Existence results for a fourth-order elastic beam equation via the variational approach
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Existence results for a fourth-order elastic beam equation via the variational approach Hadi Haghshenas1 · Ghasem A. Afrouzi1 Received: 24 November 2018 / Accepted: 12 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this work, we are concerned with the existence results for a fourth-order elastic beam equation. The proof of the main results is based on the critical point theory. Keywords Fourth-order equations · Nonlinear boundary conditions · Variational methods · Critical point theory Mathematics Subject Classification 34B15 · 58E05
1 Introduction Fourth-order boundary value problems arises in the study of deflections of elastic beams on nonlinear elastic foundations, and the existence results of at least one solution, or multiple solutions, or even infinitely many solutions for these problems, have been established by many authors using fixed point theorems, lower and upper solutions methods, and critical point theory (see, for example, [1,2,4,7,13] and the references therein). In this paper, based on two critical point theorems, we consider the elastic beam equation with nonlinear boundary conditions: ⎧ (iv) ⎨ u (t) + Au (t) + Bu(t) = λ f (t, u(t)), t ∈ [0, 1], (1.1) u(0) = u (0) = 0, ⎩ u (1) = 0, u (1) = g(u(1)), where λ is a positive parameter, and A, B are two real constants; Moreover, f : [0, 1] × R → R is L 1 -Caratheodory, and g ∈ C(R) is a real function. When the boundary conditions are nonlinear, fourth-order equations can model beams resting on elastic bearings located in their extremities; (see, for instance, [3,8,9,11,12] and
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Hadi Haghshenas [email protected] Ghasem A. Afrouzi [email protected]
1
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
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H. Haghshenas , G. A. Afrouzi
the references therein). In [11] the author obtains the existence of infinitely many solutions for problem (1.1) with A = B = 0 and an other control parameter μ ≥ 0. Also, this problem have been considered in [8,12] under the assumptions A = B = 0, λ = 1. Our main results here motivated by the papers [5,6].
2 Preliminaries Throughout this paper, the following hypotheses are needed. Let A B A B max 2 , − 4 , 2 − 4 < 1. π π π π (for example, if A ≤ 0 and B ≥ 0, then (2.1) is true). Moreover, put A B A B σ = max 2 , − 4 , 2 − 4 , 0 , π π π π and √ δ = 1 − σ.
(2.1)
(2.2) (2.3)
The variational formulation of problem (1.1) is based on the function space X = {u ∈ W 2,2 ([0, 1]) : u(0) = u (0) = 0}, where W 2,2 ([0, 1]) is the Sobolev space of all functions u : [0, 1] → R such that u and its distributional derivative u are absolutely continuous and u belongs to L 2 ([0, 1]). Then X is a Hilbert space equipped with the inner product and usual norm 1 < u, v >= u (t)v (t)dt, u = u L 2 ([0,1]) . 0
For all u ∈ X , we define u X =
1
|u (t)|2 − A|u (t)|2 + B|u(t)|2 dt
1 2
.
(2.4)
0
Since A, B satisfy (2.1), it is straightforward to ver
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