Abstract
The aim of this paper is to establish multiplicity results of nontrivial and nonnegative solutions for mixed boundary value problems with the SturmLiouville equation. The approach is based on variational methods.
MSC: 34B15.
Keywords:
boundary value problem; mixed conditions1 Introduction
The aim of this paper is to establish existence results of two and three nontrivial solutions for SturmLiouville problems with mixed conditions involving the ordinary pLaplacian. We consider the following problem:
with
The existence of at least one solution for problem (P) has been obtained in [1], where only a unique algebraic condition on the nonlinear term is assumed (see [[1], Theorem 1.1]). In the present paper, first we obtain the existence of two solutions by combining an algebraic condition on f of type contained in [1] with the classical AmbrosettiRabinowitz condition.
(AR): There exist
The role of (AR) is to ensure the boundness of the PalaisSmale sequences for the EulerLagrange functional associated to the problem. This is very crucial in the applications of critical point theory. Subsequently, an existence result of three positive solutions is obtained combining two algebraic conditions which guarantee the existence of two local minima for the EulerLagrange functional and applying the mountain pass theorem as given by Pucci and Serrin (see [2]) to ensure the existence of the third critical point.
Many mathematical models give rise to problems for which only nonnegative solutions make sense; therefore, many research articles on the theory of positive solutions have appeared. For a complete overview on this subject, we refer to the monograph [3].
In this paper, we also present, as a consequence of our main theorems, some results on the existence of nonnegative solutions for a particular problem of type
where
Multiplicity results for a mixed boundary value problem have been studied by several
authors (see, for instance, [58] and references therein). In [5], the authors establish multiplicity results for problem (P), when
In [7], by using a fixed point theorem, the existence of at least three solutions for a
mixed boundary problem with the equation
Here, as an example, we present the following result which is a particular case of Theorem 3.6.
Theorem 1.1Let
and
Then, for each
admits at least three classical nonnegative and nontrivial solutions.
2 Preliminaries and basic notations
Our main tools are Theorems 2.1 and 2.2, consequences of the existence result of a local minimum [[9], Theorem 3.1] which is inspired by the Ricceri variational principle (see [10]). For more information on this topic see, for instance, [11] and [12].
Given a set X and two functionals
for all
for all
Theorem 2.1 [[9], Theorem 5.1]
LetXbe a reflexive real Banach space;
whereβand
Then, for each
Theorem 2.2 [[9], Theorem 5.3]
LetXbe a real Banach space;
whereρis given by (2.3), and for each
Then, for each
Now, consider problem (P) and assume that
Denote by
Throughout the sequel,
Put
Moreover, it is well known that
We use the following notations:
In order to study problem (P), we introduce the functionals
Clearly, the critical points of the functional
Clearly, if f is continuous,
3 Main results
In this section we present our main results.
Given two nonnegative constants c, d such that
put
Theorem 3.1Under the following conditions:
(i) there exist three constants
such that
and
(ii) there exist
For each
Proof The proof of this theorem is divided into two steps. In the first part, by applying
Theorem 2.1, we prove the existence of a local minimum for the functional
Define the following function
and estimate
and
Fix
From (3.1), one has
Moreover, for all
Hence,
Now, arguing as before, we obtain
From hypothesis (i) and bearing in mind (3.3), (3.2), (3.5), and (3.6), we obtain
From Theorem 2.1, for each
Now, we prove the existence of the second local minimum distinct from the first one.
To this end, we must show that the functional
Clearly, the functional
From the first part of the proof, we can assume that
Now, choosing any
as
Remark 3.1 We observe that in literature the existence of at least one nontrivial solution for
differential problems is obtained associating to the classical AmbrosettiRabinowitz
condition a hypothesis on the nonlinear term of type
Now, we present an application of Theorem 2.2 which we will use to obtain multiple solutions.
Theorem 3.2Assume that there exist two constants
and
Then, for each
problem (P) admits at least one nontrivial weak solution
Proof The functionals Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.2.
Moreover, by standard computations, condition (3.8) implies that
Arguing as in the proof of Theorem 3.1, we obtain that
So, from our assumption, it follows that
Hence, from Theorem 2.2 for each
Remark 3.2 We point out that the same statement of above given result can be obtained by using a classical direct methods theorem (see [14]), but in addition we get the location of the solution, hence in particular the solution is nontrivial.
Now, we point out some results when the nonlinear term is with separable variables. To be precise, let
•
•
consider the following boundary value problem:
We observe that the following results give the existence of multiple nonnegative solutions since the nonlinear term is supposed to be nonnegative. In order to justify what has been said above, we point out the following weak maximum principle.
Lemma 3.1Suppose that
Proof We claim that a weak solution
that is,
Corollary 3.1Assume that
(i′) there exist three nonnegative constants
(ii′) there exist
Then, for each
problem (P1) admits at least two nonnegative weak solutions
Theorem 3.3Assume that there exist two positive constantsc, d, with
Further, suppose that there exist
Then, for each
Proof Our aim is to apply Corollary 3.1. To this end, we pick
On the other hand, one has
Hence, from Corollary 3.1 and taking (2.6) into account, the conclusion follows. □
A further consequence of Theorem 3.1 is the following result.
Theorem 3.4Assume that
and there are constants
Then, for each
Proof Fix
Next, as a consequence of Theorems 3.3 and 3.2, the following theorem of the existence of three solutions is obtained.
Theorem 3.5Assume that
Moreover, assume that there exist four positive constantsc, d,
and
are satisfied.
Then, for each
Proof First, we observe that
Theorem 3.6Assume that
Further, assume that there exist two positive constants
Then, for each
Proof Clearly, (3.15) implies (3.8). Moreover, by choosing d small enough and
Finally, we present two examples of problems that admit multiple solutions owing to Theorems 3.4 and 3.6.
Example 3.1 Owing to Theorem 3.4, for each
admits at least two nonnegative solutions. In fact, one has
Example 3.2 Consider the following problem:
It has three nonnegative solutions. In fact, let
Owing to Theorem 3.6, the following problem
admits three nonnegative classical solutions. In fact, one has
Moreover, taking into account that
So, it is clear that any nonnegative solution u of problem (
Competing interests
The author declares that she has no competing interests.
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