Indefinite Weight Research Articles

On a positive solution for $(p,q)$-Laplace equation with Nonlinear

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $1<q<p<\infty$ the results are completely different from those for the usua lSteklov eigenvalue problem involving the $p$-Laplacian with indefinite weight, which is retrieved when $\mu=0$. Precisely, we show that when $\mu>0$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

Global Bifurcation of Positive Solutions of Semi-Linear Elliptic Partial Differential Equations with Indefinite Weight

In this paper we consider global bifurcation of solutions of nonlinear eigenvalue problems for second order uniformly elliptic equations with an indefinite weight function and the Dirichlet boundary condition. We show the existence of an unbounded continua contained in the classes of positive and negative functions and bifurcating from the points of the line of trivial solutions corresponding to the principal eigenvalues of the linearized problem.

Some remarks on an eigenvalue problem for an anisotropic elliptic equation with indefinite weight

In this paper, we consider an eigenvalue problem for an anisotropic elliptic equation with indefinite weight, in which the differential operator involves partial derivatives with different variable exponents. Under some suitable conditions on the growth rates of the anisotropic coefficients involved in the problem, we prove some results on the existence and non-existence of a continuous family of eigenvalues by using variational methods.

Three positive solutions of N-dimensionalp-Laplacian with indefinite weight

Three positive solutions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math>-Laplacian with indefinite weight

Weighted Pseudoinversion with Indefinite Weights

We present the definition of weighted pseudoinverse matrices with nondegenerate weights of indefinite sign and study these matrices. The theorems on existence and uniqueness for these matrices are proved. The weighted pseudoinverse matrices with indefinite weights are represented in terms of the coefficients of characteristic polynomials of symmetrized matrices. The decompositions of weighted pseudoinverse matrices into matrix power series and products and their limit representations are obtained. We also propose regularized iterative methods for the determination of these matrices.

Resonant Steklov eigenvalue problem involving the (p, q)-Laplacian

In the present paper, we study the existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator (p, q)-Laplacian with indefinite weights at resonance cases.

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Abstract We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

Reversed S-Shaped Bifurcation Curve for a Neumann Problem

We study the bifurcation and the exact multiplicity of solutions for a class of Neumann boundary value problem with indefinite weight. We prove that all the solutions obtained form a smooth reversed S-shaped curve by topological degree theory, Crandall-Rabinowitz bifurcation theorem, and the uniform antimaximum principle in terms of eigenvalues. Moreover, we obtain that the equation has exactly either one, two, or three solutions depending on the real parameter. The stability is obtained by the eigenvalue comparison principle.

Spectrum theory of second-order difference equations with indefinite weight

In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

Robin problems involving the p(x)-Laplacian

By applying Mountain Pass Lemma and Ekeland’s variational principle, we prove two different situations of the existence of solutions for the following Robin problem−Δp(x)u=λV(x)|u|q(x)−2uinΩ,|∇u|p(x)−2∂u∂ν+β(x)|u|p(x)−2u=0on∂Ω,where Ω⊂RN (N ≥ 2) is a bounded smooth domain, V is an indefinite weight function which can change sign in Ω and p,q:Ω¯→(1,+∞) are continuous functions.

On the Robin Problem with Indefinite Weight in Sobolev Spaces with Variable Exponents

The present paper is concerned with a Robin problem involving an indefinite weight in Sobolev spaces with variable exponents \left\{\begin{alignedat}{2}-\text{ div}(|\nabla u|^{p(x)-2}\nabla u)&amp;=\lambda V(x)|u|^{q(x)-2}u,&amp; \quad x&amp;\in\Omega\\ |\nabla u|^{p(x)-2} \frac{\partial u}{\partial n}+a(x)|u|^{p(x)-2}u&amp;=0.&amp;\quad x&amp;\in\partial\Omega \end{alignedat}\right. By means of the variational approach and Ekeland's principle, we establish that the above problem admits a non-trivial weak solution under appropriate conditions.

Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight] {Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight under Neumann boundary conditions

In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent $\Delta^2_{p(x)} u=\lambda V(x) |u|^{q(x)-2} u$, in a smooth bounded domain,under Neumann boundary conditions, where $\lambda$ is a positive real number, $p,q: \overline{\Omega} \rightarrow \mathbb{R}$, are continuous functions, and $V$ is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.

Positive subharmonic solutions to superlinear ODEs with indefinite weight

Positive subharmonic solutions to superlinear ODEs with indefinite weight

Global Kneser solutions to nonlinear equations with indefinite weight

The paper deals with the nonlinear differential equation \begin{document}$\bigl(a(t)\Phi(x^{\prime})\bigr)^{\prime}+b(t)F(x)=0,\ \ \ t\in\lbrack1,\infty),$ \end{document} in the case when the weight $b$ has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions $x$ such that $x(t)&gt;0, {{x}'}(t)&lt;0$ on the whole closed interval $[1,\infty )$, is considered. Moreover, conditions assuring that these solutions tend to zero as $t\rightarrow\infty$ are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight $b$ is a periodic function or it is unbounded from below.

具变号权函数的多参数二阶微分系统多个正解的存在性

具变号权函数的多参数二阶微分系统多个正解的存在性

On the second eigencurve for the p-laplacian opertor with weight

abstract: In this paper we establish the existence of the second eigencuve of the p-laplacian with indefinite weights, we obtain also their asymptotic behavior and variational formulation.

Mitigation of self‐interference and multi‐user interference in downlink multi‐user MIMO system

Here, the authors develop a method of simultaneous optimisation of precoder and equaliser for the multi-user multiple-input multiple-output (MIMO) system by minimising outage probability. In order to do that, the authors first characterise the signal-to-interference-plus-noise ratio of downlink multi-user MIMO system in the presence of both multi-user interference and self-interference. The closed-form expression for the outage probability is derived in terms of the eigenvalues of an indefinite weight matrix of a quadratic norm. Next, the authors develop an active-set algorithm-based precoder and equaliser by minimising the derived outage probability expression. This work investigates the effect of various system parameters such as the number of users, the number of multi-path channels, the number of transmitting and receiving antenna elements, and the additive noise variance on the outage probability and bit error rate. Simulation results validate the theoretical analysis for multiple system configurations. Lastly, the authors show that the designed precoder and equaliser improves the overall system performance.

Positive subharmonic solutions to nonlinear ODEs with indefinite weight

We prove that the superlinear indefinite equation [Formula: see text] where [Formula: see text] and [Formula: see text] is a [Formula: see text]-periodic sign-changing function satisfying the (sharp) mean value condition [Formula: see text], has positive subharmonic solutions of order [Formula: see text] for any large integer [Formula: see text], thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

On the existence results for a class of singular elliptic system involving indefinite weight functions and asymptotically linear growth forcing term

In this work, we study the existence of positive solutions to the singular system$$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} &amp; \textrm{ in }\Omega,\\-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} &amp; \textrm{ in }\Omega,\\u = v= 0 &amp; \textrm{ on }\partial \Omega,\end{array}\right.$$where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p&gt;1$, $ \Omega \subset R^{n} $ some for $ n &gt;1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0&lt;\alpha&lt; 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.

Nonhomogeneous eigenvalue problem with indefinite weight

This paper, following the theory of partial differential equations on the Orlicz–Sobolev spaces, is mainly concerned with the nonhomogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. The results show that the spectrum of such problems contains a continuous family of eigenvalues.