We give the existence result and the vanishing order of the solution in $0$ for the following equation: $-\Delta u\left(x\right)+\left(\mu /{\left|x\right|}^2\right)u\left(x\right)=\lambda u\left(x\right)+u^{2*-1}\left(x\right)$, where $x\in B_1$, $\mu >0$, and the potential $\mu /{\left|x\right|}^2-\lambda$ is positive in $B_1$.

We give an abstract interpretation of initial boundary value problems for hyperbolic equations such that a part of initial boundary value conditions contains also a differentiation on the time $t$ of the same order as equations. The case of stable solutions of abstract hyperbolic equations is treated. Then we show applications of obtained abstract results to hyperbolic differential equations which, in particular, may represent the longitudinal displacements of an inhomogeneous rod under the action of forces at the two ends which are proportional to the acceleration.

Mathematical models describing the behavior of hypothetical species in spatially heterogeneous environments are discussed and analyzed using the fibering method devised and developed by S. I. Pohozaev.

We characterize norm-one complemented subspaces of Orlicz sequence spaces ${\ell }_M$ equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function $M$ is sufficiently smooth and sufficiently different from the square function. We measure smoothness of $M$ using $A{C}^1$ and $A{C}^2$ classes introduced by Maleev and Troyanski in 1991, and the condition for $M$ to be different from a square function is essentially a requirement that the second derivative $M^{″}$ of $M$ cannot have a finite nonzero limit at zero. This paper treats the real case; the complex case follows from previously known results.

We present sharp lower bounds for eigenvalues of the one-dimensional $p$-Laplace operator. The method of proof is rather elementary, based on a suitable generalization of the Lyapunov inequality.

We establish nonexistence results to systems of differential inequalities on the $\left(2N+1\right)$-Heisenberg group. The systems considered here are of the type $\left({\text{ES}}_m\right)$. These nonexistence results hold for $N$ less than critical exponents which depend on $p_i$ and ${\gamma }_i$, $1\le i\le m$. Our results improve the known estimates of the critical exponent.

Nonselfadjoint boundary value problems for second-order differential equations on a finite interval with nonintegrable singularities inside the interval are considered under additional sewing conditions for solutions at the singular point. We study properties of the spectrum, prove the completeness of eigen- and associated functions, and investigate the inverse problem of recovering the boundary value problem from its spectral characteristics.