On the existence of solutions for a boundary value problem on the half-line

In this note we consider Dirichlet boundary value problem on a half line. Using critical point theory we prove the existence of at least one nontrivial solution.


Introduction
In this paper we are going to prove two existence results concerning boundary value problems on a half line using critical point theory approach.Problems on a half line received lately some attention but the main approach concerning the existence issue was by fixed point theorems and the method of lower and upper solutions.The results by critical point theory are less frequent due to the lack of the Poincaré inequality and also due to the fact that the space in which the solutions are obtained is not compactly embedded into the space of continuous functions.
The above assumption is due to the fact that the space H 1 0 (0, +∞) is not compactly embedded into C[0, +∞) contrary to the case of bounded interval setting as we mentioned before.In order to overcome this problem we may take into account the embedding results contained in [6] and [7].These will allow us to have the counterpart of a definition of L 1 -Carathéodory function commonly applied in the case of bounded interval.In the literature, for example [2], the idea of L 2 -Carathéodory function is used and the embedding into the space of bounded continuous functions is utilized.
As it is common with variational problems for O.D.E.(1.1) admits two types of solutions, namely a weak and a classical one.Function u ∈ H 1 0 (0, +∞) is a weak solution of (1.1) if Function u ∈ H 1 0 (0, +∞) is a classical solution to (1.1) if both u and u are locally absolutely continuous functions on [0, +∞), and the boundary conditions u(0) = u(+∞) are satisfied.We would like to recall, following [3], that any function u ∈ H 1 0 (0, +∞) is locally absolutely continuous, i.e. absolutely continuous on any closed bounded interval contained in [0, +∞) however it is not in general absolutely continuous on the whole half line which makes the problem different from the classical bounded one.
We will look for solutions of (1.1) which are critical points to (1.2) and in order to obtain them we will apply two approaches.The first one is connected with the usage of the mountain pass geometry, see book [5] for some background.Such an approach requires that the problem under consideration satisfies some suitable geometric conditions pertaining to behaviour around 0 and also compactness condition in a form of a Palais-Smale condition.
For the second approach we will use some abstract critical point theorem derived in [8].This result provides the existence of a critical point located in some set which need not be open and was applied already to some problems in bounded domains only.This approach does not require compactness pertaining to the usage of a Palais-Smale condition but on the other hand the nonlinear part of the equation must have enough monotonicity in order to yield that the corresponding term of the action functional, namely Both methods use partially different assumptions while the common assumption concerns the issue of integrability of terms appearing in the action functional and the issue of connection between weak and classical solutions.Both approaches yield the existence of at least one non-trivial critical point.In the case of the application of the mountain pass theorem the existence of non-trivial solution follows from the abstract result without any other assumptions than those leading to the so called mountain geometry.The application of theorem from [8] provides only the existence of some critical point and that is why one must make sure that it is non-trivial by some additional assumption.Moreover critical points obtained by both methods are located in some ball around 0.
Finally, we would like to underline that there are not many results concerning solvability of problems like (1.1) when compared to the case of a bounded interval for the reasons mentioned above.Apart from [2] we would like to mention [4,6,7] where also variational approaches are used but these pertain either to the critical point type result of Ricceri or else to some non-smooth setting.In none of these sources mountain pass methodology is directly applied, while some of its ideas are hidden in the approach of three critical point theorems but with different assumptions.
To the best of our knowledge, the results in Theorem 3.4 and Theorem 3.5 are new and original as we have not found any discussion in the existing literature.Also, there exists no paper concerned with the existence of at least one nontrivial solution for our problem which is posed on the half line under assumptions similar to us.

Preliminaries
Symbol L p (0, +∞) for p ≥ 1 means the space of such measurable real valued functions defined on [0, +∞) that ∞ 0 |u (t)| p dt < +∞.Solutions to (1.1) will be considered in the space H 1 0 (0, +∞) which is defined as follows.We say that u ∈ H 1 0 (0, +∞) if u ∈ L 2 (0, +∞) and if there exists a function g ∈ L 2 (0, +∞), called a weak derivative, and such that , where C ∞ c (0, +∞) is the space of compactly supported functions from C ∞ ([0, +∞)), R).We denote g := u .We endow the space H 1 0 (0, +∞) with its natural norm , associated with the scalar product Let us also consider the space We need some definitions and lemmas which will be used later.
Definition 2.1.Let E be a Banach space.Let J ∈ C 1 (E, R).For any sequence {u n } ⊂ E, if {J(u n )} is bounded and J (u n ) → 0 as n → ∞ possesses a convergent subsequence, then we say that J satisfies the Palais-Smale condition ((PS) condition for short).
Then J has a critical value c ≥ α.Moreover, c can be characterized as where We need also the following embeddings.

Lemma 2.4 ([6, 7]).
Assume that A holds.The embedding We endow the space L ∞ (0, +∞) with the standard ess sup-norm.The constant of the continuous embedding H 1 0 (0, +∞) → L ∞ (0, +∞) is denoted by K (see [3, Remark 10, p. 214], or else Theorem 8.8 from [3]).Proposition 2.5.Let λ > 0 be fixed.Assume that A holds.The functional J is well-defined and continuously differentiable on H 1 0 (0, +∞).The derivative of J at any u ∈ H 1 0 (0, +∞) has the following form Proof.Note that the term is obviously well defined and C 1 since J 1 (u) = 1 2 u 2 .Thus we need to prove that Claim 1: J 2 is well defined and Gâteaux-differentiable.Let us take any fixed u ∈ H 1 0 (0, +∞).By Lemma 2.3 there is some r > 0 such that u ∞,p ≤ r.By assumption A and again by Lemma 2.3 we see what follows Now we turn to Gâteaux-differentiability. Indeed, let u, v ∈ H 1 0 (0, +∞) be fixed and take any t ∈ [0, +∞).Then for any θ ∈ (0, 1) and s small we have by Lemma 2.3 and Lemma 2.4, Moreover, we see by assumption A that and we see that h r (•) q(•) p(•) ∈ L 1 (0, +∞).Therefore we can apply the mean value theorem and then the Lebesgue dominated convergence theorem in order to pass to the limit s → 0 in Using A and reasoning similar to this provided in (2.3) we have by the Lebesgue dominated convergence theorem uniformly for v in the unit ball.Thus we see that Remark 2.6.We note that from the second part of the proof of the above theorem and from Lemma 2.4 it follows that J 2 is weakly continuous on H 1 0 (0, +∞).Indeed, for a sequence (u n ) ⊂ H 1 0 (0, +∞), such that u n u, as n → +∞, we have by Lemma 2.4, that u n → u, as n → +∞, in C l,p [0, +∞).Then we see that as n → +∞.Proof.We follow the same steps as in [6].If u satisfies the Euler equation J (u) = 0, i.e.
We would like to note that the counterpart of the proof of Proposition 2.5 in bounded intervals is standard but when we work on infinite intervals the assertion of the proposition is not evident and for this reason we must use hypothesis A and utilize embeddings from Lemmas 2.3 and 2.4 to prove it.
(i) Let X ⊂ E and let there exist u 0 , v ∈ X satisfying ϕ(v) = h(u 0 ), and such that Then u 0 is a critical point of J, and thus it solves (2.8).

Applications
Now we state the following hypotheses.
Remark 3.1.Note that from (H 2 )(b) we obtain a type of (H 2 )(a) only with fixed constants c 1 and c 2 .This does not suffice for our problem so the additional assumption is crucial.Relaxed version of the A-R condition, namely condition (H 2 )(b), could also be assumed but these involve some technical calculations only and do not advance our main approach.We note also that it is possible to assume convexity of F at some interval centered at 0 only.
We will show now that the functional J with the above assumptions (H 1 )-(H 3 ) has mountain pass geometry and so at least one nontrivial solution.On the other hand assuming only (H 1 ), (H 4 ) and some condition at 0, we obtain the existence of at least one solution on some arbitrarily fixed closed ball for a suitable range of numerical parameter.

Results by the mountain pass lemma
Lemma 3.2.Assume that A holds.Suppose also that (H 1 ), (H 2 ) hold.Then for any λ > 0, the functional J given by (1.2) satisfies the PS-condition.
Proof.Let us take a sequence (u k ) ⊂ H 1 0 (0, +∞) such that (J(u k )) is bounded and J (u k ) → 0, as k → ∞.We shall show that (u k ) has a convergent subsequence.
Since J (u k ) → 0, we see that for some > 0 there exists k 0 with Observe further that by a direct calculation Next, we prove that (u k ) converges strongly to some u in H 1 0 (0, +∞).Since (u k ) is bounded in H 1 0 (0, +∞), there exists a subsequence of (u k ), still denoted (u k ), such that (u k ) converges weakly to some u in H 1 0 (0, +∞) with u ≤ M 2 .As already mentioned, by Lemma 2.4, (u k ) converges to u on C l,p [0, +∞).
Assumption (1) in Lemma 3.3 is then satisfied.Now (H2)(a) guarantees that for some w 0 ∈ H 1 0 (0, +∞) with w 0 = 0 and s ∈ R + we have the following estimation Since θ > 2 we see that J(sw 0 ) → −∞ as s → +∞.Thus there is some s 0 such that for z 0 = s 0 w 0 we have J(z 0 ) < 0. Therefore Assumption (2) in Lemma 3.3 is also satisfied.Now the mountain pass lemma allows us to formulate the following existence result.

Results by the critical point theorem on a closed ball
Theorem 2.8 allows us also to obtain the existence of at least one nontrivial solution without employing mountain pass geometry.Some assumptions involved in obtaining the existence result by the mountain pass technique are to employed, namely (H 1 ).However, we need no information about the behaviour of the nonlinearity around 0 apart from some assumption concerning the sign at 0 so that to ensure that the solution is nontrivial.Note that the assumption leading to the usage of the mountain pass geometry require that f is 0 at 0. This is in contrast to the previous case and so both existence results lead to the coverage of different type of nonlinear terms.Indeed, we have the following result Theorem 3.5.Assume that A holds.Suppose also that (H 1 ), (H 4 ) hold.Then there exists λ * > 0 such that for all 0 < λ < λ * , problem (1.1) has at least one nontrivial solution provided that f (t, 0) = 0 on a subset of [0, +∞) of positive measure.
Recall that J = J 1 − λJ 2. , see (2.1), (2.2).Note that J 1 is weakly l.s.c.Since J 2 is weakly continuous, we see that J is weakly l.s.c.Since B is weakly compact, we obtain that J has at least one minimizer u 0 over B for any λ > 0.
We shall apply Theorem 2.8.Put Φ, H : H 1 0 (0, +∞) → R by formulas and note that these are convex C 1 functionals.Consider the auxiliary Dirichlet problem Note that problem (3.7) is uniquely solvable by some v ∈ H 1 0 (0, +∞).To reach this conclusion, we use the following procedure.We prove that the action functional corresponding to (3.7) J 0 (u) = Therefore problem (3.8) has at least one nontrivial solution for any λ > 0.
Concerning the usage of the theorem on a closed ball we consider the following problem for which (H 3 ) is not satisfied