We study the existence, nonexistence and exact multiplicity of positive radial solutions of the problem: $- \Delta u = f(u) $ in $ B(R_1, R)$, where $ R_1 > 0 $ is a fixed real number and $ B(R_1,R) \subset \mathbb R^n (n \geq 3) $, $ R > R_1 $, is an annulus, for the following boundary conditions: (1) $ u = 0 $ on $ \partial B(R_1,R) $. For $ f(t) = t^p - t^q $, $ 1 < p < q $ and $ p \leq (n+2)/(n-2) $, there exist $ R_0 \geq \tilde{R} > R_1 $ such that the problem admits exactly two solutions for $ R > R_0 $ and no solution for $ R < \tilde{R}$. (2) $ u = 0 $ on $ |x| = R_1 $ and $\partial u/ \partial \nu = 0 $ on $ |x| = R $. (a) For $ f(t) = t^p - t^q $, $ 1 < p < q $ and $ p \leq (n+2)/(n-2)$, there exists $ R_0 > R_1 $ such that the problem admits exactly two solutions for $ R > R_0 $, exactly one solution for $ R = R_0 $ and no solution for $ R < R_0 $. (b) Let $ f^{\prime}(t)t > f(t) $. If $ f(t) > 0$ for $ t > 0 $, then the problem admits at most one solution for any $R > R_1 $. For a large class of nonlinearities $ f $ changing sign (for example $ f(t) = t^p -t $), there exists $ R_0 > R_1 $ such that the problem admits a unique solution for all $ R < R_0 $. These results have been proved by analyzing the behaviour of the first and the second variations of $ u(\gamma,r) $, the unique solution of the corresponding initial value problem with $ u^{\prime}(R_1) = \gamma > 0 $, with respect to $ \gamma$ together with an identity (2.8) which may be regarded as variations of the generalized Pohozaev's identity with respect to $ \gamma $.

We discuss the asymptotic behaviour of solutions of the semilinear hyperbolic problem $$u_{tt} +\delta u_t -\phi (x)\Delta u = \lambda \, u|u|^{\beta -1}, \, \;\; x \in \mathbb R^N, \;\; t \geq 0,$$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$,\ in the case where $N \geq 3, $ $ \delta \geq 0$ and $(\phi (x))^{-1} =g (x)$, a positive function belonging to $L^{{N/2}}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. Under certain conditions we prove the global existence of solutions. Also we examine blow-up in finite time when the initial data are sufficiently large. The space setting of the problem is the energy space ${\mathcal X}_0={\mathcal D}^{1,2}(\mathbb R^N) \times L_{g}^{2}(\mathbb R^N)$, where $L_{g}^{2}$ is an appropriate weighted Hilbert space; see Section 2.

It is shown that solutions of the 3D Stokes resolvent problem in domains with conical boundary points, under homogeneous Dirichlet boundary conditions, do not satisfy the usual resolvent estimate in $L^p$-spaces if $p $ is close to infinity or close to $1.$

We study a nonlinear homogenization problem of harmonic maps which describes an ideal superconducting or an ideal superfluid medium with a large number of vortices and the degree conditions prescribed at the external insulating boundary. We derive the homogenized problem which describes the limiting behavior of the fluxes when the total number of vortices tends to infinity. The homogenized problem is described in terms of the effective vorticity and the effective anisotropy tensor. The calculation of this tensor amounts to solving a linear cell problem, which is well studied in the homogenization literature and can be solved by using existing numerical packages. The convergence of the fluxes is rigorously proved. We also discuss unusual features of the homogenized limit for the wave functions. The proofs are based on a variational approach which does not require periodicity and can be used in more general situations.