Abstract:
We study existence of solutions for the fractional Laplacian equation in , , with critical exponent , , , where has a potential well and is a lower order perturbation of the critical power . By employing the variational method, we prove the existence of nontrivial solutions for the equation. 1. Introduction In the last 20 years, the classical nonlinear Schr？dinger equation has been extensively studied by many authors [1–10] and the references therein. We just mention some earlier work about it. Brézis and Nirenberg [1] proved that the critical problem with small linear perturbations can provide positive solutions. In [3], Rabinowitz proved the existence of standing wave solutions of nonlinear Schr？dinger equations. Making a standing wave ansatz reduces the problem to that of studying a class of semilinear elliptic equations. Floer and Weinstein [10] proved that Schr？dinger equation with potential and cubic nonlinearity has standing wave solutions concentrated near each nondegenerate critical point of . However, a great attention has been focused on the study of problems involving the fractional Laplacian recently. This type of operator seems to have a prevalent role in physical situations such as combustion and dislocations in mechanical systems or in crystals. In addition, these operators arise in modelling diffusion and transport in a highly heterogeneous medium. This type of problems has been studied by many authors [11–18] and the references therein. Servadei and Valdinoci [11–14] studied the problem where , is an open bounded set of , , with Lipschitz boundary, is a real parameter, and is a fractional critical Sobolev exponent. is defined as follows: Here is a function such that there exists such that and for any . They proved that problem (1) admits a nontrivial solution for any . They also studied the case and , respectively. Felmer et al. [15] studied the following nonlinear Schr？dinger equation with fractional Laplacian: where , , and is superlinear and has subcritical growth with respect to . The fractional Laplacian can be characterized as , where is the Fourier transform. They gave the proof of existence of positive solutions and further analyzed regularity, decay, and symmetry properties of these solutions. In this paper, we consider the following problem: with critical exponent , , , where has a potential well, where is the fractional Laplace operator, which may be defined as is the usual fractional Sobolev space. is a lower order perturbation of the critical power . Now we give our main assumptions. In order to find weak solutions of (5),

Abstract:
We are concerned with the solution of the differential equations of the form: (1) where, = m/n, n 0. Equations of this type arise in the generalized viscoelastic constitutive equations and in fractional Brownian motion. Numerical methods for solution of Eq. (1) are well established, particularly for 0

Abstract:
In this work, we study the following problem. , where ？is the
fractional Laplacian and Ω？is a bounded
domain in R^{N}？with Lipschitz
boundary. g: R→R？is an
increasing locally Lipschitz continuous function. and f∈L^{m}(Ω), . We use Stampacchia’s theorem to study existence of
the solution u

Abstract:
In this paper we study the existence problem for the $p(x)-$Laplacian operator with a nonlinear critical source. We find a local condition on the exponents ensuring the existence of a nontrivial solution that shows that the Pohozaev obstruction does not holds in general in the variable exponent setting. The proof relies on the Concentration--Compactness Principle for variable exponents and the Mountain Pass Theorem.

Abstract:
In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$.

Abstract:
In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the $p(x)-$Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration--compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.

Abstract:
In the paper, we considered the existence and uniqueness of the global solution in the space of continuously differentiable functions for a nonlinear differential equation with the Caputo fractional derivative of general form. Our main method is to derive an integral equation corresponding to the original nonlinear fractional differential equation and to prove their eqivalence. Once we prove the equivalence, then the proof of existence and uniquness of solution can be done by standard way of using Banach fixed point theorem.

Abstract:
We prove, using a fixed point theorem in a Banach algebra, an existence result for a fractional functional differential equation in the Riemann-Liouville sense. Dependence of solutions with respect to initial data and an uniqueness result are also derived.

Abstract:
In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon$ is a positive parameter, $f$ has a subcritical growth, $V$ possesses a local minimum, $N > 2s,$ $s \in (0,1),$ and $ (-\Delta)^{s}u$ is the fractional laplacian.

Abstract:
We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.