ON THE EXISTENCE OF BUBBLE-TYPE SOLUTIONS OF NONLINEAR

Considered in this paper is a class of singular boundary value problem, arising in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are sought: (p(t)u) = c(t)p(t)f(u), u(0) = 0, u(+∞) = L > 0 in the half-line [0, +∞), where p(0) = 0. We are interested in strictly increasing solutions of this problem in [0,∞) having just one zero in (0, +∞) and finite limit at zero, which has great importance in applications or pure and applied mathematics. Sufficient conditions of the existence of such solutions are obtained by applying the critical point theory and by using shooting argument [1, 2] to better analysis the properties of certain solutions associated with the singular differential equation. To the authors’ knowledge, for the first time, the above problem is dealt with when f satisfies non-Lipschitz condition. Recent results in the literature are generalized and significantly improved.


Introduction
The singular problem which we investigate in this paper appears when Cahn−Hillard theory has been developed to study the behavior of nonhomogeneous fluid (fluid− fluid, fluid−vapor, fluid−gas, etc., see, e.g., [3,5] and references therein). If ρ is the density of the medium, µ(ρ) the chemical potential of a nonhomogeneous fluid and the motion of the fluid is absent, the state of the fluid in R N is described by the equation where γ and µ 0 are suitable constants. This equation can describe the formation of microscopical bubbles in a nonhomogeneous fluid, in particular, vapor inside one liquid. With this purpose, we add to Eq. (1) the boundary conditions for the bubbles. Follows from the central symmetry, it is necessary for the smoothness of solutions of (1) at the origin: ρ ′ (0) = 0. (2) Since the bubble is surrounded by an external liquid with density ρ l , the following condition holds at infinity: lim r→∞ ρ(r) = ρ l > 0.
From (3) it follows that µ 0 = µ(ρ l ). Whenever a strictly increasing solution to problem (1)−(3) exists, for some ρ(0) = ρ v , with 0 < ρ v < ρ l , then ρ v is the density of the gas at the center of the bubble and the solution ρ determines an increasing mass density profile [12]. In the case of plane or spherical bubbles Eq. (1) takes the form where N = 2 or N = 3, respectively, and is known as the density profile equation [3,6].
More general situation is that the constant coefficient 4λ 2 depends on the variable r, r N −1 and the nonlinear term are generalized by p(r) and f (ρ), respectively. Noting that the nonlinear boundary value problem (5), (6) has at least the solution ρ(r) ≡ ξ > 0. We are interested in solutions which are strictly increasing and have just exactly one zero in (0, ∞). If such solutions exist, many important physical properties of the bubbles depend on them (in particular, the gas density inside the bubble, the bubble radius and the surface tension). It is also interesting to remark that boundary value problems of same kind arise in nonlinear field theory [4]. Therefore, it becomes an interesting and challenging problem to study the more general system which we will discuss in this paper. We investigate in this paper a generalization of problem (5), (6), which refer to as the second order singular boundary value problem (BVP for short) in the half-line: u(0) = 0, u(+∞) = L > 0, (8) where p, c and f are given continuous functions satisfying some assumptions and p(0) = 0. We consider the existence of a strictly increasing solution of problem (7), (8) having just one zero in (0, ∞) and belonging to C 1 ([0, ∞)) C 2 ((0, ∞)). When c(t) ≡ 1, the singular BVP (7), (8) has been investigated in [15,16] and [17] by means of differential and integral inequalities, upper and lower functions approach, respectively. We mention here that if c(t) ≡ 1, some arguments in [15]- [17] are unavailable. Problem (7), (8) can be transformed into a problem about the existence of a strictly decreasing and positive solution in the positive half-line which is of significant importance in many disciplines of science such as engineering or pure and applied mathematics. For p(t) = t k , k ∈ N or k ∈ (1, ∞), such a problem was solved by shooting argument combined with variational methods in [1] and [2], respectively. It is worth pointing out that in this paper, if p(t) reduces to t k , we can extend k ∈ (0, ∞). As for BVP (5) and (6), analytical−numerical investigation and numerical simulation of the problem can be found in [12] and [3,8], respectively. We emphasize, if BVP (7), (8) reduces to BVP (5), (6), some sufficient conditions obtained in this paper are also necessary. It should be mentioned here that the critical point theory is a powerful tool to deal with the boundary value problems of differential equations on the bounded and unbounded domain, see for instance [11,13,14]. In particular, the existence of homoclinic solutions of differential equations has been extensive and intensive studied, see [7,9,18,19] and the references listed therein for information on this subject. Note that the strictly increasing solutions of BVP (7), (8) having just one zero in (0, ∞) and finite limit at zero can also be called homoclinic solutions [15]- [17].
When f satisfies non-Lipschitz condition, as far as authors know, there is no research about the existence of monotone solutions of BVP (7), (8) or some other similar problems. In the present paper, we are interested in the case that f satisfies non-Lipschitz condition, and motivated mainly by the papers [2,15,16], we consider the more general problem (7), (8) by applying the critical point theory and by using shooting argument [1,2] to better analysis the properties of certain solutions associated with the singular differential equation. Recent results in the literature are generalized and significantly improved. Now we present the basic assumptions in order to obtain the main results in this paper: In addition, we need the following hypothesis on the function p to announce the first result in this paper: where c 2 and F are given by (H 1 ) and (9), respectively.
Remark 1.1. We will show in Appendix that under the condition (H 4 ), the explicit condition is a sufficient condition for (H 6 ). There are many functions satisfy (H 4 ) − (H 6 ), for example, Up until now, we can state our first main result. (7), (8) possesses at least one strictly increasing solution u with just one zero and u(0) ∈ [L 0 , 0).

Remark 1.2.
In the particular case that c(t) ≡ 1, assume that f ∈ Lip([L 0 , L]), (H 3 ), (H 4 ) and (11) hold. In [16], Rachůnková et al. obtained the existence of escape solutions of BVP (7), (8) which can be used to find the strictly increasing solution having just one zero in (0, ∞) (also called homoclinic solution) for BVP (7), (8). Note that, the homoclinic solutions were obtained under similar conditions in [15] by means of differential and integral inequalities and also obtained under stronger conditions in [17] by using upper and lower functions approach. However, in Theorem 1.1, we do not require f satisfies Lipschitz condition. Moreover, as we will show in Appendix, if function p satisfies (11), then (H 6 ) holds, but the reverse is not true. In fact, if p(t) = e t − 1 for e L−L0 < 1/F 0 + 1, then it is easy to check that (H 4 ) − (H 6 ) hold but (11) does not. Therefore, we generalize and improve the results in [15]- [17] in some sense.
In addition, if we substitute c 2 /c 1 < 1 + F (L)/F (L 0 ) by f ′ (0) < 0, then we have the following theorem.   (7), (8) can be transformed into problems concerning the existence of a strictly decreasing and positive solution which have been considered in [1] and [2] with p(t) = t k for k ∈ N and k ∈ (1, ∞), respectively. However, in this paper, if p(t) reduces to t k , we do not need any requirement on k expect that k ∈ (0, ∞) by using an original decomposition technique to better estimate functions in a new function space which we construct in Section 2. In this point of view, we improve and generalize the results in [1,2]. Remark 1.4. When BVP (7), (8) reduces to the problem (5), (6), after a simple calculation, we get that the conditions of Theorem 1.1 and 1.2 are satisfied if and only if 0 < ξ < 1. On the other hand, according to [12](Proposition 4), 0 < ξ < 1 is also a necessary condition for the existence of at least a strictly increasing solution having exactly one zero in (0, ∞) for problem (5), (6). Now we give the main idea of this paper. Similar to [15,16], consider an auxiliary equation wheref It is obvious that if BVP (12) and (8) has a strictly increasing solution u having just one zero with u(0) ∈ [L 0 , 0), then u is also a solution of BVP (7), (8) with required properties. Therefore, we only need to consider the problem (12) and (8) in the rest of the paper. Firstly, we consider the case that f satisfies Lipschitz condition. Motivated mainly by the papers [15,16], we discuss the initial value problem (IVP for short) (12) with initial value condition by means of contraction mapping theorem, differential and integral inequalities. On the other hand, motivated by [2], by using variational method and estimating the values of the variational functional (24) at critical points, we carry out a study of the existence and properties of solutions to BVP (12) with boundary value condition It is suffices then to invoke the shooting argument of [1] to obtain the existence results for BVP (12), (8) and thus for BVP (7), (8) in the case that f satisfies Lipschitz condition. Let us describe the main idea of the proof. Set I = (L 0 , 0) and let I i (i = 1, 2) be the subset of I, consisting of all B such that the solution of IVP (12), (14) corresponding to B is type (i), i = 1, 2, see Proposition 4.1 for the precise definitions of type (i), i = 1, 2, 3. We then prove that I i (i = 1, 2) are disjoint, nonempty open sets, from which we can conclude that there exist elements B ∈ I which belong neither to I 1 nor to I 2 . We conclude the proof by showing that such an element B yields a solution of BVP (12) and (8) with required properties.
Finally, we study BVP (12), (8) under non-Lipschitz condition. Motivated by [10], we first construct a sequence of Lipschitz functionsf n which gives a nice approximation of continuous functionf in (−∞, +∞) as n → ∞. Then we consider the problem (12) and (8) withf replaced byf n . By using the results obtained in Section 4, we have a certain sequence of strictly increasing functions u n in (0, ∞) with just one zero and u n (0) ∈ (L 0 , 0). We prove that, as n → ∞ the limit of u n exists and is the solution of BVP (12), (8) with required properties.
Part of the difficulty in treating the non-Lipschitz case is caused by the fact that in order to use the results obtained in Section 4, we need that the properties off n are similar tof in (−∞, +∞). Moreover, the limit of the functions u n should also be a solution of BVP (12), (8) which satisfies required properties.
The remaining of this paper is organized as follows. In Section 2, we develop a Hilbert space and exhibit a variational functional for BVP (12), (15). Some inequalities and properties are proven which yield the basis for the subsequent use of critical point theory in what follows. In Section 3, in the case that f ∈ Lip([L 0 , L]), some basic properties of solutions of IVP (12), (14) and BVP (12), (15) are discussed, and also in Section 4, several existence criteria of strictly increasing solutions of BVP (12), (8) having just one zero with initial values belong to (L 0 , 0) are obtained under the Lipschitz condition. In Section 5, the case that f satisfies non-Lipschitz condition is discussed and the proofs of Theorem 1.1 and 1.2 are given.
Throughout of this paper, we denote by C some positive constant that may change from line to line.
2. Variational structure for BVP (12) and (15) We shall make use of a variational setting, where solutions of BVP (12), (15) are in correspondence with critical points of a functional. The main idea of this section comes from [2], in which the Sobolev space with weight t k , k > 1 has been considered. As we pointed out in the Introduction, if p(t) reduces to t k , we can extend k ∈ (0, ∞).
Given T ∈ (0, ∞). We introduce a function space H(0, T ) consisting of u absolutely continuous in [0, T ] such that is finite and u(T ) = 0. The right-hand side of (16) defines the square of a norm in this space.
It is easy to verify that H(0, T ) is a reflexive and separable Banach space. In fact, Let {u n } be a Cauchy sequence in H(0, T ). Then the completeness of , which means that u n → u in H(0, T ) as n → +∞. The reflexivity and separability of H(0, T ) can be verified by standard theories.
Here we define the inner product over H(0, T ) by and H(0, T ) is a Hilbert space respect to the inner product.
We can now give some useful estimates.
holds, where α is given in (H 5 ) and C > 0 is a constant depends on T .
Proof. For any u ∈ H(0, T ), noting that u(T ) = p(0) = 0 and 0 < α < 1, we have Therefore, by using Cauchy−Schwartz inequality we obtain According to (H 4 ) an (H 5 ), we know that there exists a constant C such that Combining this with (18), we obtain (17) and the proof is complete.
Proof. For any u ∈ H(0, T ) and t ∈ [0, T ], we compute by applying Cauchy− Schwartz inequality and (17) The proof is complete.
Proof. For any u ∈ H(0, T ) and t ∈ [0, T ], since (H 4 ) and (H 5 ), we may write as product of two functions. Notice that the second function of the right-hand side is bounded for t ∈ [0, T ], we then apply (17) to conclude. The proof is complete. These properties have as an immediate consequence the following proposition. Then Proof. Consider the function set with the norm · p defined by Then (C p (0, T ), · p ) is a Banach space by the similar arguments we used to discuss H(0, T ). According to (19), the injection of . By the Banach−Steinhaus theorem, {u k } is bounded in H(0, T ) and, hence, in C p (0, T ). Moreover, the sequence {pu k } is equi-uniformly continuous since, for 0 ≤ t 1 < t 2 ≤ T , by applying (17) and in view of (20), we have By the Ascoli−Arzela theorem, {pu k } is relatively compact in C([0, T ]), and thus going to a subsequence if necessary, we may assume that pu k → u * in C([0, T ]). Hence, u k → u * /p in C p (0, T ). By the uniqueness of the weak limit in C p (0, T ), every uniformly convergent subsequence of {u k } converges uniformly on [0, T ] to u in C p (0, T ), which means that pu k → pu in C([0, T ]) and this completes the proof.
We are now in a position to establish a variational structure which enables us to reduce the existence of solutions of BVP (12), (15) to the one of finding critical points of corresponding functional defined on the space H(0, T ).
First of all, Consider BVP , it is obvious that u is a solution of BVP (22) if and only if u + L is a solution of BVP (12), (15). Therefore, we seek a solution of BVP (12), (15) If (H 1 ), (H 4 ) and (H 5 ) are satisfied, then (i) the functional ϕ : is continuously differentiable on H(0, T ) and for any u, v ∈ H(0, T ), we have (ii) ϕ is weakly lower semicontinuous and coercive on H(0, T ); and is a solution of BVP (22).
Proof. The Proof of (i) follows from the standard line (see, e.g., [13](Theorem 1.4)), using Proposition 2.2−2.4, so we omit it. For the proof of (ii), ϕ is weakly lower semi-continuous functional on H(0, T ) as the sum of a convex continuous function [13](Theorem 1.2) and of a weakly continuous one [13](Proposition 1.2). In fact, according to On the other hand, it follows from Proposition 2.2 that Hence, ϕ is coercive. For the proof of (iii), we first note that a critical point u of ϕ satisfies u(T ) = 0 and < ϕ ′ (u), v >= 0 for any v ∈ H(0, T ), and of course for Integrating (26) between t 1 > 0 and t 2 > t 1 , using the boundness of functions c and f , we conclude that pu ′ satisfies the Cauchy condition at t = 0 and t = T , so that p(t)u ′ (t) has a finite limit as t → 0 + or t → T − . We shall show that p(t)u ′ (t) → 0 as t → 0 + . Multiplying (26) by u and integrating between 0 and T , we get Noting that p(t) is increasing in (0, ∞) and the boundness of c,f , we have where C > 0 is a constant and the assertion follows. The proof is complete.
Let us conclude Section 2 with some remarks. Consider the F andF which we defined by (9) and (23), respectively. Assumptions (H 2 ) and (H 3 ) yield the following results.
(iii) For each b > 0 and each ǫ > 0, there exists δ > 0 such that for any Here u i is the unique solution of IVP (12), (14) with B = B i , i = 1, 2.
Proof. Noting thatf is Lipschitz and bounded in (−∞, +∞) and (H 1 ) implies the boundness of c. The proof of (i) is similar to that of [16](Lemma 4) and the method is contraction mapping theorem. The proof of (ii) follows the arguments of step 2 and step 3 in [15](Lemma 3). The proof of (iii) is similar to that of [16](Lemma 7) and the technical tool is Gronwall inequality. The proof is complete.
We can prove as in the proof of (i) in Proposition 3.1 that IVP (12), (31) has a unique solution in [a, +∞). In particular, for C = 0, C ≥ L or C ≤ L 0 , the unique solution of IVP (12), (31) is u ≡ C.
The following result is similar to [16](Lemma 6), while the main idea of the proof borrowed from [2](Proposition 11) which is different from that of [16](Lemma 6).
If u satisfies (ii), then u has a unique zero θ > 0. Multiplying u ′ and integrating (32) over [0, θ], we get and thus we have On the other hand, integrating (32) over [θ, t], we obtain for t > θ Therefore, letting t → ∞, we get u ′ 2 (θ) ≥ 2c 1 F (L) by (35). This together with (34) implies c 1 F (L) ≤ c 2 F (B), which is a contradiction as B is sufficiently close to 0. The proof is complete.
We are now in a position to consider BVP (12), (15). As we pointed out in Section 2, we only need consider BVP (22) and according to Proposition 2.5, we know that in order to find a solution of BVP (22), it suffices to obtain the critical point of functional ϕ given by (24).
In what follows we shall make use of a function w : [0, ∞) → R defined by whereb is given by (H 6 ). According to Remark 2.2,F (L 0 − L) < 0. Proof. Since ϕ is coercive and weakly lower semicontinuous according to Proposition 2.5(ii), it follows from [13](Theorem 1.1) that ϕ attains its minimum at some point in H(0, T ), say u. Hence, by Proposition 2.5(iii) and Remark 3.1, it suffices to show that the critical point u is a nonzero function so that BVP (22) is solvable. In fact, for any T ≥b − L 0 + L, according to Remark 2.1 and 2.2 and noting thatF (L 0 − L) < 0, we compute by (H 6 ) where w is given by (36). It is easy to see that (37) together with ϕ(0) = 0 implies that u is nonzero. The proof is complete.  . Let t 2 ∈ (t 0 , T ) be the smallest number in this interval with u(t 2 ) = u(t 0 ). we can assume that u(t 1 ) = min t∈[t0,t2] u(t). If otherwise, we conclude again ϕ(v) < ϕ(u) which lead to a contradiction. The proof is complete.
Proof. Noting that (33), (13), (H 1 ) − (H 4 ) imply u is strictly increasing for t > 0 as long as u(t) ∈ (L 0 , 0), Remark 3.1 and (H 3 ) indicate that u can not be a constant in any interval of (0, ∞). These together with proposition 3.2 show the properties stated. The proof is complete. Let I i (i = 1, 2) be the set of all B ∈ (L 0 , 0) such that the corresponding solutions of IVP (12), (14) are type (i)(i = 1, 2) in Proposition 4.1. It is obvious that I i (i = 1, 2) are disjoint. Then, we have the following result, and some ideas of the proof are taken from that of [15](Theorem 14 and Theorem 20). Proof. We divide the proof into two steps.
Step 1 . Let B 0 ∈ I 1 and u 0 be a solution of IVP (12), (14) with B = B 0 . So, u 0 is the first type in Proposition 4.1. By proposition 3.1(iii), if B ∈ (L 0 , 0) is sufficiently close B 0 , then the corresponding solution u of IVP (12), (14) must be the first type, as well.
Let B 0 ∈ I 2 and u 0 be a solution of IVP (12), (14) with B = B 0 . So, u 0 is the second type in Proposition 4.1. In the case that u attains a local maximum which belongs to (0, L) at some pointt > 0 and u is strictly increasing in (0,t), then proposition 3.1(iii) and (33) guarantee that if B is sufficiently close to B 0 , the corresponding solution u of IVP (12), (14) has also its first local maximum in (0, L) at some pointt 1 > 0 and u is strictly increasing in (0,t 1 ).
Step 2 . We are now in a position to consider the case that u 0 is strictly increasing in (0, ∞) with lim t→∞ u 0 (t) = 0. Noting that c 2 /c 1 < 1 + F (L)/F (L 0 ), we can choose c 0 > 0 sufficiently small such that Since u 0 fulfils (32) and noting thatf (u 0 (t)) ≥ 0, u ′ 0 (t) ≥ 0, for t ∈ (0, ∞), we get by integration over [0, t] For t → ∞ we get, by the fact that u 0 (t) → 0, u ′ 0 (t) → 0 as t → ∞ Therefore we can find b > 0 such that Let δ > 0 and M = M (b, B 0 , δ) be the constants from Proposition 3.1(ii). Choose ǫ ∈ (0, c 0 /2M ). Assume that B ∈ (L 0 , 0) and u is a corresponding solution of IVP (12), (14). Using Proposition 3.1 and the continuity of F , we can findδ ∈ (0, δ) such that if |B − B 0 | <δ, then Suppose that u is not the second type in Proposition 4.1. Then there exists θ > 0 which is the first zero (in fact the only one zero) of u in (0, ∞). Then there are two possibilities. If u is the first type, there is b 0 > 0 such that u(b 0 ) = L and by Remark 3.1, If u is the third type, then We now rule out possibilities (43) and (44). Integrating (32) over [0, t] and using (39) − (42), we get for t > max{b 0 , b} In view of (38) and the Monotonicity of F , we get for t > max{b 0 , b}. Duo to (28) in Remark 2.1, we have sup{u(t), t > max{b, b 0 }} < L, which contradicts (43) and (44). The proof is complete.
Proof. The arguments are similar to that of Lemma 4.1. Therefore, we just briefly sketch it. It follows from Proposition 3.2 that in the second type of Proposition 4.1, the case that the solution u of IVP (12), (14) is strictly increasing in (0, ∞) with lim t→∞ u(t) = 0 is impossible. Therefore, same arguments as Step 1 in the proof of proposition 4.2 show that I i (i = 1, 2) are nonempty. The rest of the proof is the same as that of Lemma 4.1 and the proof is complete.
Therefore, according to Lemma A and (H 2 ), we can construct a sequence of functions {f n , n ≥ K} given by (45) and combing with (H 3 ), we have following properties off n .
We are now in a position to prove thatf n (L 0 ) = 0 for n sufficiently large. Suppose thatf n (L 0 ) = 0, thenf n (L 0 ) < 0 by (H 3 ) and Lemma A(ii). Therefore, since |f n (x)| ≤ K, x ∈ R and for n ≥ (K + 1)/L 0 we have Our task is now to provef n (x) = 0 for x ∈ (−∞, L 0 ]. In fact, noting that f n (L 0 ) = 0, by using the similar arguments as we did to obtain (49), we can get thatf Therefore by Lemma A(ii), we have 0 ≤f n (x) ≤f (x) = 0, x ∈ (−∞, L 0 ] and the result follows.

This together with Lemma
The proof is complete. According to Proposition 5.1 and 5.2, we can consider the auxiliary problems for n sufficiently large We are now in a position to prove our main results given in Introduction. Proofs of Theorem 1.1 and Theorem 1.2. According to Lemma A(iv) and Proposition 5.1, we can get that for n sufficiently large, there exists a constant b > 0, such that b 0 p(t)dt > 1 + 2c 2Fn (L n ) where c 2 andF n are given by (H 1 ) and (46), respectively.
It is obvious that u n is bounded uniformly in n. We now show that the sequence {u n } is equicontinuous on any bounded interval. Noting that (53) and (54) imply that u n satisfies the following integral equation For any bounded interval [0, b] and 0 ≤ t 1 < t 2 ≤ b, we have by the boundness of c,f n and the monotonicity of p u n (t 2 ) − u n (t 1 ) = t2 t1 1 p(s) s 0 c(τ )p(τ )f n (u n (τ ))dτ ds ≤ C(t 2 − t 1 ), where C depends on b. Therefore, take a sequence {T k } k≥1 such that T k < T k+1 and T k → ∞ as k → ∞, we conclude that the sequence {u n } is equicontinuous and uniformly bounded on every interval [0, T k ]. Hence, it has a uniformly convergent subsequence on every [0, T k ].
So let {u 1 ni } be a subsequence of {u n } that converges on [0, T 1 ]. Consider this subsequence on [0, T 2 ] and select a further subsequence {u 2 ni } of {u 1 ni } that converges uniformly on [0, T 2 ]. Repeat this procedure for all k, and then take a diagonal sequence {u ni }, which consists of u 1 n1 , u 2 n2 , u 3 n3 , · · · . Since the diagonal sequence u p np , u p+1 np+1 , · · · is a subsequence of {u p ni } for any p ≥ 1, it follows that it converges uniformly on any bounded interval to a function u. Without loss of generality, we still denote {u ni } by {u n }.
Finally, we need to show that lim t→∞ u(t) = L and u(t) = L for t ∈ (0, ∞). In fact, lim t→∞ u(t) = L is obvious since lim t→∞ u n (t) = L n and u n → u, L n → L as n → ∞. Suppose that there exists t 1 ∈ (0, ∞) such that u(t 1 ) = L. There are two possibilities. If u ′ (t 1 ) ≤ 0, contradicts to u ′ (t) > 0 on any bounded interval of (0, ∞). If u ′ (t 1 ) > 0, the fact that u is strictly increasing in (0, ∞) implies u(t) > L for t > t 1 , contradicts to lim t→∞ u(t) = L. The proof is complete.