Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition

The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{2}$\end{document}. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.


Introduction
In [], Lyapunov proved that, if p(x) is a nonnegative and continuous function and u(x) ∈ C(I, R), a necessary condition for the following boundary value problem: u (x) + p(x)u(x) = , u(x) = , ∀x ∈ I, u(a  ) =  = u(b  ), (.) to have nontrivial solutions is Since Lyapunov's study, because the inequality of (.) plays a key role for the qualitative properties, such as oscillatory and disconjugacy etc., of differential equations' solutions, several authors focused on the inequality of (.). Those authors improved and generalized the inequality of (.) in R. In this work the literature of the one-dimensional case is not studied in detail but it is listed in the references for the interested reader. See [-] and the references cited therein.
In addition to studies in R, several authors [-] have extended the inequality of (.) in R n recently. To the best of our knowledge, it was extended by Cañada where ⊂ R N is a smooth bounded domain with N ≥  and the function a : → R belongs to the set Their main result is as follows.
Theorem A The following statements hold.
In this case, any function a ∈ ∧ ∩ L q ( ) from which β q is attained has one of the following forms: where u is a solution of the problem as follows: The limits lim p→∞ β p and lim p→( N  ) + β p always exist and take the values Here, we also note that in the study Cañada et al. they proved that the relation between the p and N  plays a crucial role. They also considered the equation in (.) with zero Dirichlet boundary condition. They presented similar inequalities at their study. Then others established Lyapunov-type inequalities for different equations with boundary conditions. For more information about the studies in R n , the interested reader can refer to [-] and the references cited therein.
The aim of this paper is to prove a Lyapunov-type inequality for the two-dimensional quasilinear elliptic problem as follows: r(x, y) is a measurable function on , and p (u(x, y)) = |u(x, y)| p- u(x, y) for p > . In addition to this, we note that by a solution of the problem As usual, L p ( ) is a space of Lebesgue measurable functions.

Main results
Now, we give a key lemma as a proof of our main conclusions.
Lemma  Assume that u(x, y) ∈ W ,p ( ), it satisfies the boundary conditions in (.) and to the right-hand side of (.), we get Applying Hölder's inequality to the right hand side of (.) again, we obtain Thereby, the proof of (.) is completed. Similarly we have and Adding (.) and (.), we get Applying Hölder's inequality to the right hand side of (.) and integrating from a  to b  , we have Consequently, the proof of (.) is completed.
Theorem  If u(x, y) ∈ W ,p ( ) is a nontrivial solution of the problem (.), then the following inequality: holds, where q is the Hölder conjugate of p.
Proof Let u(x, y) ∈ W ,p ( ) is a nontrivial solution of the problem (.). Multiplying the equation in (.) by u x and integrating on ,we obtain Then, applying partial integration in b  a  -(|u xy | p- u xy ) y u x dy and using the boundary By taking the absolute value on right hand side of (.), we get Hence, applying Hölder's inequality to the right hand side of (.), we find Now, considering only the second term of right hand side in (.) from Fubini's theorem, we can rewrite the inequality (.) as follows: Hence, using the inequality (.) in (.), we obtain Then, replacing the point of (x, y), which is used in Lemma , with the maximum point of |u(x, y)|, from (.), we get  max u(x, y) Then, using the inequality (.) in the inequality (.), we have Since u(x, y) is a nontrivial solution, we have max |u(x, y)| = . Therefore, we obtain Thus, the proof is completed.
Corollary  Let λ  be the first eigenvalue of the equation that is defined on as follows: where is a domain, which is defined in the beginning of the paper, and with the boundary conditions in (.). Then we have  p+ /(b a  ) p- (b a  ) p r(x, y) q L q ( ) ≤ λ  . (  .   ) Remark  If we take the Dirichlet boundary conditions, which are u(a  , y) =  = u(b  , y) and u(x, a  ) =  = u(x, b  ), instead of the Robin boundary conditions in the problem (.), then we obtain the identical conclusions given above.
Remark  The result, which is obtained in this study, is also the necessary condition for the problem of (.) to have a nontrivial solution.