Narrow Results

This paper proves the existence and regularity of weak solutions for a class of mixed local–nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent.

This paper is concerned with an optimization problem related to the pseudo p-Laplacian eigenproblem, with Robin boundary conditions. The principal eigenvalue is minimized over a rearrangement class generated by a fixed positive function. Existence and optimality condition are proved. The popular case where the generator is a characteristic function is also considered. In this case the method of domain derivative is used to capture qualitative features of the optimal solutions.

The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -Δ_pu = λk(x)u^q ± h(x)u^σ if x ∈ Ω, subject to the Dirichlet condition u = 0 on ∂Ω. In the proof of our results we use the sub-solution, super-solution methods and variational arguments.

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C^∞ metric.

The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs.26(1/2) (2013) 105–118]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper.

The aim of this paper is to present a convex curve evolution problem which is determined by both local (curvature $\kappa$) and global (area $A$) geometric quantities of the evolving curve. This flow will decrease the perimeter and the area of the evolving curve and make the curve more and more circular during the evolution process. And finally, as $t$ goes to infinity, the limiting curve will be a finite circle in the $C^{\infty }$ metric.

In this paper, we introduce two $1/{\kappa }^n$-type ($n\ge 1$) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. Finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

The notion of horizontal energy minimizers between C-C spaces is introduced. We prove the existence of such energy minimizers when the domain is a C², noncharacteristic bounded open set in a C-C space and the target is a C-C space of Carnot type.

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type

We consider the Cauchy problem in R^N × R₊ with arbitrary singular initial data u₀ ≠ 0 such that u₀(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and , this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ₀, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation

This paper studies the existence of positive solutions for a class of singular boundary value problems of p-Laplacian. By using Global Continuation Theorem and the fixed point index technique, criteria of the existence, multiplicity and nonexistence of positive solutions are established.

We study existence of solutions to

This paper deals with a non-local curve evolution problem in the plane which will increase both the length of the evolving curve and the area it bounds and make the evolving curve more and more circular during the evolution process. And the limiting shape of the evolving curve will be a finite circle (i.e. a circle with finite radius) as the time t goes to infinity.

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of ℝ^N. The first one, of the form

Let Ω be a smooth bounded domain in ℝ^N with N ≥ 1. In this paper we study the Hardy–Poincaré inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and necessary conditions on the existence of minimizers.

In this paper we present a simplified version of a coercivity inequality due to Gianazza, Savaré, and Toscani [The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal.194 (2009) 133–220]. Then we use the inequality to construct a weak solution to the initial-boundary value problem for the viscous quantum Euler model.

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

We consider the existence of positive singular solutions of the problem

In this paper, we construct a weak solution to the unipolar quantum drift–diffusion equations coupled with initial and mixed boundary conditions in up to four space dimensions. We only assume that the boundary of the domain is Lipschitz and the interface between the Dirichlet boundary and the Neumann boundary is a Lipschitzian hypersurface, and thus the lack of high regularity for solutions of such problems is an issue. The convergence of a sequence of approximate solutions is established via an application of a recent theorem by the author on the logarithmic upper bound for solutions of elliptic equations with the same type of boundary conditions [Logarithmic up bounds for solutions of elliptic partial differential equations, Proc. Amer. Math. Soc.139 (2011) 3485–3490].

Let $({ℳ},g)$ be a smooth compact Riemannian manifold of dimension $N\ge 3$ and let $\mathrm{\Sigma }$ to be a closed submanifold of dimension $1\le k\le N-2$. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $\mathrm{\Sigma }$ within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal.171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.