We find multiple solutions for semilinear boundary value problems when the corresponding functional exhibits local splitting at zero.

In this paper, we are dealing with the following superlinear elliptic problem: $$\begin{cases} -\Delta u = \lambda u+h(x)u^p &\text{in }{\mathbb R}^N,\\ u\geq 0,\end{cases}\tag{P} $$ where $h$ is a $C^2$ function from ${\mathbb R}^N$ to ${\mathbb R}$ changing sign such that $\Omega^+ :=\{x\in {\mathbb R}^N\mid h(x)> 0\}$, $\Gamma :=\{x\in {\mathbb R}^N\mid h(x)=0 \}$ are bounded.

For $1< p< {(n+2)}/{(n-2)}$ we prove the existence of global and connected branches of solutions of (P) in ${\mathbb R}^-\times H^1({\mathbb R}^N)$ and in ${\mathbb R}\times L^{\infty}({\mathbb R}^N)$. The proof is based upon a local approach.

We are concerned with the asymptotic behaviour of classical solutions of systems of the form $$ \begin{cases} u_{t} = A u_{xx} + f(u, u_{x}) &\text{for } x \in {\mathbb R},\ t> 0,\ u(x,t) \in {\mathbb R}^N,\\ u(x,0) = \varphi (x),& \end{cases}\tag{1} $$ where $A$ is a positive-definite diagonal matrix and $f$ is a "bistable" nonlinearity satisfying conditions which guarantee the existence of a comparison principle for (1). Suppose that (1) has a travelling-front solution $w$ with velocity $c$, that connects two stable equilibria of $f$. (There are hypotheses on $f$ under which such a front is known to exist [E. C. M. Crooks and J. F. Toland, Travelling waves for reaction-diffusion-convection systems, Topol. Methods Nonlinear Anal. 11 (1998), 19–43].) We show that if $\varphi$ is bounded, uniformly continuously differentiable and such that $\Vert w(x) - \varphi (x) \Vert $ is small when $|x|$ is large, then there exists $\chi \in {\mathbb R}$ such that $$ \Vert u(\cdot, t) - w(\cdot + \chi - ct) \Vert _{BUC^{1}} \rightarrow 0 \quad\text{as } t \rightarrow \infty.\tag{2} $$ Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where $f$ is independent of $u_{x}$. First $\varphi$ is assumed to be increasing in $x$, and (2) proved via a homotopy argument. Then we deduce the result for arbitrary $\varphi$ by showing that there is an increasing function in the $\omega$-limit set of $\varphi$.

An orthogonal representation $V$ of a group $G$ is said to have the Borsuk-Ulam property if the existence of an equivariant map $f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ into a sphere of representation $V$ implies that $\dim W \leq \dim V$. It is known that a sufficient condition for $V$ to have the Borsuk-Ulam property is the nontriviality of its Euler class ${\text{\bf e}}(V)\in H^{*} (BG;\mathcal R)$. Our purpose is to show that ${\text{\bf e}}(V) \neq 0 $ is also necessary if $G$ is a cyclic group of odd and double odd order. For a finite group $G$ with periodic cohomology an estimate for $G$-category of a $G$-space $X$ is also derived.

The relative Lefschetz and Nielsen fixed-point theorems are generalized for compact absorbing contractions on ANR-spaces and nilmanifolds. The nontrivial Lefschetz number implies the existence of a fixed-point in the closure of the complementary domain. The relative Nielsen numbers improve the lower estimate of the number of coincidences on the total space or indicate the location of fixed-points on the complement. Nontrivial applications of these topological invariants (under homotopy) are given to admissible semi-flows and differential inclusions.

In the paper a sufficient condition for the existence of an $R_{\delta}$-set of solutions to a generalized boundary value problem on a compact interval is established. The proof is based on the Browder-Gupta theorem on the existence of an $R_{\delta}$-set of solutions of an operator equation and on the relation between boundary value problems and Fredholm operators. Similar result is obtained by means of the Vidossich theorem.

The paper is devoted to study the existence of periodic solutions for retarded differential inclusions. The nonsmooth guiding potential method is used and topological degree theory for multivalued maps is applied.

In the paper we examine nonlinear evolution hemivariational inequality defined on a Gelfand fivefold of spaces. First we show that the problem with multivalued and $L$-pseudomonotone operator and zero initial data has a solution. Then the existence result is established in the case when the operator is single valued of Leray-Lions type and the initial condition is nonzero. Finally, the asymptotic behavior of solutions of hemivariational inequality with operators of divergence form is considered and the result on upper semicontinuity of the solution set is given.

The nonlinear second order differential equation $$ \frac{d}{dt} h(t,x'(t))+g(t,x(t))=0, \quad t\in[0,T] \text{ a.e. } \quad x'(0)=x'(T)=0 $$ with superlinear function $g$ is investigated. Based on dual variational method the existence of solution is proved. Dependence on parameters and approximation method are also presented.

A general abstract theorem on existence of solutions to optimal shape design problems for systems governed by partial differential equations, or variational inequalities or hemivariational inequalities is formulated and two main properties (conditions) responsible for the existence are discussed. When one of them fails one have to make "relaxation" in order to get some generalized optimal shapes. In particular, some relaxation "in state", based on $\Gamma$ convergence, is presented in details for elliptic, parabolic and hyperbolic PDEs (and then for optimal shape design problems), while the relaxation "in cost functional" is discussed for some special classes of functionals.

Let $\Omega\subset \mathbb R^3$ be a bounded domain. Denote by $\lambda_1(m)$ the principal eigenvalue of the Schrödinger operator $L_m(u)=-\nabla^2 u-mu$ defined on $H^1_0(\Omega)\cap W^{2,1}(\Omega)$. We prove that $\lambda_1: L^{3/2}(\Omega)\to \mathbb R$ is continuous.

From the KKM theorem for the "closed" and "open" valued cases, we deduce a generalization of the Alexandroff-Pasynkoff theorem, existence theorems for almost fixed points of lower semicontinuous multimaps, and a partial solution of the Ben-El-Mechaiekh conjecture.