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## Concavity maximum principle for viscosity solutions of singular equations

### Nonlinear Differential Equations and Applications NoDEA (2010-10-01) 17: 601-618 , October 01, 2010

We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.

## Non-local heat flows and gradient estimates on closed manifolds

### Journal of Evolution Equations (2009-08-26) 9: 787-807 , August 26, 2009

In this paper, we study two kinds of *L*^{2} norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability, and asymptotic behavior of such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

## Stability of radial symmetry for a Monge-Ampère overdetermined problem

### Annali di Matematica Pura ed Applicata (2008-09-17) 188: 445-453 , September 17, 2008

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.

## The existence of semiclassical states for some p-Laplacian equation with critical exponent

### Acta Mathematicae Applicatae Sinica, English Series (2017-04-01) 33: 417-434 , April 01, 2017

In this paper, we study the existence of semiclassical states for some *p*-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that *ε* ≤ *E*; for any *m* ∈ ℕ, it has m pairs solutions if *ε* ≤ *E*_{m}, where *E*, *E*_{m} are sufficiently small positive numbers. Moreover, these solutions are closed to zero in *W*^{1,p}(ℝ^{N}) as *ε* → 0.

## Radial entire solutions for supercritical biharmonic equations

### Mathematische Annalen (2006-04-01) 334: 905-936 , April 01, 2006

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions *n* ≤ 12 and *n* ≥ 13.

## Convex hull property and maximum principle for finite element minimisers of general convex functionals

### Numerische Mathematik (2013-08-01) 124: 685-700 , August 01, 2013

The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for $$\mathbb{P }_1$$ conforming finite elements on simplicial non-obtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the $$p$$ -Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.

## Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

### Manuscripta Mathematica (2012-11-01) 139: 291-319 , November 01, 2012

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class
$${\mathcal {S}^{\mathfrak {L}_0}}$$
of uniformly elliptic nonlinear equations with 1 < *σ* < 2 (subcritical case) and to their subclass
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
with 0 < *σ* ≤ 1. We show that
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and *C*^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets in each cases.

## Estimates for solutions of quasilinear problems with dead cores

### Zeitschrift für angewandte Mathematik und Physik ZAMP (2003-09-01) 54: 815-821 , September 01, 2003

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense.

## Prescribing curvature problem of Bakry-Émery Ricci tensor

### Science China Mathematics (2013-09-01) 56: 1935-1944 , September 01, 2013

We consider the problem of deforming a metric in its conformal class on a closed manifold, such that the *k*-curvature defined by the Bakry-Émery Ricci tensor is a constant. We show its solvability on the manifold, provided that the initial Bakry-Émery Ricci tensor belongs to a negative cone. Moveover, the Monge-Amp`ere type equation with respect to the Bakry-Émery Ricci tensor is also considered.

## Nonautonomous and non periodic Schrödinger equation with indefinite linear part

### Journal of Fixed Point Theory and Applications (2017-03-01) 19: 17-36 , March 01, 2017

The existence of solution of the nonlinear Schrödinger equation
$$\begin{aligned} \begin{array}{lc} -\Delta u + V(x) u = f(x,u),&\end{array} \end{aligned}$$
is stablished in
$$\mathbb {R}^N$$
, where *V* changes sign and *f* is an asymptotically linear function at infinity, with *V* and *f* non periodic in *x*. Spectral theory, a classical linking theorem and interaction between translated solutions of the problem at infinity are employed.