Abstract
The paper presents results on the solvability and parameter dependence for problems driven by weakly continuous potential operators with continuously differentiable and coercive potential. We provide a parametric version on the existence result to nonlinear equations involving coercive and weakly continuous operators. Applications address a variant of elastic beam equation.
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1 Introduction
In [4] the author gives a very general theorem about the existence of solutions to nonlinear problems which involve some partial linearity:
Theorem 1
Let E be a separable reflexive Banach space. Assume that the operator \(A:E\rightarrow E^{*}\) is (i) weakly continuous, i.e. \(u_{n}\rightharpoonup u_{0}\) in E implies \( A\left( u_{n}\right) \rightharpoonup A\left( u_{0}\right) \) in \(E^{*}\), and (ii) coercive, i.e.
Then for any \(b\in E^{*}\) the equation
has a solution.
Remark 2
It follows from the proof contained in [4] that instead of assuming (i) one can impose a slightly relaxed condition, namely given a weakly convergent sequence \(\left( u_{n}\right) \) with a weak limit \(u_{0}\) it follows for some subsequence \( \left( u_{n_{k}}\right) \) that \(A\left( u_{n_{k}}\right) \rightharpoonup A\left( u_{0}\right) \) in \(E^{*}\). Moreover, we can replace \(\left( u_{n}\right) \) with some bounded sequence. In fact we need to choose another subsequence in Step 4 of the proof of Theorem 1.2 in [4]. Such a remark applies to all other subsequent results and we will not repeat it so that not to complicate the formulation of results.
In this work we aim at putting some further insight into the solvability of nonlinear equations inspired by Theorem 1 by providing:
-
a)
some information about the convergence of Galerkin schemes in Theorem 1;
-
b)
the parametric version of Theorem 1 considering the situation of equation
$$\begin{aligned} A\left( u,v\right) =b, \end{aligned}$$with v from a parameter space Y, and where the dependence of solutions as parameter varies is investigated;
-
c)
the variational counterpart of Theorem 1 (together with its parametric version) in case where A is potential with lack of coercivity but its potential is coercive;
-
d)
applications of the above mentioned results to a fourth order problem being a variant of the elastic beam equation.
Theorem 1 has been revived as of late as an important abstract tool, see for example: [15] and also [14] with some additions in [12].
We would like to mention that problems involving partial linearity of the problem under consideration are very common in the literature and pertain to both second order, see for example [7], and higher order problems among which there appears the boundary value problem connected to the fourth order elastic beam equation with either simply supported or rigidly fastened ends. In this direction there exist a vast research by variational methods pertaining to the use of various multiplicity criteria, like the Ricceri three critical point theorem and also fixed point arguments pertaining to Krasnosel’skiĭ-Guo and the Leggett-Williams fixed point theorems, see [1, 2, 10, 11, 13].
For the background on nonlinear analysis tools uses here we refer to [5] and [8].
The paper is organized as follows. In Sect. 2 we gather abstract results developed in this paper connected to the parametric and variational versions of Theorem 1 as well as some comments and additions. In Sect. 3 we give applications to both non-variational and variational parametric version of Theorem 1 to the fourth order boundary value problem related to the beam equation and containing an unbounded perturbation.
2 Abstract Results
Let us recall that operator \(A:E\rightarrow E^{*}\), where E is a reflexive and separable Banach space, satisfies condition (S) if \( u_{n}\rightharpoonup u_{0} \ \)in\( \ E \ \) and\( \ \left\langle A\left( u_{n}\right) -A\left( u_{0}\right) ,\right. \)\(\left. u_{n}-u_{0}\right\rangle \rightarrow 0\) imply \( u_{n}\rightarrow u_{0}\ \)in E.
Since E is separable it contains a dense and countable set \(\left\{ h_{1},...,h_{n},...\right\} \). Define \(E_{n}\) for \(n\in \mathbb {N} \) as a linear hull of \(\left\{ h_{1},...,h_{n}\right\} \). The sequence of subspaces \(E_{n}\) has the approximation property: for each \(u\in E\) there is a sequence \(\left( u_{n}\right) _{n=1}^{\infty }\) such that \(u_{n}\in E_{n}\) for \(n\in \mathbb {N} \) and \(u_{n}\rightarrow u\). Let \(b\in E^{*}\) be fixed. By \(b_{n}\) we denote the restriction of functional b to space \(E_{n}\). Similarly by \( A_{n}\) we understand the restriction of A to space \(E_{n}\). We call the sequence \(\left( u_{n}\right) \) with \(u_{n}\in E_{n}\) of solutions to
the Galerkin scheme connected with equation
We begin with remarking on the convergence of Galerkin type schemes in Theorem 1:
Corollary 3
Let E be a separable reflexive Banach space. Assume that the operator \(A:E\rightarrow E^{*}\) is
(i) weakly continuous,
and
(ii) coercive
Then for any \(b\in E^{*}\) the equation
has a solution \(u_{0}\in E\) such that \(u_{n}\rightharpoonup u_{0}\) in E, where \(\left( u_{n}\right) \) stands for the Galerkin scheme. In case A satisfies additionally condition (S), we have that \(u_{n}\rightarrow u_{0}\) in E.
Proof
The assertions about the weak convergence of Galerkin type scheme follows directly from the proof contained in [4], while the remark about the norm convergence under condition (S) then follows as in [5, Chapter 6.2]. \(\square \)
Now we proceed to the parametric version of Theorem 1. This relies on a type of uniform coercivity subject to a parameter.
Theorem 4
Assume that Y is a normed space and E is a reflexive and separable Banach space. Assume that \(A:E\times Y\longrightarrow E^{*}\) is an operator satisfying the following conditions: (i) A is (weakly,norm)\(\rightarrow \)weakly continuous, i.e. \( u_{n}\rightharpoonup u_{0}\) in E and \(v_{n}\rightarrow v_{0}\) in Y imply \(A\left( u_{n},v_{n}\right) \rightharpoonup A\left( u_{0},v_{0}\right) \) in \( E^{*}\); (ii) there exists a function \(\rho :[0,\infty )^{2}\longrightarrow \mathbb {R} \) such that
and that
uniformly with respect to y from every bounded interval. Let \(b\in E^{*}\) be fixed. Then for every \(v\in Y\) there exists an element \(u_{v}\in E\) solving the equation
Moreover \(v_{n}\rightarrow v_{0}\) in Y implies up to a subsequence that \( u_{v_{n}}\rightharpoonup u_{0}\) in E with \(A(u_{0},v_{0})=b\).
Proof
For any fixed \(v\in Y\) it follows by Theorem 1 that there exists an element \(u_{v}\in E\) such that \(A(u_{v},v)=b\). Now let us consider a sequence \(\left( v_{n}\right) \subset Y\) norm convergent to some \(v_{0}\in Y\). Then for any \(n\in \mathbb {N}\) it holds
It follows by assumption (ii) that the sequence \(\left( u_{v_{n}}\right) \) is bounded and therefore up to a subsequence which we do not renumber, weakly convergent to some \(u_{0}\in E\). By assumption (i) we see passing to the limit in \(A(u_{v_{n}},v_{n})=b\) that \(A(u_{v_{0}},v_{0})=b\), so the assertion follows. \(\square \)
According to Remark 2 the assumptions may slightly be relaxed. We note that we can rephrase the above result as follows: Given a sequence \(\left( v_{n}\right) \subset Y\) norm convergent to some \(v_{0}\in Y\) we can find a sequence \(\left( S_{n}\right) \) of sets where each \(S_{n}\) consists of solutions to (1) corresponding to \(y_{n}\). Denote by \(S_{0}\) the set of solutions to (1) corresponding to \(v_{0}\). Then any sequence \(\left( u_{n}\right) \) such that \(u_{n}\in S_{n}\) contains a weak cluster point in \(S_{0}\). Thus we would obtain the upper limit of the sequence of sets \(\left( S_{n}\right) \) in the Painleve-Kuratowski sense should we would be able to demonstrate that \( u_{v_{n}}\rightarrow u_{v_{0}}\) in E. In the following result - in which we impose a parametric version of condition (S)- we consider such a situation:
Corollary 5
If in addition to the assumptions of Theorem 4 the following condition about operator A is satisfied: (iii) \(u_{n}\rightharpoonup u_{0}\) in E, \(v_{n}\rightarrow v_{0}\) in Y, and \(\left\langle A\left( u_{n},v_{n}\right) -A\left( u_{0},v_{0}\right) ,u_{n}-u_{0}\right\rangle \rightarrow 0\) imply \(u_{n}\rightarrow u_{0}\) in E, then the conclusion in Theorem 4 is that \( v_{n}\rightarrow v_{0}\) in Y implies \(u_{v_{n}}\rightarrow u_{v_{0}}\) in E.
Proof
From Theorem 4 we get that \(u_{v_{n}}\rightharpoonup u_{v_{0}}\) in E. Since \(A(u_{v_{n}},v_{n})=b\) and \(A(u_{v_{0}},v_{0})=b\), we see that also \(\left\langle A\left( u_{v_{n}},v_{n}\right) -A\left( u_{v_{0}},v_{0}\right) ,u_{n}-u_{v_{0}}\right\rangle \rightarrow 0\). Now, using condition (iii) we obtain the assertion. \(\square \)
Now we proceed to a variational counterpart of Theorem 1 which involves neither monotonicity nor its generalizations. This is why we do not expect to have sequential weak lower semicontinuity of the Euler action functional. For the proof of our next result we will need the celebrated Ekeland Variational Principle in the differential form (see, e.g., [8]) that we recall:
Theorem 6
(Ekeland Variational Principle—differentiable form) Let \( \mathcal {I}:E\rightarrow \mathbb {R}\) be a Gâteaux differentiable functional which is bounded from below and lower semicontinuous. Then there exists a minimizing sequence \((u_n)\) of \(\mathcal {I}\) consisting of almost critical points, i.e., such that \(\mathcal {I}(u_n)\rightarrow \inf _{u\in E}\mathcal {I}(u)\) and \(\mathcal {I}^{\prime }(u_n)\rightarrow 0\).
We state the following result.
Theorem 7
Let E be a separable reflexive Banach space. Let \(b\in E^{*}\) be fixed. Assume that:
(i) the operator \(A:E\rightarrow E^{*}\) is potential with the \(C^{1}\) potential \(\mathcal {A}:E\rightarrow \mathbb {R}\), i.e., \(\mathcal {A}\) is continuously differentiable with \(\mathcal {A}^{\prime }=A\);
(ii) the operator A is weakly continuous;
(iii) the functional \(J:E\rightarrow \mathbb {R}\) given by
is coercive and bounded from below.
Then the equation
has a solution \(u_{0}\) (equivalently, \(J^{^{\prime }}\left( u_{0}\right) =0\) ). Moreover there is a (minimizing) sequence \(\left( u_{n}\right) \) with
Proof
The functional J is continuously differentiable due to (i). By (iii), J is bounded from below, so Theorem 6 can be applied providing a minimizing sequence \(\left( u_{n}\right) \) with \(J^{\prime }(u_n)\rightarrow 0\) in \( E^*\) as \(n\rightarrow \infty \).
We show that J has a critical point solving (2). By the coercivity postulated in (iii), the minimizing sequence \(\left( u_{n}\right) \) is bounded. Through the reflexivity of the space E, passing to a subsequence it can be assumed to be weakly convergent to some \(u_{0}\in E\). Since by (ii) the operator A is weakly continuous, we see that
thus \(J^{\prime }(u_{0})=0\), and the proof is completed. \(\square \)
Remark 8
We emphasize that with the assumptions of Theorem 7 we may not use Theorem 1 since the coercivity of the potential need not imply the coercivity of its differential as seen by the example of the coercive function
whose derivative is not coercive.
Remark 9
Theorem 7 is related to the existence result from [3, Theorem 4], where it is assumed about the operator that it is bounded, coercive and continuous and satisfies some compactness condition related to the one which we employ. Our advantage is again that we do not impose the coercivity on the operator, while we can impose exactly the same compactness condition.
In accordance with Corollary 3 we can also consider the case when the minimizing sequence obtained in Theorem 7 is norm convergent:
Corollary 10
In addition to the assumptions of Theorem 7 impose that operator A satisfies property (S). Then a (minimizing) sequence \( \left( u_{n}\right) \) is norm convergent.
Next, we proceed to formulate a parameter dependent version of Theorem 7 which is in turn a variational counterpart of Theorem 4.
Theorem 11
Assume that Y is a normed space and E is a reflexive Banach space. Let \(b\in E^{*}\) be fixed. Assume that \( A:E\times Y\longrightarrow E^{*}\) is an operator satisfying the following conditions:
-
(i)
for each \(v\in Y\) the operator \(A\left( \cdot ,v\right) :E\longrightarrow E^{*}\) is potential with continuously differentiable potential \(\mathcal {A}\left( \cdot ,v\right) ;\)
-
(ii)
A is (weakly,norm)\(\rightarrow \)weakly continuous;
-
(iii)
there exists a function \(\rho :[0,\infty )^{2}\longrightarrow \mathbb {R }\) such that
$$\begin{aligned} \mathcal {A}\left( u,v\right) -\left\langle b,u\right\rangle \ge \rho (\Vert u\Vert ,\Vert v\Vert )\quad \text {for all }v\in Y\text { and }u\in E, \end{aligned}$$and that
uniformly with respect to y from every bounded interval. Then for every \(v\in Y\) there exists an element \(u_{v}\in E\) solving the equation
Moreover, \(v_{n}\rightarrow v_{0}\) in Y implies \(u_{v_{n}}\rightharpoonup u_{v_{0}}\) in E with \(u_{v_{0}}\) being a solution to (4 ) for \(v=v_{0}\). If we additionally assume condition (iii) from Corollary 5, then \(u_{v_{n}}\rightarrow u_{v_{0}}\) in E.
Proof
For any fixed \(v\in Y\), Theorem 7 yields a critical point \(u_{v}\in E\) of the functional \(u\mapsto \mathcal {A}\left( u\right) -\left\langle b,u\right\rangle \). Now let \(v_{n}\rightarrow v_{0}\) in Y. Then by assumption (iii) it follows that the sequence \(\left( u_{v_{n}}\right) \) with \(u_{v_{n}}\) solution to (4) corresponding to \(v_{n}\) is bounded in E. Therefore one has up to a subsequence that \(u_{v_{n}}\rightharpoonup u_{0}\) in E for some \(u_{0}\in E \). By assumption (ii) we see passing to the limit that
The remaining assertions follow as in Corollary 5. \(\square \)
Remark 12
From Theorem 7 we know that corresponding to \(v_{0}\) there is a critical point of the functional \(u\mapsto \mathcal {A} \left( u,v_{0}\right) -\left\langle b,u\right\rangle \) which solves (4). However, we do not know if the limit solution obtained in Theorem 11 is a minimizer.
We illustrate the insight of our abstract results with the following model problem: given \(v\in Y\) find \(u\in E\) such that
The data in (5) are required to fulfill the hypotheses below.
\((H_{1})\) There are given the Hilbert spaces E, X, and Y and \(E\subset X\) compactly and densely (thus, \(X^{*}\subset E^{*}\)).
\((H_{2})\) The map \(B:E\rightarrow E^{*}\) is continuous, linear, self-adjoint and positive definite, i.e. there is a constant \(c_{0}>0\) such that
for all \(u\in E\) (thus B is strongly monotone).
\((H_{3})\) The map \(F:X\rightarrow X^{*}\) is of potential type with \(F= \mathcal {F}^{\prime }\) for some continuously differentiable functional \( \mathcal {F}:X\rightarrow \mathbb {R}\) satisfying
with constants \(C>0\) and \(\alpha \in [0,2)\).
\((H_{4})\) The map \(G:Y\rightarrow E^{*}\) is continuous and satisfies \( \Vert G(v)\Vert _{E^{*}}\le \varphi (\Vert v\Vert _{Y})\) for all \(v\in Y \) with \(\varphi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) being continuous.
Theorem 13
Assume that the conditions \((H_{1})\)–\((H_{4})\) hold. Let \(b\in E^{*}\) be fixed. Then for every \(v\in Y\) there exists \(u_{v}\in E\) solving problem (5). Moreover, \(v_{n}\rightarrow v_{0}\) in Y implies \(u_{v_{n}}\rightarrow u_{v_{0}}\) in E with \(u_{v_{0}}\) being solution to (5).
Proof
We show that Theorem 11 applies. Let us define the map \(A:E\times Y\longrightarrow E^{*}\) by
From assumptions \((H_{1})\), \((H_{2})\), and \((H_{3})\), we see for every \(v\in Y\) that the map \(A(\cdot ,v)\) in (6) is potential with the potential \( \mathcal {A}\left( \cdot ,v\right) :E\rightarrow \mathbb {R}\) as
so condition (i) in Theorem 11 is verified.
In order to check condition (ii) in Theorem 11, let \(u_{n}\rightharpoonup u_{0}\) in E and \(v_{n}\rightarrow v_{0}\) in Y. The linearity and the continuity of the operator B imply \( Bu_{n}\rightharpoonup Bu_{0}\) in \(E^{*}\), while hypothesis \((H_{1})\) ensures \(u_{n}\rightarrow u_{0}\) in X, thus \(F(u_{n})\rightarrow F(u_{0})\) in \(E^{*}\). As the continuity of G provides \(G(v_{n})\rightarrow G(v_{0})\) in \(E^{*}\), from (6) we find that \(A\left( u_{n},y_{n}\right) \rightharpoonup A\left( u_{0},y_{0}\right) \) in \(E^{*} \). Therefore condition (ii) in Theorem 11 is satisfied.
From (7), \((H_{1})\), \((H_{2})\), \((H_{3})\), and \((H_{4})\), we get the estimate
with a constant \(c_{1}>0\) depending on the embedding and independent of u. Define the function \(\rho :[0,\infty )^{2}\longrightarrow \mathbb {R}\) by setting
Since \(\alpha <2\) and since \(\varphi \) is continuous, it follows that (3) holds true. In addition, we have
This amounts to saying that condition (iii) in Theorem 11 is fulfilled.
Note that a strongly monotone operator B satisfies condition (S) which is not violated by perturbation F. Indeed, if we assume that \( u_{n}\rightharpoonup u_{0}\) in E then \(F\left( u_{n}\right) \rightarrow F\left( u_{0}\right) \) in \(E^{*}\) and
implies that \(\langle Bu_{n}-Bu_{0},u_{n}-u_{0}\rangle \rightarrow 0\). Since \(\langle Bu_{n}-Bu_{0},u_{n}-u_{0}\rangle \ge c_{0}\Vert u_{n}-u_{0}\Vert _{E}^{2}\) we have the assertion.
Consequently, Theorem 11 can be applied to resolve problem (5) obtaining the desired conclusion. \(\square \)
It easily follows from Theorem 13 that we can consider the case of weakly convergent sequence of parameters under some structure assumptions as shown below:
Corollary 14
Assume that the conditions \((H_{1})\)-\((H_{3})\) hold and that \(Y=X\), where X is identified with its dual. Then for every \(v\in X\) there exists \(u_{v}\in E \) solving problem
Moreover, \(v_{n}\rightharpoonup v_{0}\) in X implies \(u_{v_{n}}\rightarrow u_{v_{0}}\) in E with \(u_{v_{0}}\) being solution to (8) corresponding to \(v_{0}\).
Without the assumption about the potentiality of operator F we can consider the direct application of Theorem 1 to problem (5) under the following version of \(\left( H_{3}\right) :\)
\(\left( H_{3}^{^{\prime }}\right) \) The map \(F:X\rightarrow X^{*}\) is continuous and satisfies that
with constants \(C>0\) and \(\alpha \in [0,2)\).
Now we proceed to the formulation of relevant result:
Theorem 15
Assume that the conditions \((H_{1})\), \((H_{2}),\left( H_{3}^{^{\prime }}\right) ,\) \((H_{4})\) hold. Let \(b\in E^{*}\) be fixed. Then for every \( v\in Y\) there exists \(u_{v}\in E\) solving problem (5). Moreover, \( v_{n}\rightarrow v_{0}\) in Y implies \(u_{v_{n}}\rightarrow u_{v_{0}}\) in E with \(u_{v_{0}}\) being solution to (5).
Proof
We follow the lines of the proof of Theorem 13, so define the map \( A:E\times Y\longrightarrow E^{*}\) by (6).
We check that condition (i) in Theorem 7 is satisfied exactly as in the proof of Theorem 13. In order to see condition (ii) holds, we note that from \((H_{1})\), \((H_{2})\), \(\left( H_{3}^{^{\prime }}\right) \), and \((H_{4})\), we get the estimate
with a constant \(c_{1}>0\) depending on the embedding and independent of u.
Consequently, Theorem 7 can be applied to resolve problem (5) obtaining the desired conclusion. \(\square \)
3 Applications
We present applications of our abstract results to non-potential and potential versions of the beam equation extending the method developed in [6].
3.1 Results by Theorem 4
We are interested in the following variant of the elastic beam equation expressed as the fourth order problem with perturbation g and a functional parameter \(v\in L^{2}\left( 0,1\right) \)
where
are functions which are subject to some conditions provided below. We seek weak solutions in the space
normed by
As is the case of the well known space \(H_{0}^{1}\left( 0,1\right) \), the Sobolev and Poincaré inequalities read as follows: for any \( u\in H_{0}^{2}\left( 0,1\right) \) it holds
and
where
Let us recall that \(f:\left[ 0,1\right] \times \mathbb {R}\times \mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) is a Carathéodory function if the following conditions are satisfied:
(i) \(t\mapsto f\left( t,x,y,z\right) \) is measurable on \(\left[ 0,1\right] \) for each fixed \(x,y,z\in \mathbb {R}\),
(ii) \(\left( x,y,z\right) \mapsto f\left( t,x,y,z\right) \) is continuous on \(\mathbb {R}\times \mathbb {R}\times \mathbb {R}\) for a.e. \(t\in \left[ 0,1 \right] .\)
The assumptions are as follows:
- A1:
-
\(g:\left[ 0,1\right] \times \mathbb {R}_{+}\rightarrow \mathbb {R} \) is a continuous function for which there are a constant \(g_{0}\ge 0\) and a function \(g_{1}:\mathbb {R} _{+}\rightarrow \mathbb {R} \) such that \(g\left( t,x\right) \ge g_{1}\left( x\right) \ge g_{0}\) for all \(t\in \left[ 0,1\right] \) and \(x\in \mathbb {R}_+\) and \(\lim _{x\rightarrow \infty }g_{1}\left( x\right) =+\infty \).
- A2:
-
\(f:\left[ 0,1\right] \times \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}\) is a Carathéodory function such that there exist \(a_{1},b_{1}\in L^{2}\left( 0,1;\mathbb {R} _{+}\right) \), \(c_{1}\in L^{1}\left( 0,1\right) \) for which
$$\begin{aligned} \left| f(t,x,y,z)\right| \le a_{1}\left( t\right) \left| x\right| +b_{1}\left( t\right) \left| y\right| +c_{1}\left( t\right) \end{aligned}$$for a.e. \(t\in \left[ 0,1\right] \) and all \(x,y,z\in \mathbb {R}.\)
- A3:
-
There exist \(a,b\in L^{\infty }\left( 0,1; \mathbb {R}_{+}\right) \), \(c\in L^{1}\left( 0,1\right) \) such that
$$\begin{aligned} \pi ^{4}>\left\| a\right\| _{L^{\infty }}+\pi ^{2}\left\| b\right\| _{L^{\infty }} \end{aligned}$$(10)and that for a.e. \(t\in \left[ 0,1\right] \) and all \( x,y,z\in \mathbb {R}\) it holds
$$\begin{aligned} f(t,x,y,z)x\le a\left( t\right) \left| x\right| ^{2}+b\left( t\right) \left| y\right| ^{2}+c\left( t\right) . \end{aligned}$$
We consider weak solutions, namely we say that \(u\in H_{0}^{2}\left( 0,1\right) \) solves (9) in the weak sense provided that
for all \(w\in H_{0}^{2}\left( 0,1\right) \). From A1, A2 we note that the above formula makes sense. Now we demonstrate that any solution to (9) is necessarily bounded.
Lemma 16
Assume that conditions A1, A2, A3 are satisfied. Let \(v\in L^{2}\left( 0,1\right) \) be fixed. Then there is some \(R>0\) such that \(\left\| u\right\| _{H_{0}^{2}}\le R\) and \(\left\| \dot{u} \right\| _{C}\le R\) for every \(u\in H_{0}^{2}\left( 0,1\right) \) which solves problem (9).
Proof
Assume that \(u\in H_{0}^{1}\left( 0,1\right) \) solves problem (9). Testing it with \(w=u\) we have
Then we obtain concerning the left hand side of (11) that
Estimating the right hand side of (11) we have by A3 what follows
Summing up we arrive at
which implies the assertion \(\left\| u\right\| _{H_{0}^{2}}\le R\) since (10) holds. We see that we can take
The remaining assertion follows by the Sobolev inequality. \(\square \)
With \(R>0\) in Lemma 16, we introduce the continuous function \( g_{R}:\left[ 0,1\right] \times \mathbb {R}_{+}\mathbb {\rightarrow }\mathbb {R}\)
We see that for all \(t\in \left[ 0,1\right] \) and \(x\in \mathbb {R}_{+}\),
Consider the following truncated problem with a functional parameter
Before looking for weak solutions to problem (14) we set forth the regularity of the weak solution by means of the higher order regularity in du Bois-Reymond Lemma. We follow the pattern in [6] but rewritten to fit our problem. By [9, Proposition 4.5], we have the following result.
Lemma 17
If \(h\in L^{2}\left( 0,1\right) \) satisfies
for all \(w\in H_{0}^{2}\left( 0,1\right) \), then there exist constants \( c_{0},c_{1}\in \mathbb {R}\) such that \(h\left( t\right) =c_{0}+c_{1}t\) a.e. on \(\left[ 0,1\right] \).
The following regularity result regarding problem (14) is available.
Proposition 18
Assume that conditions A1, A2, A3 are satisfied. Let \(v\in L^{2}\left( 0,1\right) \) be fixed. Then any \(u\in H_{0}^{2}\left( 0,1\right) \) which is a weak solution to (14) is such that \(u,\frac{d}{dt}u,\frac{d^{2}}{dt^{2}}u\), \(\frac{d^{3}}{dt^{3}}u\) are absolutely continuous and \(\frac{d^{4}}{dt^{4}}u\in L^{2}\left( 0,1\right) \), and u satisfies (14) a.e. on \(\left[ 0,1\right] .\)
Proof
Since \(u\in H_{0}^{2}\left( 0,1\right) \) is a weak solution to (14), we see that \(u,\frac{d}{dt}u\) are absolutely continuous. Next, using the definition of the weak solution to (14) and integrating by parts twice, which makes sense due to (13), we see that the following holds for any \(w\in H_{0}^{2}\left( 0,1\right) \)
Now using Lemma 17 and differentiating twice, we obtain the assertion. \(\square \)
We say that a function \(u\in H_{0}^{2}\left( 0,1\right) \) is a classical solution to (14), if it is a weak solution, if it satisfies (14) a.e. on \(\left[ 0,1\right] \) and if \(u,\frac{d}{dt}u,\frac{ d^{2}}{dt^{2}}u\), \(\frac{d^{3}}{dt^{3}}u\) are absolutely continuous and \( \frac{d^{4}}{dt^{4}}u\in L^{2}\left( 0,1\right) \).
Remark 19
Proposition 18 implies that under conditions A1, A2, A3 any weak solution is a classical one.
In order to proceed further we define the operator \(A:H_{0}^{2}\left( 0,1\right) \times L^{2}\left( 0,1\right) \rightarrow \left( H_{0}^{2}\left( 0,1\right) \right) ^{*}\) by
Lemma 20
Under conditions A1, A2, A3, the operator A satisfies the assumptions of Corollary 5.
Proof
Let us define \(A_{1},A_{2}:H_{0}^{2}\left( 0,1\right) \rightarrow \left( H_{0}^{2}\left( 0,1\right) \right) ^{*}\) by
and \(A_{3}:H_{0}^{2}\left( 0,1\right) \times L^{2}\left( 0,1\right) \rightarrow \left( H_{0}^{2}\left( 0,1\right) \right) ^{*}\) by
Then (15) results in
The weak continuity of \(A_{1}\) follows from its linearity and continuity. As shown in [6], \(A_{1}\) is coercive, bounded and satisfies condition (S). Concerning the operator \(A_{2}\), from [6] we know that it is bounded and strongly continuous, thus it is also weakly continuous. The operator \(A_{3}\) is bounded due to assumption A2. We prove that it is (weakly,norm)\(\rightarrow \)weakly continuous. Take a sequence \(\left( u_{n}\right) _{n=1}^{\infty }\) which is weakly convergent to some \(u_{0}\) in \(H_{0}^{2}\left( 0,1\right) \) and a sequence \(\left( v_{n}\right) _{n=1}^{\infty }\) norm convergent to some \(v_{0}\in L^{2}\left( 0,1\right) \). Then both \(\left( u_{n}\right) _{n=1}^{\infty }\) and \(\left( \dot{u} _{n}\right) _{n=1}^{\infty }\) are norm convergent in \(L^{2}\left( 0,1\right) \). Therefore by assumption A2 it holds for a.e. \(t\in \left[ 0,1 \right] \) that
for some function \(g\in L^{1}\left( 0,1\right) \). Hence we can apply the Lebesgue Dominated Convergence Theorem reaching the continuity claim.
Now we focus on the uniform coercivity. Arguing as for (12) we get for all \(u\in H_{0}^{2}\left( 0,1\right) \) and \(v\in L^{2}\left( 0,1\right) \) that
which proves the thesis. \(\square \)
From Lemma 20, Remark 19 and Corollary 5 it follows that:
Theorem 21
Assume that conditions A1, A2, A3 are satisfied. Let \(\left( v_{n}\right) _{n=1}^{\infty }\) be a sequence of parameters which is norm convergent to some \(v_{0}\) in \(L^{2}\left( 0,1\right) \). Then for each \(n\in \mathbb {N}\cup \left\{ 0\right\} \) there is at least one classical solution \(u_{n}\) to problem (14) corresponding to \(v_{n}\). Moreover, there is subsequence of \(\left( u_{n}\right) \) convergent weakly to \(u_{0}\) in \(H_{0}^{2}\left( 0,1\right) \).
We observe that any solution to problem (14) solves in fact (9). This is true because of the choice of \(R>0\) in (13) complying with Lemma 16.
We can state the main existence result of this subsection.
Theorem 22
Assume that conditions A1, A2, A3 are satisfied. Let \(\left( v_{n}\right) _{n=1}^{\infty }\) be a sequence of parameters which is norm convergent to some \(v_{0}\) in \(L^{2}\left( 0,1\right) \). Then for each \(n\in \mathbb {N}\cup \left\{ 0\right\} \) there is at least one classical solution \( u_{n}\) to problem (9) corresponding to \(v_{n}\). Moreover, there is subsequence of \(\left( u_{n}\right) \) convergent weakly to \(u_{0}\) in \( H_{0}^{2}\left( 0,1\right) \).
3.2 Results by Theorem 11
Now we consider a potential version of (9), specifically the Dirichlet problem
with a functional parameter \(v\in L^{2}\left( 0,1\right) \) and functions \(g: \left[ 0,1\right] \times \mathbb {R}_{+}\mathbb {\rightarrow }\mathbb {R}\) satisfying A1 and \(f:\left[ 0,1\right] \times \mathbb {R}\times \mathbb {R}\mathbb {\rightarrow }\mathbb {R}\) that is subject to the following conditions where \(F:\left[ 0,1\right] \times \mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) is defined by
- A4:
-
\(f:\left[ 0,1\right] \times \mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) is a Carathéodory function such that there exist \(a_{1}\in L^{\infty }\left( 0,1\right) \) and \(b_{1}\in L^{2}\left( 0,1\right) \) for which
$$\begin{aligned} \left| f(t,x,z)\right| \le a_{1}\left( t\right) \left| x\right| +b_{1}\left( t\right) \end{aligned}$$for a.e. \(t\in \left[ 0,1\right] \) and all \(x,z\in \mathbb {R}.\)
- A5:
-
There exist \(a\in L^{\infty }\left( 0,1\right) \) and \(b\in L^{1}\left( 0,1\right) \) such that
$$\begin{aligned} \frac{1}{2}-\frac{\Vert a\Vert _{L^{\infty }}}{\pi ^4}>0, \end{aligned}$$(17)and for a.e. \(t\in \left[ 0,1\right] \) and all \(x,z\in \mathbb {R}\) it holds
$$\begin{aligned} F(t,x,z)\le a\left( t\right) \left| x\right| ^{2}+b\left( t\right) . \end{aligned}$$
We will follow the same pattern as in the results of the previous subsection with the necessary changes arising from the fact that now the associated nonlinear operator is not coercive. For the same reason we cannot invoke [6].
Our goal is to develop a variational approach for problem (16). Let us fix parameter \(v\in L^{2}\left( 0,1\right) \). Notice that assumption A1 does not guarantee that the term
can define a potential operator on \(H_{0}^{2}\left( 0,1\right) \). To overcome this difficulty we set
which is a positive number due to (17), and define the cut-off function \(g_{R}\) by formula (13). It follows from [6] that (18) with \(g_{R}(t,\cdot )\) in place of \(g(t,\cdot )\) defines a potential operator with the potential
Instead of problem (16) we consider the problem
whose weak solutions are exactly the critical points of the Euler action integral \(J_{v}:H_{0}^{2}\left( 0,1\right) \rightarrow \mathbb {R}\) given by
with
The next results establish properties of the functional \(J_{v}\) in (21).
Lemma 23
Assume that conditions A1, A4 are satisfied. Let \(v\in L^{2}\left( 0,1\right) \) be fixed. The functional \(J_{v}:H_{0}^{2}\left( 0,1\right) \rightarrow \mathbb {R}\) is continuously differentiable with the differential at any \(u\in H_{0}^{2}(0,1)\) given by
Proof
The conclusion is achieved by standard arguments that we omit here. \(\square \)
Lemma 24
Under assumptions A1, A4 the operator \(J_{v}^{\prime }:H_{0}^{2}(0,1)\rightarrow \left( H_{0}^{2}(0,1)\right) ^{*}\) expressed in (22) is weakly continuous.
Proof
This is the consequence of the results in [6] (see also the proof of Lemma 20). \(\square \)
Lemma 25
Assume that conditions A1, A4, A5 are satisfied. Let \( v\in L^{2}\left( 0,1\right) \) be fixed. Then the functional \(J_{v}\) in (21) is coercive and bounded from below. Moreover, for the number \(R>0\) introduced in (19), we have that \(\left\| u\right\| _{H_{0}^{2}}\le R\) and \(\left\| \dot{u}\right\| _{C}\le R\) whenever \( u\in H_{0}^{2}\left( 0,1\right) \) is a global minimizer of \(J_{v}\) in (21).
Proof
We see by hypothesis A1 that
Using hypothesis A5 we obtain
for all \(u\in H_{0}^{2}(\Omega )\). The preceding estimates show for any \( u\in H_{0}^{2}(\Omega )\) that
By (17) we have \(\frac{1}{2}-\frac{\Vert a\Vert _{L^{\infty }}}{\pi ^{4}}>0\), so the functional \(J_{v}\) is coercive.
Let now \(u\in H_{0}^{1}\left( 0,1\right) \) be a solution to problem (16) which is a global minimizer of \(J_{v}\) in (21). Then we obtain
which implies that
and therefore we can take R indicated in (19). \(\square \)
We are led to the following existence result.
Proposition 26
Assume that conditions A1, A4, A5 are satisfied. Then for every \(v\in L^{2}(0,1)\) there exists a classical solution to problem (16).
Proof
Lemmas 23 and 24 enable us to apply Theorem 7 ensuring the existence of a classical solution \(u_R\) to problem (20). This is valid because the operator \(J_{v}^{\prime }\) in ( 22) satisfies condition (S) (see [6, Theorem 2] ). By Lemma 25 we know that \(u_R\) is a classical solution to problem (16), too. \(\square \)
We can state the main existence result regarding the dependence on parameters.
Theorem 27
Assume that conditions A1, A4, A5 are satisfied. Let \(\left( v_{n}\right) _{n=1}^{\infty }\) be a sequence of parameters which is norm convergent to some \(v_{0}\in L^{2}\left( 0,1\right) \). Then for each \(n\in \mathbb {N}\cup \left\{ 0\right\} \) there is at least one classical solution \( u_{n}\) to problem (16) corresponding to \(v_{n}\). Moreover, there is a subsequence of \(\left( u_{n}\right) \) norm convergent to \(u_{0}\).
Proof
We show that the assumptions of Theorem 11 are satisfied. Condition (i) has been verified in Lemma 23, while condition (iii) follows from Lemma 24. The proof of condition (ii) can be done arguing with the operator \(A:H_{0}^{2}\left( 0,1\right) \times L^{2}\left( 0,1\right) \rightarrow \left( H_{0}^{2}\left( 0,1\right) \right) ^{*}\) defined in (15) as in the proof of Lemma 20. Note that operator \(J_{v}^{\prime }:H_{0}^{2}(0,1)\rightarrow \left( H_{0}^{2}(0,1)\right) ^{*}\) defined in (22) satisfies condition (S). Application of Theorem 11 completes the proof. \(\square \)
3.3 Final Comments and Examples
Concerning the concrete models about the fourth order boundary value problem connected with the beam equation, in the sources mentioned in the Introduction the authors mainly considered, as we do here, rigidly fastened beams, i.e. fourth order equation
pertaining to boundary conditions
or simply supported beams, i.e. the Eq. (23) with conditions
are considered. Equation (23) is a simplified version of the following one
with suitable assumptions placed on f and where \(E:\left[ 0,1\right] \rightarrow R\) is Young’s modulus of elasticity for the beam, \(I:\left[ 0,1 \right] \rightarrow R\) is the moment of inertia of cross section of the beam and w is the load density (force per unit length of a beam). It is usually assumed that that \(w\left( t\right) >0\), \(E\left( t\right) \ge E_{0}>0\), \( I\left( t\right) \ge I_{0}>0\) for \(t\in \left[ 0,1\right] \) and that E, \( I,w\in L^{\infty }\left( 0,1\right) \). Connected to this model we have the following direct result about the existence and continuous dependence on parameters:
Theorem 28
Assume that conditions A4, A5 are satisfied. Let \(\left( v_{n}\right) _{n=1}^{\infty }\) be a sequence of parameters which is norm convergent to some \(v_{0}\in L^{2}\left( 0,1\right) \). Then for each \(n\in \mathbb {N}\cup \left\{ 0\right\} \) there is at least one classical solution \( u_{n}\) to problem
corresponding to \(v_{n}\). Moreover, there is subsequence of \(\left( u_{n}\right) \) norm which is convergent to \(u_{0}\).
We note that in case functions E, I are not constant we cannot apply Theorem 13 here.
Now we provide some examples of nonlinear terms which satisfy our assumptions.
Example 29
Concerning the nonlinear perturbation \(g:\left[ 0,1\right] \times \mathbb {R} _{+}\mathbb {\rightarrow } \mathbb {R} \) we may consider the following unbounded from above function
which is bounded from below.
Example 30
Concerning the nonlinear term \(f: \mathbb {R} \times \mathbb {R} \mathbb {\rightarrow } \mathbb {R} \) satisfying A2, A3 we may consider the following function (where we drop the dependence on t for clarity)
which satisfies the required growth conditions with \(0<a<\pi ^{4}-\pi ^{2}\).
Example 31
Concerning the nonlinear term \(f: \mathbb {R} \times \mathbb {R} \mathbb {\rightarrow } \mathbb {R} \) satisfying conditions A4, A5 we propose the function related to Remark 8, namely we put
and therefore
where \(a\in \left( 0,\frac{\pi ^{4}}{2}\right) \).
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Andrzejczak, G., Galewski, M. & Motreanu, D. Existence Theorems for Parameter Dependent Weakly Continuous Operators with Applications. Results Math 79, 160 (2024). https://doi.org/10.1007/s00025-024-02189-1
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DOI: https://doi.org/10.1007/s00025-024-02189-1