A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is...

We consider a class of fourth order elliptic systems which include the Euler-Lagrange equations of biharmonic mappings in dimension 4 and we prove that a weak limit of weak solutions to such systems is again a weak solution to a limit system.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where $N\ge 4;V,K,g$ are periodic in $x_j$ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K\>0$. If $0\<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

In the present paper, we prove the existence and uniqueness of weak solution to a class of nonlinear degenerate elliptic $p$-Laplacian problem with Dirichlet-type boundary condition, the main tool used here is the variational method combined with the theory of weighted Sobolev spaces.

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{ccc}\hfill \overline{DA(x,u(x),Du(x\left)\right)}=& B(x,u(x),Du(x\left)\right),x\in \Omega ,u\left(x\right)=\hfill & \hfill 0,x\in \partial \Omega ,\end{array}$$ where $D$ is a Dirac operator in Euclidean space, $u\left(x\right)$ is defined in a bounded Lipschitz domain $\Omega$ in ${ℝ}^n$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned...

We investigate the class of functions associated with the complex Hessian equation ${\left(d{d}^{c}u\right)}^m\wedge {\omega }^{n-m}=0$.

We study the composite membrane problem in all dimensions. We prove that the minimizing solutions exhibit a weak uniqueness property which under certain conditions can be turned into a full uniqueness result. Next we study the partial regularity of the solutions to the Euler–Lagrange equation associated to the composite problem and also the regularity of the free boundary for solutions to the Euler–Lagrange equations.