ξ is defined. We have u ξ ∈ C 2 ([0, R ξ )). In the following we simply write u, R instead of u ξ , R ξ , for brevity. Moreover, being g an odd function, by (g2), we can reduce ourselves to consider only the case ξ 0.
By the assumptions on g, (i) follows immediately.
If R ∈ R, then, by the maximality of R, we conclude that L = +∞. Let us consider the case R = +∞ and assume by contradiction that L ∈ R. Then, being u bounded, by (3.7) there exists δ > 0 such that δ 1 − (u ′ (r)) 2 1, for any r 0, and so Hence by (3.8) and by (g3), we infer that u is strictly convex definitively and so L = +∞ reaching a contradiction.
Let us prove (iii). As in the previous section, we divide the proof into intermediate steps.
Step 1: u decreases to a first zero. By (3.5) and (g3), for all r > 0 such that 0 < u < α on [0, r] we have and so u is strictly decreasing as long as it remains positive. Suppose by contradiction that u(r) > 0, for all r > 0, then, being u bounded, since by (3.7) there exists δ > 0 such that δ 1 − (u ′ (r)) 2 1, for any r 0, by (3.8), we have where we have used the fact that u ′ < 0 and g(u) > 0. Therefore, if we set v = r N−1 2 u, we get the following By (g4), we infer that there exists c 0 > 0 such that σ − 2 (r) c 0 , definitively, and so v ′′ is definitively negative. Therefore, arguing as in the Euclidean case, we reach a contradiction.
Step 2: u oscillates and u L ∞ (R + ) = ξ. and we observe that Z − decreases as Moreover Z ′ − (r) = 0 if and only if u ′ (r) = 0. Arguing as in the Euclidean case, we show that there are 0 = r 0 < r 1 < r 2 < r 3 < · · · such that all r 4j are local maximizers, all r 4j+2 are local minimizers and all r 2j+1 are zeros of u and ξ = u(r 0 ) > −u(r 2 ) > u(r 4 ) > · · · . Moreover there are no further zeros or critical points and u L ∞ (R + ) = ξ.
From now on, we can adapt easily the arguments of the Euclidean case to conclude.
Acknowledgment. The author is partially supported by a grant of the group GNAMPA of INdAM.
The author wishes to express his more sincere gratitude to the anonymous referee: his/her acute comments and suggestions have been crucial to improve strongly the quality and the clarity of paper and to find and to fill a gap present in the previous version of this manuscript.