Indefinite Weight Research Articles

On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].

Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria

This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. The authors employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an O(1/N)-Nash equilibrium for the obtained decentralized strategies.

Three positive nonconstant radial solutions of nonlinear Neumann problems with indefinite weight

In this article, we consider the global behavior of components of positive nonconstant radial solutions for the Neumann problem with indefinite weight where is a parameter, is the unit open ball, and satisfying is the sign-changing function. We determine the intervals of λ in which the above problem has one, two or three positive nonconstant radial solutions by using the directions of a bifurcation.

Stability analysis of a stage‐structure model with spatial heterogeneity

A stage‐structure competition‐diffusion model with spatial heterogeneity is investigated in this paper. Under some suitable assumptions, the existence of spatially non‐homogeneous steady‐state solutions is established by investigating eigenvalue problems with indefinite weight and employing Lyapunov‐Schmidt reduction. The stability of spatially nonhomogeneous steady‐state solutions is obtained by analyzing the set of the spectrum of the associated infinitesimal generator. In particular, the global stability of the steady‐state solutions is described in weak competition case.

Existence of nonnegative solutions for discrete Robin boundary value problems with sign-changing weight

In this paper,~we are concerned with the following discrete problem first $$\left\{ \begin{array}{ll} -\Delta^{2}u(t-1)=\lambda p(t)f(u(t)), &t\in[1,N-1]_{\mathbb{Z}},\\ \Delta u(0)=u(N)=0,\\ \end{array} \right. $$ where $N>2$~is an integer,~$\lambda>0$~is a parameter,~$p:[1,N-1]_{\mathbb{Z}}\rightarrow\mathbb{R}$~is a sign-changing function,~$f:[0,+\infty)\rightarrow[0,+\infty)$~is a continuous and nondecreasing function.~$\Delta u(t)=u(t+1)-u(t)$,~$\Delta^{2}u(t)=\Delta(\Delta u(t))$.~By using the iterative method and Schauder's fixed point theorem,~we will show the existence of nonnegative solutions to the above problem. Furthermore, we obtain the existence of nonnegative solutions for discrete Robin systems with indefinite weights.

Stabilization analysis for Markov jump systems with multiplicative noise and indefinite weight costs

Stabilization analysis for Markov jump systems with multiplicative noise and indefinite weight costs

On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

Existence of solutions for critical fractional p-Laplacian equations with indefinite weights

This article concerns the critical fractional p-Laplacian equation with indefinite weights $$ (-\Delta_p)^su=\lambda g(x)|u|^{p-2}u+h(x)|u|^{p_s^*-2}u \quad \text{in }\mathbb{R}^N, $$ where \(0<s<1<p<\infty\), \(N>sp\) and \(p_s^*=Np/(N-sp)\), the weight functions \(g\) may be indefinite, and \(h\) changes sign. Specifically, based on the results of asymptotic estimates for an extremal in the fractional Sobolev inequality and the discrete spectrum of fractional p-Laplacian operator, we establish an existence criterion for a nontrivial solution to this problem.
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S-Shaped Connected Component of Positive Solutions for a Minkowski-Curvature Dirichlet Problem with Indefinite Weight

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter $$\lambda $$ . In particular, we obtain the existence of at least three positive solutions for $$\lambda $$ being in a certain interval.

Formula omitted] bounds of eigenfunctions with an indefinite weight function

This paper is concerned with the L∞ bounds of eigenfunctions to the eigenvalue problem Δu=−λm(x)uin Ω,u=0on ∂Ω,where Ω is a bounded domain and m(x) is an indefinite weight function. For the Dirichlet eigenvalue problem when m(x)≡1, the L∞ bound of eigenfunction is obtained by classical heat kernel estimate. However, the case is quite different for the weighted problem, additional difficulty appears if m(x) is indefinite. In this paper, we investigate the L∞ bounds of eigenfunctions by the means of comparison argument and fundamental solution of heat equation.

Recovering Differential Operators on the Half-Line with Indefinite Discontinuous Weights

Non-self-adjoint second-order differential operators on the half-line with indefinite discontinuous weights are studied. Properties of spectral characteristics are established and inverse problems of recovering operators from their spectral characteristics are investigated. For these class of nonlinear inverse problems algorithms for constructing the global solutions are developed, and uniqueness theorems are proved

Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight.

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive-contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré-Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré-Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Optimal Ergodic Control of Linear Stochastic Differential Equations with Quadratic Cost Functionals Having Indefinite Weights

Optimal Ergodic Control of Linear Stochastic Differential Equations with Quadratic Cost Functionals Having Indefinite Weights

Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

In this paper, we consider bifurcation from zero or infinity of nontrivial solutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontrivial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line R × {∞} (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight

<p style='text-indent:20px;'>We prove the existence of infinitely many sign changing radial solutions for a <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary.

Nodal solutions of weighted indefinite problems

This paper analyzes the structure of the set of nodal solutions, i.e., solutions changing sign, of a class of one-dimensional superlinear indefinite boundary value problems with indefinite weight functions in front of the spectral parameter. Quite surprisingly, the associated high-order eigenvalues may not be concave as is the case for the lowest one. As a consequence, in many circumstances, the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried out is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminates the analysis.

Inverse Problems for Differential Operators with Indefinite Discontinuous Weights

Non-self-adjoint second-order differential operators on a finite interval with indefinite discontinuous weights are studied. Properties of spectral characteristics are established and inverse problems of recovering operators from their spectral characteristics are investigated. For these class of nonlinear inverse problems algorithms for constructing the global solutions are developed, and uniqueness theorems are proved.

On a Class of Critical p-Biharmonic Kirchhoff Type Problems with Indefinite Weights

Using the genus theory introduced by Krasnoselskii and a variant of the mountain pass theorem due to Rabinowitz [24], we study the existence of solutions for the following Kirchhoff type problem: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \int _\Omega |\Delta u|^p\,\mathrm{d}x\right) \Delta \Big (|\Delta u|^{p{-}2}\Delta u\Big ) {=} \lambda |u|^{p^{**}{-}2}u{+}a(x)|u|^{p{-}2}u{+}f(x,u), \,\, x\in \Omega ,\\ u = \frac{\partial u}{\partial \nu } = 0,\,\,x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^N$$ ( $$N\ge 3$$ ) with $$C^2$$ boundary, $$1<p<\frac{N}{2}$$ , $$p^{**} = \frac{Np}{N-2p}$$ is the critical exponent, $$\Delta $$ is the Laplace operator and $$\frac{\partial u}{\partial \nu }$$ is the outer normal derivative, $$\lambda $$ is a positive parameter, $$M: [0,+\infty ) \rightarrow \mathbb {R}$$ and $$f: \overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$$ are continuous functions, and $$a\in L^\infty (\Omega )$$ is a weight function.

Non-real eigenvalues of a class of fourth order discontinuous differential operators with indefinite weight function

The spectral theory of differential operators is one of the basic problems of differential operator theory. It has been an important tool in quantum mechanics and other fields. For the Sturm-Liouville Problems with weighted functions, the spectral theory of right-definite problems with self-adjoint boundary condition has been accomplished. But the spectral structure of indefinite problems, is quite different from and more complicated than that of right-definite problems. Compare with second order case, much less known for the fourth order indefinite differential operators. The present paper discusses a class of fourth order discontinuous differential operators. We obtained the estimate on the non-real eigenvalues, under separated boundary condition and the condition that weighted functions change signs for only once or any times.

Bounds of Non-real Eigenvalues for Indefinite p-Laplacian Problems

The present paper deals with the non-real eigenvalues of one dimensional p-Laplacian problem with an indefinite weight. The upper bounds on the imaginary parts and the absolute values of non-real eigenvalues for this indefinite p-Laplacian problem with Dirichlet boundary conditions are obtained through a new method, partly inspired by the estimates obtained in Kikonko and Mingarelli (J Differ Equ 261:6221–6232, 2016).