Existence and multiplicity for Hamilton-Jacobi-Bellman equation

<p style='text-indent:20px;'>This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded regular domain with <inline-formula><tex-math id="M4">\begin{document}$ N\geq3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document}</tex-math></inline-formula> are general Hamilton-Jacobi-Bellman operators, <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is a real parameter. By using bifurcation theory, we determine the range of parameter <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the above problem which has one or multiple constant sign solutions according to the behaviors of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>, and whether <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies the signum condition <inline-formula><tex-math id="M12">\begin{document}$ f(s)s&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ s\neq0 $\end{document}</tex-math></inline-formula>.</p>

A geometrical criterion for nonexistence of constant-sign solutions for some third-order two-point boundary value problems

We give a simple geometrical criterion for the nonexistence of constant-sign solutions for a certain type of third-order two-point boundary value problem in terms of the behavior of nonlinearity in the equation. We also provide examples to illustrate the applicability of our results.

Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

Constant Sign Solutions to Linear Fractional Integral Problems and Their Applications to the Monotone Method

This manuscript provides some results concerning the sign of solutions for linear fractional integral equations with constant coefficients. This information is later used to prove the existence of solutions to some nonlinear problems, together with underestimates and overestimates. These results are obtained after applying suitable modifications in the classical process of monotone iterative techniques. Finally, we provide an example where we prove the existence of solutions, and we compute some estimates.

Unilateral Global Bifurcation, Half-Linear Eigenvalues and Constant Sign Solutions for a Fractional Laplace Problem

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation [Formula: see text] with nondifferentiable nonlinearity [Formula: see text]. It shows that there are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text], and two unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of this unilateral global bifurcation results, we present the existence of the principal half-eigenvalues of the half-linear fractional eigenvalue problem. Finally, we deal with the existence of constant sign solutions for a class of fractional nonlinear problems. Main results of this paper generalize the known results on classical Laplace operators to fractional Laplace operators.