We consider reaction-diffusion equations on a family of domains depending on a parameter $\varepsilon > 0$. As $\varepsilon\to 0$, the domains degenerate to a lower dimensional manifold. Using some abstract results introduced in the recent paper [M.C. Carbinatto and K.P. Rybakowski, Localized singularities and Conley index, Topol. Methods Nonlinear Anal. 37 (2011), 1-36] we show that there is a limit equation as $\varepsilon\to 0$ and obtain various convergence and admissibility results for the corresponding semiflows. As a consequence, we also establish singular Conley index and homology index continuation results. Under an additional dissipativeness assumption, we also prove existence and upper-semicontinuity of global attractors. The results of this paper extend and refine earlier results of [M.C. Carbinatto and K.P. Rybakowski, Conley index continuation and thin domain problems, Topol. Methods Nonlinear Anal. 16 (2000), 201-251] and [M. Prizzi, M. Rinaldi and K.P. Rybakowski, Curved thin domains and parabolic equations, Studia Math. 151 (2002), 109-140].

The existence of both constant and sign-changing (namely, nodal) solutions to a Neumann boundary-value problem with $p$-Laplacian and reaction term depending on a positive parameter is established. Proofs make use of sub- and super-solution techniques as well as critical point theory.

We consider nonlinear periodic systems driven by the vector $p$-Laplacian. An existence and a multiplicity theorem are proved. In the existence theorem the potential function is $p$-superlinear, but in general does not satisfy the AR-condition. In the multiplicity theorem the problem is strongly resonant with respect to the principal eigenvalue $\lambda_0=0$. In both of the cases the Euler-Lagrange functional is noncoercive and the method is variational.

In this paper, using Brouwer degree arguments, we prove some existence results for nonlinear problems of the type $$ -\nabla[\phi(\Delta x_m)]=g_m(x_m,\Delta x_m) \quad (1\leq m\leq n-1), $$ submitted to Dirichlet, Neumann or periodic boundary conditions, where $\phi(x)=|x|^{p-2}x$ $(p> 1)$ or $\phi(x)={x}/{\sqrt{1-|x|^2}}$ and $g_m\colon \mathbb{R}^N\to\mathbb{R}^N$ $(1\leq m\leq n-1)$ are continuous nonlinearities satisfying some additional assumptions.

We study the existence of homoclinic solutions for a class of non-homogeneous difference equation with periodic coefficients. Our proofs rely on the critical point theory combined with adequate variational techniques, which are mainly based on the mountain-pass lemma.

We obtain an estimate for the covering dimension of the set of bifurcation points for solutions of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch of solutions.

Using variational methods we establish existence of multi-peak solutions for the following class of quasilinear problems $$ -\varepsilon^{p}\Delta_{p}u + V(x)u^{p-1}= f(u), \quad u> 0, \text{ in } {\mathbb{R}}^{N} $$ where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$, $\varepsilon > 0$ and $f$ is a continuous function with subcritical growth.

We establish the existence of smooth invariant stable manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a nonuniform exponential dichotomy for the linear equation $u'=A(t)u$. One of the main advantages of our work is that the results are optimal, in the sense that the invariant manifolds are of class $C^k$ if the vector field is of class $C^k$. To the best of our knowledge, in the nonuniform setting this is the first general optimal result (for a large family of perturbations and not for some specific perturbations). Furthermore, in contrast to some former works, we do not require a strong nonuniform exponential behavior (we note that contrarily to what happens for autonomous equations, in the nonautonomous case a nonuniform exponential dichotomy need not be strong). The novelty of our proofs, in this setting, is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, we can also consider linear perturbations, and our results have thus immediate applications to the robustness of nonuniform exponential dichotomies.

The symmetry of minimisers for the best constant in the trace inequality in a ball, $S_q(\rho)=\inf_{u\in W^{1,p}(B_\rho)} \|u\|^p_{W^{1,p}(B_\rho)}/ \|u\|^{p}_{L^q(\partial B(\rho))}$ has been studied by various authors. Partial results are known which imply radial symmetry of minimisers, or lack thereof, depending on the values of trace exponent $q$ and the radius of the ball $\rho$. In this work we consider a one dimensional analogue of the trace inequality and the corresponding minimisation problem for the best constant. We describe the exact values of $q$ and $\rho$ for which minimisers are symmetric. We also consider the behaviour of minimisers as the symmetry breaking threshold for $q$ and $\rho$ is breached, and show a case in which both symmetric and nonsymmetric minimisers coexist.

In this paper we study stable isolated invariant sets and show that the zeroth singular homology of the Conley index characterizes stability completely. Furthermore, we investigate isolated mountain pass points of gradient-like semiflows introduced by Hofer in [Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493-514] and show that the first singular homology characterizes them completely.

The result of the last section shows that for reaction-diffusion equations \begin{align*} u_{t}-\Delta u& = f(u),\\ u_{|\partial\Omega}& = 0, \end{align*} the Conley index of isolated mountain pass points is equal to $\Sigma^{1}$ - the pointed $1$-sphere. Finally we generalize the result of [T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000), 655-677, Proposition 3.3] about mountain pass points to Alexander-Spanier cohomology.

The notions of uniform boundedness and equidistant uniform boundedness of an operator (both weaker than usual boundedness) are introduced. The main results say that the generator of any uniformly bounded (or equidistantly uniformly bounded) composition Nemytskiĭ operator acting between general Lipschitzian normed function spaces must be affine with respect to the function variable.

Any orientation-preserving homeomorphism of the plane having a two cycle has also a fixed point. This well known result does not provide any hint on how to locate the fixed point, in principle it can be anywhere. J. Campos and R. Ortega in Location of fixed points and periodic solutions in the plane consider the class of Lipschitz-continuous maps and locate a fixed point in the region determined by the ellipse with foci at the two cycle and eccentricity the inverse of the Lipschitz constant. It will be shown that this region is not optimal and a sub-domain can be removed from the interior. A curious fact is that the ellipse mentioned above is relevant for the optimal location of fixed point in a neighbourhood of the minor axis but it is of no relevance around the major axis.