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Multiplicity results for nonlinear mixed boundary value problem
 Giuseppina D’Aguì^{1}Email author
https://doi.org/10.1186/168727702012134
© D’Aguì; licensee Springer 2012
 Received: 20 July 2012
 Accepted: 26 October 2012
 Published: 14 November 2012
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 conditions
1 Introduction
with $p>1$, $q,r\in {L}^{\mathrm{\infty}}([a,b])$, with ${q}_{0}={ess\hspace{0.17em}inf}_{[a,b]}q>0$ and ${r}_{0}={ess\hspace{0.17em}inf}_{[a,b]}r\ge 0$. Here the nonlinearity $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is an ${L}^{1}$Carathéodory function and λ is a real positive parameter.
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.
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].
where $\alpha \in {L}^{1}([a,b])$ is such that $\alpha (x)\ge 0$ a.e. $x\in [a,b]$, $\alpha \not\equiv 0$, and $g:\mathbb{R}\to \mathbb{R}$ is a nonnegative continuous function. In particular, we obtain for such a problem the existence of at least three nonnegative solutions by requiring that the function g has a superlinear behavior at zero, a sublinear behavior at infinity, and a particular growth in a suitable interval $[c,d]$. By a similar approach, in [4], the authors obtain the existence of multiple solutions for a Neumann elliptic problem.
Multiplicity results for a mixed boundary value problem have been studied by several authors (see, for instance, [5–8] and references therein). In [5], the authors establish multiplicity results for problem (P), when $p=2$, and, in particular, they obtain the existence of three solutions, one of which can be trivial. On the contrary, our results (Theorems 3.5 and 3.6) guarantee the existence of three nonnegative and nontrivial solutions.
In [7], by using a fixed point theorem, the existence of at least three solutions for a mixed boundary problem with the equation ${({{u}^{\prime}}^{p2}{u}^{\prime})}^{\prime}=q(x)f(u)$ is obtained, by requiring, among other things, the boundness of f in a right neighborhood of zero (hypothesis (H6), Theorem 3.1), instead in our results (Theorems 3.5 and 3.6) the nonlinearity can blow up at zero.
Here, as an example, we present the following result which is a particular case of Theorem 3.6.
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].
for all $r\in \mathbb{R}$.
Theorem 2.1 [[9], Theorem 5.1]
where β and ${\rho}_{2}$ are given by (2.1) and (2.2).
Then, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]\frac{1}{{\rho}_{2}({r}_{1},{r}_{2})},\frac{1}{\beta ({r}_{1},{r}_{2})}[$, there is ${u}_{0,\lambda}\in {\mathrm{\Phi}}^{1}(]{r}_{1},{r}_{2}[)$ such that ${I}_{\lambda}({u}_{0,\lambda})\le {I}_{\lambda}(u)$ for all $u\in {\mathrm{\Phi}}^{1}(]{r}_{1},{r}_{2}[)$ and ${I}_{\lambda}^{\prime}({u}_{0,\lambda})=0$.
Theorem 2.2 [[9], Theorem 5.3]
where ρ is given by (2.3), and for each $\lambda >\frac{1}{\rho (r)}$, the function ${I}_{\lambda}=\mathrm{\Phi}\lambda \mathrm{\Psi}$ is coercive.
Then, for each $\lambda >\frac{1}{\rho (r)}$, there is ${u}_{0,\lambda}\in {\mathrm{\Phi}}^{1}(]r,+\mathrm{\infty}[)$ such that ${I}_{\lambda}({u}_{0,\lambda})\le {I}_{\lambda}(u)$ for all $u\in {\mathrm{\Phi}}^{1}(]r,+\mathrm{\infty}[)$ and ${I}_{\lambda}^{\prime}({u}_{0,\lambda})=0$.
Throughout the sequel, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is an ${L}^{1}$Carathéodory function. We recall that a function $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is said to be an ${L}^{1}$Carathéodory function if $x\to f(x,t)$ is measurable for all $t\in \mathbb{R}$, $t\to f(x,t)$ is continuous for almost every $x\in [a,b]$, and for all $M>0$, one has ${sup}_{t\le M}f(x,t)\in {L}^{1}([a,b])$. Clearly, if f is continuous in $[a,b]\times \mathbb{R}$, then it is ${L}^{1}$Carathéodory.
Clearly, if f is continuous, $q\in {C}^{1}([a,b])$, and $r\in {C}^{0}([a,b])$, the weak solutions for (P) are classical solutions.
3 Main results
In this section we present our main results.
 (i)there exist three constants ${c}_{1}$, ${c}_{2}$, d, with${\left(\frac{{q}_{0}^{p2}}{{2}^{p1}}\right)}^{\frac{1}{p}}{c}_{1}<d<{\left(\frac{{q}_{0}^{p1}}{{2}^{p1}}\right)}^{\frac{1}{p}}\frac{1}{{[{\parallel q\parallel}_{\mathrm{\infty}}+\frac{p+2}{p+1}{(\frac{ba}{2})}^{p}{\parallel r\parallel}_{\mathrm{\infty}}]}^{\frac{1}{p}}}{c}_{2},$(3.1)
 (ii)
For each $\lambda \in \phantom{\rule{0.2em}{0ex}}]\frac{1}{{a}_{d}({c}_{1})},\frac{1}{{a}_{d}({c}_{2})}[$, problem (P) admits at least two nontrivial weak solutions ${\overline{u}}_{1}$, ${\overline{u}}_{2}$, with ${\overline{u}}_{1}$ such that ${(\frac{{q}_{0}}{ba})}^{\frac{p1}{p}}{c}_{1}<\parallel {\overline{u}}_{1}\parallel <{(\frac{{q}_{0}}{ba})}^{\frac{p1}{p}}{c}_{2}$.
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 $\mathrm{\Phi}\lambda \mathrm{\Psi}$, where $\mathrm{\Phi}(u)$ and $\mathrm{\Psi}(u)$ are functionals given in (2.7) for all $u\in X$. Obviously, Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.1, and the critical points in X of the functional $\mathrm{\Phi}\lambda \mathrm{\Psi}$ are exactly the weak solutions of problem (P). To this end, we verify condition (2.4) of Theorem 2.1.
Fix ${c}_{1}$, d, ${c}_{2}$ satisfying (3.1) and put ${r}_{1}=\frac{1}{p}{(\frac{{q}_{0}}{ba})}^{p1}{c}_{1}^{p}$ and ${r}_{2}=\frac{1}{p}{(\frac{{q}_{0}}{ba})}^{p1}{c}_{2}^{p}$.
From (3.1), one has ${r}_{1}<\mathrm{\Phi}({u}_{0})<{r}_{2}$.
Now, we prove the existence of the second local minimum distinct from the first one. To this end, we must show that the functional $\mathrm{\Phi}\lambda \mathrm{\Psi}$ satisfies the hypotheses of the mountain pass theorem.
Clearly, the functional $\mathrm{\Phi}\lambda \mathrm{\Psi}$ is of class ${C}^{1}$ and $(\mathrm{\Phi}\lambda \mathrm{\Psi})(0)=0$.
From the first part of the proof, we can assume that ${\overline{u}}_{1}$ is a strict local minimum for $\mathrm{\Phi}\lambda \mathrm{\Psi}$ in X. Therefore, there is $\rho >0$ such that ${inf}_{\parallel u{u}_{1}\parallel =\rho}(\mathrm{\Phi}\lambda \mathrm{\Psi})(u)>(\mathrm{\Phi}\lambda \mathrm{\Psi})({\overline{u}}_{1})$, so condition [[13], (${I}_{1}$), Theorem 2.2] is verified.
as $t\to +\mathrm{\infty}$, so condition [[13], (${I}_{2}$), Theorem 2.2] is verified. Moreover, by standard computations, $\mathrm{\Phi}\lambda \mathrm{\Psi}$ satisfies the PalaisSmale condition. Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point ${\overline{u}}_{2}$ of $\mathrm{\Phi}\lambda \mathrm{\Psi}$ such that $(\mathrm{\Phi}\lambda \mathrm{\Psi})({\overline{u}}_{2})>(\mathrm{\Phi}\lambda \mathrm{\Psi})({\overline{u}}_{1})$. So, ${\overline{u}}_{1}$ and ${\overline{u}}_{2}$ are two distinct weak solutions of (P) and the proof is complete. □
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 $f(x,t)=o(t)$ as $t\to 0$. This implies that the problem possesses also the trivial solution $u\equiv 0$. In Theorem 3.1, we find a nontrivial solution of the problem that actually is a proper local minimum of the EulerLagrange functional associated to the problem different from zero.
Now, we present an application of Theorem 2.2 which we will use to obtain multiple solutions.
problem (P) admits at least one nontrivial weak solution $\tilde{u}$ such that $\parallel \tilde{u}\parallel >\frac{\overline{c}}{{(\frac{{q}_{0}}{ba})}^{\frac{p1}{p}}}$.
So, from our assumption, it follows that $\rho (r)>0$.
Hence, from Theorem 2.2 for each $\lambda >\tilde{\lambda}$, the functional $\mathrm{\Phi}\lambda \mathrm{\Psi}$ admits at least one local minimum $\tilde{u}$ such that $\parallel \tilde{u}\parallel >\frac{\overline{c}}{{(\frac{{q}_{0}}{ba})}^{\frac{p1}{p}}}$ and our conclusion is achieved. □
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

$\alpha \in {L}^{1}([a,b])$ such that $\alpha (x)\ge 0$ a.e. $x\in [a,b]$, $\alpha \not\equiv 0$, and

$g:\mathbb{R}\to \mathbb{R}$ be a nonnegative continuous function,
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 3.1 Suppose that $\overline{u}\in X$ is a weak solution of problem (P1), then $\overline{u}$ is nonnegative.
that is, ${\parallel \overline{u}\parallel}_{{W}^{1,2}(A)}=0$ which is absurd. Hence, our claim is proved. □
Corollary 3.1 Assume that
problem (P1) admits at least two nonnegative weak solutions ${\overline{u}}_{1}$ and ${\overline{u}}_{2}$ such that $\frac{1}{\sqrt{ba}}{c}_{1}<\parallel {\overline{u}}_{1}\parallel <\frac{1}{\sqrt{ba}}{c}_{2}$.
Then, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]\frac{2{d}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([\frac{a+b}{2},b])}G(d)},\frac{{c}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([a,b])}G(c)}[$, problem (P1) admits at least two nonnegative weak solutions.
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.
Then, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]0,{\lambda}^{\ast}[$, where ${\lambda}^{\ast}=\frac{1}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([a,b])}}{sup}_{c>0}\frac{{c}^{2}}{G(c)}$, problem (P1) admits at least two nonnegative weak solutions.
Proof Fix $\lambda \in \phantom{\rule{0.2em}{0ex}}]0,{\lambda}^{\ast}[$. Then there is $c>0$ such that $\lambda <\frac{1}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([a,b])}}\frac{{c}^{2}}{G(c)}$. From (3.11), there is $d<\frac{c}{\sqrt{2}}$ such that $\frac{2(ba){\parallel \alpha \parallel}_{{L}^{1}([\frac{a+b}{2},b])}G(d)}{2{d}^{2}}>\frac{1}{\lambda}$. Hence, Theorem 3.3 ensures the conclusion. □
Next, as a consequence of Theorems 3.3 and 3.2, the following theorem of the existence of three solutions is obtained.
are satisfied.
Then, for each $\lambda \in \mathrm{\Lambda}=\phantom{\rule{0.2em}{0ex}}]max\{\tilde{\lambda},\frac{2{d}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([\frac{a+b}{2},b])}G(d)}\},\frac{{c}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([a,b])}G(c)}[$, problem (P1) admits at least three weak nonnegative solutions.
Proof First, we observe that $\mathrm{\Lambda}\ne \mathrm{\varnothing}$ owing to (3.13). Next, fix $\lambda \in \mathrm{\Lambda}$. Theorem 3.3 ensures a nontrivial weak solution $\overline{u}$ such that $\parallel \overline{u}\parallel <c$ which is a local minimum for the associated functional $\mathrm{\Phi}\lambda \mathrm{\Psi}$, as well as Theorem 3.2 guarantees a nontrivial weak solution $\tilde{u}$ such that $\parallel \tilde{u}\parallel >c$ which is a local minimum for $\mathrm{\Phi}\lambda \mathrm{\Psi}$. Hence, the mountain pass theorem as given by Pucci and Serrin (see [2]) ensures the conclusion. □
Then, for each $\lambda \in \phantom{\rule{0.2em}{0ex}}]\frac{2{\overline{d}}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([\frac{a+b}{2},b])}G(\overline{d})},\frac{{\overline{c}}^{2}}{2(ba){\parallel \alpha \parallel}_{{L}^{1}([a,b])}G(\overline{c})}[$, problem (P1) admits at least three weak nonnegative solutions.
Proof Clearly, (3.15) implies (3.8). Moreover, by choosing d small enough and $c=\overline{c}$, simple computations show that (3.14) implies (3.10). Finally, from (3.16) we get (3.7) and also (3.13). Hence, Theorem 3.5 ensures the conclusion. □
Finally, we present two examples of problems that admit multiple solutions owing to Theorems 3.4 and 3.6.
admits at least two nonnegative solutions. In fact, one has ${lim}_{u\to {0}^{+}}\frac{g(u)}{u}={lim}_{u\to {0}^{+}}\frac{{u}^{4}+1}{u}=+\mathrm{\infty}$ and (AR) is satisfied as a simple computation shows. Moreover, one has ${\lambda}^{\ast}=\frac{1}{2(ba){\parallel \alpha \parallel}_{1}}{sup}_{c>0}\frac{{c}^{2}}{G(c)}=\frac{1}{2}$.
Moreover, taking into account that $G(u)=u\frac{{\sum}_{i=0}^{7}\frac{7!}{i!}{u}^{i}}{{e}^{u}}+7!$, by choosing $\overline{c}=2$ and $\overline{d}=9$, one has $\frac{G(2)}{{2}^{2}}<\frac{1}{4}\frac{G(9)}{{9}^{2}}$ and $\frac{2\phantom{\rule{0.2em}{0ex}}{9}^{2}}{G(9)}<\frac{1}{10}<\frac{{2}^{2}}{2G(2)}$.
So, it is clear that any nonnegative solution u of problem (${P}_{g}$) is also a solution of problem (${P}_{E}$).
Declarations
Authors’ Affiliations
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