Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$ where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. $\lambda\in\Lambda$ the map $x\mapsto u(x,\lambda)$ is continuous from $S^n$ into $S^n$ and therefore its (Brouwer) degree is well defined. We prove that this degree is independent of $\lambda$ a.e. in $\Lambda$. This result is extended to a more general setting, as well to fractional Sobolev spaces $W^{s,p}$ with $sp\geq n+1$.

We are interested in solutions $u\in H^1({\mathbb R}^N)$ of the linear Schrödinger equation $-\delta u +b_{\lambda} (x) u =f(x,u)$. The nonlinearity $f$ grows superlinearly and subcritically as $\vert u\vert \to\infty$. The potential $b_{\lambda}$ is positive, bounded away from $0$, and has a potential well. The parameter $\lambda$ controls the steepness of the well. In an earlier paper we found a positive and a negative solution. In this paper we find third solution. We also prove that this third solution changes sign and that it is concentrated in the potential well if $\lambda \to \infty$. No symmetry conditions are assumed.

We construct solutions of the Dirichlet problem (1.1)-(1.3) concentrating at strict local maximum point of the coefficient $Q$ either at the boundary or in the interior of $\Omega$. We also prove the existence of solutions concentrating at an interior strict local minimum point of $Q$.

Of concern is the following quasilinear parabolic equation with a nonlinear monotone boundary condition: \begin{equation} \begin{cases} u_{t} (x, t) = \frac{\partial \alpha (x, u_{x})}{\partial x} + g(x, u), \quad (x, t) \in (0, 1) \times (0, \infty), \\ (\alpha (0, u_{x}(0, t)), - \alpha (1, u_{x}(1, t))) \in \beta (u(0, t),u(1, t)), \\ u(x, 0) = u_{0}(x). \end{cases} \tag{*} \end{equation} Here $ \beta $ is a maximal monotone graph in $ {\mathbb R} \times {\mathbb R}$, which contains the origin $(0, 0)$. It is showed that (*) has a unique strong solution $ u $, with the property that $$ \sup_{t \in [0, T]}\|u(x, t)\|_{C^{1+ \nu}[0, 1]} $$ is uniformly bounded for $ 0 < \nu < 1 $ and finite $ T > 0 $.

In this paper we establish the existence of multiple solutions for a semilinear elliptic equation with competing convex and concave nonlinearities. With either a subcritical or critical exponent in the nonlinearity, the existence of solutions is determined with critical point theorems based on the symmetric mountain pass theorem.

We study the existence of many nonradial sign-changing solutions of a superlinear Dirichlet boundary value problem in an annulus in $\mathbb R^N$. We use Nehari-type variational method and group invariance techniques to prove that the critical points of an action functional on some spaces of invariant functions in $H_{0}^{1,2}(\Omega_{\varepsilon})$, where $\Omega_{\varepsilon}$ is an annulus in $\mathbb R^N$ of width $\varepsilon$, are weak solutions (which in our case are also classical solutions) to our problem. Our result generalizes an earlier result of Castro et al. (See [A. Castro, J. Cossio and J. M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differential Equations 2 (1998), 1–18]).

Aim of this paper is to provide conditions in order to guarantee that the periodic solutions in time and in the space variables of the Navier-Stokes equations bifurcate. Specifically, we study this problem when the considered state domain has one dimension which is small with respect to the others which we let to tend to zero. The thinness of the domain represents the bifurcation parameter in our situation.

The algebraic structure of a factor ring and the topological structure of a semi-normed algebra permit the construction of many algebras of generalized functions. Some of them are well suitable to solve a non linear Dirichlet problem with irregular data.

The purpose of the paper is to define a fixed point index for equivariant maps of $G$-ENR's and to state and prove some of its properties, such as the compactly fixed $G$-homotopy property, the Lefschetz property, its converse, and the retraction property. At the end, some examples are given of equivariant self-maps which have a nonzero index (hence cannot be deformed equivariantly to be fixed point free) but have a zero $G$-Nielsen invariant.

It is proved that there exists a fixed point index theory for operators which are condensing on the countable subsets of the space only. Even weaker compactness assumptions on countable subsets suffice, e.g. conditions with respect to classes of measures of noncompactness, or if measures of noncompactness of countable noncompact sets are not preserved (not necessarily decreased). As an application, we prove a generalization of the Fredholm alternative.

We introduce a method based on the Conley index theory for proving the existence of a periodic trajectory in a smooth dynamical system in $\mathbb R^n$ where an attracting periodic orbit is numerically observed. We apply this method to prove the existence of a periodic orbit in the Rössler equations, as announced in [Computer assisted proof of the existence of a periodic orbit in the Rössler equations].