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.

In this paper we consider Neumann noncoercive hemivariational inequalities, focusing on nontrivial solutions. We use the critical point theory for locally Lipschitz functionals.

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.

The proof of the Friedrichs' inequality on the class of finite dimensional spaces used in the finite element method is given. In particular, the approximate spaces generated by simplicial isoparametric elements are considered.

In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis.

We study the nonexistence result of radial solutions to -Δu + c u/(|x|2) + |x|σ|u|qu ≤ 0 posed in B or in B {0} where B is the unit ball centered at the origin in RN, N ≥ 3. Moreover, we give a complete classification of radial solutions to the problem -Δu + c u/(|x|2) + |x|σ|u|qu = 0. In particular we prove that the latter has exactly one family of radial solutions.

In this work we study non-negative singular infinity-harmonic functions in the half-space. We assume that solutions blow-up at the origin while vanishing at infinity and on a hyperplane. We show that blow-up rate is of the order |x|-1/3.

In this Note we extend Gibbons conjecture to Carnot groups using the sliding method and the maximum principle in unbounded domains.