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A variational approach of SturmLiouville problems with the nonlinearity depending on the derivative
 Ghasem A Afrouzi^{1},
 Armin Hadjian^{2} and
 Vicenţiu D Rădulescu^{3, 4}Email author
 Received: 17 April 2015
 Accepted: 1 May 2015
 Published: 20 May 2015
Abstract
In this paper, we are concerned with the existence of positive classical solutions for a class of secondorder differential equations with the nonlinearity dependent on the derivative. We also provide a range of the parameter in order to obtain the existence of multiple solutions. The approach is based on variational methods. An example illustrates the abstract results of this paper.
Keywords
 Dirichlet problem
 critical points
 variational methods
MSC
 34B15
 35B38
 58E05
1 Introduction
The classical SturmLiouville theory does not depend upon the calculus of variations but stems from the theory of ordinary linear or nonlinear differential equations. Linear SturmLiouville equations can be also studied in the context of functional analysis by means of selfadjoint operators or integral operators with a continuous symmetric kernel (the Green’s function of the problem). Certain applications involving linear partial differential equations can be treated with the help of the SturmLiouville theory, for instance the normal modes of vibration of a thin membrane. We also refer to [1], where a perturbed nonlinear SturmLiouville problem with superlinear convex nonlinearity is studied. In the recent paper [2], the authors study a class of discrete anisotropic SturmLiouville problems. We also refer to [3–5] for related properties of solutions of SturmLiouville problems.
In the present paper, we are concerned with a class of nonlinear SturmLiouville problems and we establish some qualitative properties of the eigenvalues by using variational principles. A feature of our work is the presence of the derivative in the nonlinear term, which creates further technical constraints.
Our main purpose in this paper is to establish a range of eigenvalues in a suitable interval in order to create at least one eigenfunction. As a consequence, we establish sufficient conditions for the existence of two or three solutions.
Dirichlet boundary value problems have been widely studied because of their applications to various fields of applied sciences such as mechanical engineering, control systems, computer science, economics, artificial or biological neural networks and many others.
In this connection, several existence and multiplicity results for solutions to secondorder ordinary differential nonlinear equations, with the nonlinearity dependent on the derivative and Dirichlet conditions at the ends, have been investigated making use of variational methods.
Successively, in [8], the authors considered the system (1.3) in the case \(\mu=0\), finding the existence of infinitely many classical solutions for certain values of the parameter λ by using variational methods.
In the present paper, first we obtain the existence of at least one solution for problem (1.2). It is worth noticing that, usually, to obtain the existence of one solution, asymptotic conditions both at zero and at infinity on the nonlinear term are requested, while here it is assumed only a unique algebraic condition (see (A_{4}) in Theorem 3.3). As a consequence, by combining with the classical AmbrosettiRabinowitz condition (see [10]), the existence of two solutions is obtained (see Theorem 4.1). Subsequently, an existence result of three nonnegative 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 [11]) to ensure the existence of the third critical point (see Theorem 4.5).
As an example, we state here the following special case of Theorem 4.5.
Theorem 1.1
Finally, we point out that Theorem 1.1 ensures a precise conclusion in the sense that a location of the parameter λ in order to obtain at least three distinct solutions is also provided.
2 Preliminaries
Theorem 2.1
([7])
Then, for each \(\lambda\in ]\frac{1}{\rho_{2}(r_{1},r_{2})},\frac{1}{\beta(r_{1},r_{2})} [\), there is \(u_{0,\lambda}\in\Phi^{1}(]r_{1},r_{2}[)\) such that \(I_{\lambda}(u_{0,\lambda})\leq I_{\lambda}(u)\) for all \(u\in \Phi^{1}(]r_{1},r_{2}[)\) and \(I'_{\lambda}(u_{0,\lambda})=0\).
Theorem 2.2
([7])
Then, for each \(\lambda>\frac{1}{\rho(r)}\), there is \(u_{0,\lambda}\in\Phi^{1}(]r,+\infty[)\) such that \(I_{\lambda}(u_{0,\lambda})\leq I_{\lambda}(u)\) for all \(u\in \Phi^{1}(]r,+\infty[)\) and \(I'_{\lambda}(u_{0,\lambda})=0\).
 (a)
\(x\mapsto f(x,\xi)\) is measurable for every \(\xi\in \mathbb{R}\);
 (b)
\(\xi\mapsto f(x,\xi)\) is continuous for almost every \(x\in[a,b]\);
 (c)for every \(\rho>0\), there is a function \(l_{\rho}\in L^{1}([a,b])\) such thatfor almost every \(x\in[a,b]\).$$\sup_{\xi\leq\rho}\biglf(x,\xi)\bigr\leq l_{\rho}(x), $$
The following lemma is taken from [6], Lemma 2.2.
We cite recent monographs [13–15] as general references for the basic notions used in the paper.
3 Main results
In this section we present our main results. To be precise, we establish an existence result of at least one solution, Theorem 3.1, which is based on Theorem 2.1, and we point out some consequences, Theorems 3.2, 3.3 and 3.4. Finally, we present another existence result of at least one solution, Theorem 3.7, which is based in turn on Theorem 2.2.
Theorem 3.1
 (A_{1}):

\(F(x,t) \geq0\) for all \((x,t)\in([a,a+\alpha]\cup [b\beta,b])\times[0,d]\);
 (A_{2}):

\(a_{d}(c_{2})< a_{d}(c_{1})\).
Proof
Now, we point out an immediate consequence of Theorem 3.1.
Theorem 3.2
 (A_{3}):

$$\frac{ \int_{a}^{b}\max_{t\leq c}F(x,t)\,dx}{ m(2c)^{p}}< \frac{ \int_{a+\alpha}^{b\beta }F(x,d)\,dx}{ Dd^{p}(ba)^{p1}pM}. $$
Proof
Theorem 3.3
 (A_{4}):

$$\frac{G(c)}{c^{p}} < \biggl(\frac{ 2^{p}m\int_{a+\alpha}^{b\beta}\gamma(x)\,dx}{ D(ba)^{p1}pM\\gamma\_{1}} \biggr)\frac{G(d)}{d^{p}}. $$
Proof
We now give a special case of our main result as follows.
Theorem 3.4
 (A_{5}):

\(\lim_{t\to0^{+}}\frac {g(t)}{t^{p1}}=+\infty\).
Proof
Remark 3.5
Remark 3.6
Finally, we present an application of Theorem 2.2 which we will use in the next section to obtain multiple solutions.
Theorem 3.7
 (A_{6}):

\(\int_{a}^{b}\max_{t\leq\bar {c}}F(x,t)\,dx<\int_{a+\alpha}^{b\beta}F(x,\bar{d})\,dx\);
 (A_{7}):

\(\limsup_{t\rightarrow+\infty}\frac {F(x,t)}{t^{p}}\leq 0\) uniformly in x.
Proof
4 Multiplicity results
The main aim of this section is to present multiplicity results. First, as a consequence of Theorem 3.1, taking into account the classical theorem of Ambrosetti and Rabinowitz, we have the following multiplicity result.
Theorem 4.1
 (A_{8}):

there exist positive constants ν and R such that \(\nu m>pM\), and for all \(t\geq R\) and \(x\in[a,b]\), one has$$ 0< \nu F(x,t)\leq t\cdot f(x,t). $$
Proof
Fix λ as in the conclusion. So, Theorem 3.1 ensures that problem (1.2) admits at least one nontrivial classical solution \(\bar{u}_{1}\) satisfying condition (4.1) which is a local minimum of the functional \(I_{\lambda}\).
Now, we prove the existence of the second solution distinct from the first one. To this end, we must show that the functional \(I_{\lambda}\) satisfies the hypotheses of the mountain pass theorem.
Clearly, the functional \(I_{\lambda}\) is of class \(C^{1}\) and \(I_{\lambda}(0)=0\).
We can assume that \(\bar{u}_{1}\) is a strict local minimum for \(I_{\lambda}\) in X. Therefore, there is \(\rho>0\) such that \(\inf_{\u\bar{u}_{1}\=\rho}I_{\lambda}(u)>I_{\lambda}(\bar{u}_{1})\), so condition [18], \((I_{1})\), Theorem 2.2, is verified.
Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point \(\bar{u}_{2}\) of \(I_{\lambda}\) such that \(I_{\lambda}(\bar{u}_{2})>I_{\lambda}(\bar{u}_{1})\). So, \(\bar{u}_{1}\) and \(\bar{u}_{2}\) are two distinct classical solutions of (1.2) and the proof is complete. □
Corollary 4.2
 (A_{9}):

there exist positive constants ν and R such that \(\nu m>pM\), and for all \(t\geq R\), one has$$0< \nu G(t)\leq t\cdot g(t). $$
Corollary 4.3
Assume that (A_{5}) and (A_{9}) are satisfied. Then, for each \(\lambda\in]0,\lambda^{\star}[\), problem (3.3) admits at least two nonnegative classical solutions.
Next, as a consequence of Theorems 3.7 and 3.2, the following theorem of the existence of three classical solutions is obtained, and its consequence for the nonlinearity with separable variables is presented.
Theorem 4.4
 (A_{10}):

$$\frac{ \int_{a}^{b}\max_{t\leq c} F(x,t)\,dx}{ (2c)^{p}} < \frac{ m (\int_{a+\alpha}^{b\beta} F(x,\bar{d})\,dx\int_{a}^{b}\max_{t\leq\bar{c}}F(x,t)\,dx )}{ D\bar{d}^{p}(ba)^{p1}pMm(2\bar{c})^{p}} $$
Proof
Theorem 4.5
 (A_{11}):

\(\limsup_{t\rightarrow0^{+}}\frac{G(t)}{t^{p}}=+\infty\);
 (A_{12}):

\(\limsup_{t\rightarrow+\infty}\frac{G(t)}{t^{p}}=0\).
 (A_{13}):

$$\frac{G(\bar{c})}{\bar{c}^{p}} < \biggl(\frac{ 2^{p}m\int_{a+\alpha}^{b\beta}\gamma(x)\,dx}{ D(ba)^{p1}pM\\gamma\_{1}} \biggr) \frac{G(\bar{d})}{\bar{d}^{p}}. $$
Proof
Clearly, (A_{12}) implies (A_{7}). Moreover, by choosing d small enough and \(c=\bar{c}\), simple computations show that (A_{11}) implies (A_{3}). Finally, from (A_{13}) we get (A_{6}) and, arguing as in the proof of Theorem 3.2, also (A_{10}). Hence, Theorem 4.4 ensures the conclusion. □
Remark 4.6
If \(g(0)\neq0\), Corollaries 4.2 and 4.3 ensure two positive classical solutions while Theorem 4.5 ensures three positive classical solutions (see the proof of Theorem 3.3).
Finally, we present the following example to illustrate our results.
Example 4.7
Declarations
Acknowledgements
V Rădulescu acknowledges the support through Grant CNCSPCE47/2011.
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Authors’ Affiliations
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