Half-linear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval
- PDF / 323,896 Bytes
- 10 Pages / 595.276 x 793.701 pts Page_size
- 21 Downloads / 204 Views
ESEARCH
Open Access
Half-linear eigenvalue problem: Limit behavior of the first eigenvalue for shrinking interval Gabriella Bognár1 and Ondˇrej Došlý2* * Correspondence: [email protected] 2 Department of Mathematics and Statistics, Masaryk University, Kotláˇrská 2, Brno, 611 37, Czech Republic Full list of author information is available at the end of the article
Abstract We investigate the limit behavior of the first eigenvalue of the half-linear eigenvalue problem when the length of the interval tends to zero. We show that the important role is played by the limit behavior of ratios of primitive functions of coefficients in the investigated half-linear equation. MSC: 34C10
1 Introduction We consider the eigenvalue problem associated with the half-linear second order differential equation – r(t) x + c(t)(x) = λw(t)(x),
(x) := |x|p– x,
p > ,
()
p being the conjugate exponent with t ∈ (a, b), –∞ < a < b < ∞, r–q , c, w ∈ L (a, b), q = p– of p, and r(t) > , w(t) > . Equation () can be written as the first order system
x = r–q (t)– (u),
u = c(t) – λw(t) (x),
()
– (u) = |u|q– u being the inverse function of , and the integrability assumption on the functions r–q , c, w implies the unique solvability of this system. The original paper of Elbert [], where the existence and uniqueness results are proved via the half-linear version of the Prüfer transformation, deals with continuous functions in (), but the idea of the proof applies without change to integrable coefficients when, as a solution x, u, absolutely continuous functions are considered (which satisfy () a.e. in (a, b)). Along with (), we consider the separated boundary conditions x(a) cosp α – rq– (a)x (a) sinp α = ,
x(b) cosp β – rq– (b)x (b) sinp β = ,
()
where sinp , cosp are the half-linear goniometric functions, which will be recalled in the next section. Motivated by the paper [], where the linear Sturm-Liouville differential equation (which is the special case of ()) is considered, we investigate the limit behavior (as b → a+) of the first eigenvalue of (), () in dependence on α, β. We show that this ©2013 Bognár and Došlý; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Bognár and Došlý Boundary Value Problems 2013, 2013:221 http://www.boundaryvalueproblems.com/content/2013/1/221
Page 2 of 10
limit behavior is, in a certain sense, the same as for an eigenvalue problem when boundary conditions () are associated with an equation with constant coefficients. The investigation of half-linear eigenvalue problems is motivated, among others, by the fact that the partial differential equation with the p-Laplacian (which models, e.g., the flow of non-Newtonian fluids, while the linear case p = corresponds to the Newtonian fluid) –p u + c(x)(u) = λ(u),
p– p
Data Loading...
