Abstract:
We prove the existence of positive continuous solutions to the nonlinear fractional system $$displaylines{ (-Delta|_D) ^{alpha/2}u+lambda g(.,v) =0, cr (-Delta|_D) ^{alpha/2}v+mu f(.,u) =0, }$$ in a bounded $C^{1,1}$-domain $D$ in $mathbb{R}^n$ $(ngeq 3)$, subject to Dirichlet conditions, where $0

Abstract:
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}\xi^{p}\leq f\left(\xi\right) \leq k_{2}\xi^{p}$ for all $\xi\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-\Delta u=m\left(x\right) f\left(u\right) $ in $\Omega$, $u=0$ on $\partial\Omega$.

Abstract:
We study nonlinear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in the (possibly nonlinear) Neumann boundary conditions. We provide, for bounded domains, several results on the existence, uniqueness and properties of this ergodic constant.

Abstract:
Let $D$ be an unbounded domain in $mathbb{R}^{n}$ ($ngeq 2$) with a nonempty compact boundary $partial D$. We consider the following nonlinear elliptic problem, in the sense of distributions, $$displaylines{ Delta u=f(.,u),quad u>0quad hbox{in }D,cr uig|_{partial D}=alpha varphi ,cr lim_{|x|o +infty }frac{u(x)}{h(x)}=eta lambda , }$$ where $alpha ,eta,lambda $ are nonnegative constants with $alpha +eta >0$ and $varphi $ is a nontrivial nonnegative continuous function on $partial D$. The function $f$ is nonnegative and satisfies some appropriate conditions related to a Kato class of functions, and $h$ is a fixed harmonic function in $D$, continuous on $overline{D}$. Our aim is to prove the existence of positive continuous solutions bounded below by a harmonic function. For this aim we use the Schauder fixed-point argument and a potential theory approach.

Abstract:
In this paper we prove global existence for solutions of the Vlasov-Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary.

Abstract:
We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schr\"odinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-supercritical regime. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L^{p+1}-norm among functions having prescribed L^2 and H^1_0-norm.

Abstract:
The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an energy-minimal diffeomorphism. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.

Abstract:
This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem in subject to some Dirichlet conditions, where is a regular domain in ？ with compact boundary. The nonlinearity is nonnegative continuous and the potential belongs to some Kato class . So we prove the existence of a positive continuous solution depending on by the use of a potential theory approach. 1. Introduction In this paper, we study the existence of positive bounded solution of semilinear elliptic problem where is a -domain in with compact boundary, and are fixed nonnegative constants such that , and when is bounded. The parameter is nonnegative, and the function is nontrivial nonnegative and continuous on . Numerous works treated semilinear elliptic equations of the type For the case of nonpositive function , many results of existence of positive solutions are established in smooth domains or in , for instance, see [1–5] and the references therein. In the case where changes sign, many works can be cited, namely, the work of Glover and McKenna [6], whose used techniques of probabilistic potential theory for solving semilinear elliptic and parabolic differential equations in . Ma and Song [7] adapted the same techniques of those of Glover and McKenna to elliptic equations in bounded domains. More precisely, the hypotheses in [6, 7] require in particular that and on each compact, there is a positive constant such that . In [8], Chen et al. used an implicit probabilistic representation together with Schauder's fixed point theorem to obtain positive solutions of the problem ( ). In fact, the authors considered a Lipschitz domain in , with compact boundary and imposed to the function to satisfy on , where is nonnegative Borel measurable function defined on and the potentials are nonnegative Green-tight functions in . In particular, the authors showed the existence of solutions of ( ) bounded below by a positive harmonic function. In [9], Athreya studied ( ) with the singular nonlinearity , , in a simply connected bounded -domain in . He showed the existence of solutions bounded below by a given positive harmonic function , under the boundary condition , where is a constant depending on , , and . Recently, Hirata [20] gave a Chen-Williams-Zhao type theorem for more general regular domains . More precisely, the author imposed to the function to satisfy where the potential belongs to a class of functions containing Green-tight ones. We remark that the class of functions introduced by Hirata coincides with the classical Kato class introduced for

Abstract:
We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

Abstract:
We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.