We obtain upper bounds for the number of nodal domains of sign changing solutions of semilinear elliptic Dirichlet problems using suitable min-max descriptions. These are consequences of a generalization of Courant's nodal domain theorem. The solutions need not to be isolated. We also obtain information on the Morse index of solutions and the location of sub- and supersolutions.

This paper is a sequel to our previous work [Morse decompositions in the absence of uniqueness, Topol. Methods Nonlinear Anal. 18 (2001), 205–242]. We first extend the concept of $\mathcal{T}$-Morse decompositions to the partially ordered case and prove a generalization of a result by Franzosa and Mischaikow characterizing partially ordered $\mathcal{T}$-Morse decompositions by the so-called $\mathcal{T}$-attractor semifiltrations. Then we extend the (regular) continuation result for Morse decompositions from [Morse decompositions in the absence of uniqueness, Topol. Methods Nonlinear Anal. 18 (2001), 205–242] to the partially ordered case. We also define singular convergence of families of "solution" sets in the spirit of our previous paper [On a general Conley index continuation principle for singular perturbation problems, Ergodic Theory Dynam. Systems 22 (2002), 729–755] and prove various singular continuation results for attractor-repeller pairs and Morse decompositions. We give a few applications of our results, e.g. to thin domain problems. The results of this paper are a main ingredient in the proof of regular and singular continuation results for the homology braid and the connection matrix in infinite dimensional Conley index theory. These topics are considered in the forthcoming publications [Continuation of the connection matrix in infinite-dimensional Conley index theory] and [Continuation of the connection matrix in singular perturbation problems].

In this paper we consider the semilinear problem with critical growth in the Heisenberg group $-\Delta_{_{\mathbb{H}^n}} u = u^{(Q+2)/(Q-2)}+\lambda u$ in $\Omega$, $u> 0$ in $\Omega$, $u=0$ in $\partial\Omega$, and we provide a multiplicity existence result involving Lusternik-Schnirelmann category.

We consider the elliptic problem $$ -\Delta u-\lambda u=a(x) g(u), $$ with $a(x)$ sign-changing and $g(u)$ behaving like $u^p$, $p> 1$. Under suitable conditions on $g(u)$ and $a(x)$, we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space $\mathbb R^N$. More precisely, we show that there exists $\Lambda> 0$ such that this equation on $\mathbb R^N$ has no positive solution for $\lambda> \Lambda$, at least two positive solutions for $\lambda\in (0,\Lambda)$, and at least one positive solution for $\lambda\in (-\infty,0]\cup\{\Lambda\}$.

Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.

The aim of this paper is twofold. On one hand we establish a three critical points theorem for functionals depending on a real parameter $\lambda \in \Lambda$, which is different from the one proved by B. Ricceri in [On a three critical points theorem, Arch. Math. 75 (2000), 220-226] and gives an estimate of where $\Lambda$ can be located. On the other hand, as an application of the previous result, we prove an existence theorem of three classical solutions for a two-point boundary value problem which is independent from the one by J. Henderson and H. B. Thompson [Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000), 443-454]. Specifically, an example is given where the key assumption of [J. Differential Equations 166 (2000), 443-454] fails. Nevertheless, the existence of three solutions can still be deduced using our theorem.

We consider sufficient conditions of local removability of coincidences of maps $f,g\colon N\rightarrow M$, where $M,N$ are manifolds with dimensions $\dim N\geq\dim M$. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. We also address the normalization property of the index and coincidence-producing maps.

We show in the paper the method of isolating chains in proof of chaotic dynamics. It is based on the earlier notion of an isolating segment but gives more powerful tool for exploring dynamics of periodic ODE's. As an application we show that the processes generated by the equations in the complex plane $ \dot{z} = {\overline z}^k + e^{i\phi t} {\overline z}$, where $k\geq 3$ is an odd number and $\phi$ is close to $0$, has chaotic behaviour.

A deformation lemma for functionals which are the sum of a locally Lipschitz continuous function and of a concave, proper and upper semicontinuous function is established. Some critical point theorems are then deduced and an application to a class of elliptic variational-hemivariational inequalities is presented.

Let $f$ be a local planar homeomorphism with an isolated fixed point at $0$. We study the form of the sequence $\{{\rm ind}(f^n,0)\}_{n\neq 0},$ where ${\rm ind}(f,0)$ is a fixed point index at $0$.

Nontrivial periodic solutions of Lagrange Equations are investigated. Sublinear and superlinear nonlinearity are included. Convexity assumptions are significiently relaxed. The method used is the duality developed by the authors.

Let $X$ be a Banach space, $Y$ a metric space, $A\subseteq X$, $C\colon A\to Y$ a compact operator and $T$ an operator defined at least on the set $A\times C(A)$ with values in $X$. By assuming that the family $\{T(\cdot,y):y\in C(A)\}$ is equicontractive we present two fixed point theorems for the operator of the form $Ex:=T(x,C(x))$. Our results extend the well known Krasnosel'skiĭ's fixed point theorem for contractions and compact mappings. The results are used to prove the existence of (global) solutions of integral and integrodifferential equations.

Some new fixed point theorems for approximable maps are obtained in this paper. Homotopy results, via essential maps, are also presented for approximable maps.