A priori bounds and existence of positive solutions to a p-kirchhoff equations

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

Extremal solution and Liouville theorem for anisotropic elliptic equations

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^{N} $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) is a bounded domain, and the operator <inline-formula><tex-math id="M4">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as Finsler-Laplacian or anisotropic Laplacian, is defined by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here, <inline-formula><tex-math id="M5">\begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $\end{document}</tex-math></inline-formula> is a convex function of <inline-formula><tex-math id="M7">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}</tex-math></inline-formula>, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if <inline-formula><tex-math id="M8">\begin{document}$ N\leq9 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also concern the Hénon type anisotropic Liouville equation, </p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ \alpha&gt;-2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ F^{0} $\end{document}</tex-math></inline-formula> is the support function of <inline-formula><tex-math id="M12">\begin{document}$ K: = \{x\in\mathbb{R}^{N}:F(x)&lt;1\} $\end{document}</tex-math></inline-formula>. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for <inline-formula><tex-math id="M13">\begin{document}$ 2\leq N&lt;10+4\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ 3\leq N&lt;10+4\alpha^{-} $\end{document}</tex-math></inline-formula> respectively, where <inline-formula><tex-math id="M15">\begin{document}$ \alpha^{-} = \min\{\alpha, 0\} $\end{document}</tex-math></inline-formula>.</p>

An Itô Formula for rough partial differential equations and some applications

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u$∂tu−Atu−f=(Ẋt(x)⋅∇+Ẏt(x))u on $[0,T]\times \mathbb {R}^{d}.$[0,T]×ℝd. To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u↦F(u), where F is C2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F(u) = |u|p with p ≥ 2, but also when Y ≠ 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of Lp-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem:\left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\frac{% f(u)}{u^{q}},&&\displaystyle u>0\text{ in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right.where {(-\Delta)^{s}} denotes the fractional Laplace operator for {s\in(0,1)}, {n>2s}, {q\in(0,1)}, {\lambda>0} and Ω is a smooth bounded domain in {\mathbb{R}^{n}}. Here {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying\lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0.We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when {f=0} , we show some explicit solutions.