In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear...

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and...

We prove the existence of cylindrical solutions to the semilinear elliptic problem $-\Delta u+\frac{u}{{\left|y\right|}^2}=f\left(u\right)$, $u\in H^1\left({ℝ}^N\right)$, $u\ge 0$, where $(y,z)\in {ℝ}^k\times {ℝ}^{N-k}$, $N>k\ge 2$ and $f$ has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $-{\sum }_{i,j=1}^n{D}_i\left(a_{ij}{D}_{j}u\right)=f(u,{\int }_{\Omega }g\left(u^p\right))$, in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $-\Delta u-{\Lambda }_N\frac{u}{{\left|x\right|}^2}={\left|\nabla u+\frac{N-2}{2}\frac{u}{{\left|x\right|}^2}x\right|}^2{\left|x\right|}^{\left(N-2\right)/2}+\lambda f\left(x\right)$ in $\Omega$, $u=0$ on $\partial \Omega$, ${\Lambda }_N={(\left(N-2\right)/2)}^2$. This problem is a particular case of problem (2). Notice that $\left(N-2\right)/2$ is optimal as coefficient and exponent on the right hand side.