We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they...

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct...

We study vortices for solutions of the perturbed Ginzburg–Landau equations $\Delta u+(u/{\epsilon }^2){\left(1-\right|u|}^2)=f_{\epsilon }$ where $f_{\epsilon }$ is estimated in $L^2$. We prove upper bounds for the Ginzburg–Landau energy in terms of $\parallel f_{\epsilon }{\parallel }_{L^2}$, and obtain lower bounds for $\parallel f_{\epsilon }{\parallel }_{L^2}$ in terms of the vortices when these form “unbalanced clusters” where ${\sum }_i{d}_i^2\ne {\left({\sum }_i{d}_i\right)}^2$. These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including...

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N\ge 3$. We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if $N=3$).

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.