Semilinear elliptic operators in unbounded cylinders are considered. It is shown that under some conditions the operators are Fredholm and proper. A topological degree for these operators is defined and shown to be unique. The results are applied to operators describing traveling waves.

We prove existence, multiplicity and bifurcation results for a family of semilinear Neumann problems with nonlinear terms that are indefinite in sign and exhibit sublinear growth near zero. The solutions are non-negative, but the combined effect of indefiniteness and the non-Lipschitz character of the nonlinear term yields solutions which may vanish on large sets. Combining variational methods with bifurcation analysis and the sub- and super-solution

Results on (a) the existence and (b) asymptotic stability of mild and of strong solutions to the nonlinear partial functional differential equation with delay $ (FDE) \; \, \dot{u} (t) + B u(t) \ni F(u_t), \; t \geq 0 , \; u_0 = \varphi \in E, $ are presented. The `partial differential expression' $B$ will be a, generally multivalued, accretive operator, and the history-responsive operator $F$ will be allowed to be (defined and) Lipschitz continuous on `thin' subsets of the initial-history space $E $ of functions from an interval $I \subset (-\infty,0] $ to the state Banach space $X.\,$ As one of the main results, it is shown that the well-established solution theory on strong, mild and integral solutions to the undelayed counterpart to (FDE) of the nonlinear initial-value problem $ (CP) \; \, \dot{u} (t) + B u(t) \ni f(t), \; t \geq 0 , \; u(0) = u_0 \in X, $ can fully be extended to the more general initial-history problem (FDE). The results are based on the relation of the solutions to (FDE) to those of an associated nonlinear Cauchy problem in the initial-history space $E. $ Applications to models from population dynamics and biology are presented.

The author considers the semilinear elliptic equation $$ -\Delta^{m}u=g(x,u), $$ subject to Dirichlet boundary conditions $u=Du=\cdots=D^{m-1}u=0$, on a bounded domain $\Omega\subset\mathbb{R}^{2m}$. The notion of nonlinearity of critical growth for this problem is introduced. It turns out that the critical growth rate is of exponential type and the problem is closely related to the Trudinger embedding and Moser type inequalities. The main result is the existence of non trivial weak solutions to the problem.

In this paper, we study the stability of Serrin's classical symmetry result for overdetermined boundary value problems [13]. We prove that if there exists a positive solution of $\Delta u +f(u) =0$ in $\Omega$ with $u=0$ on $\partial\Omega$ and if $\partial u / \partial \nu$ on $\partial\Omega$ is close to a constant, then the domain $\Omega$ is close to a ball. Additionally, we give an explicit estimate for the distance of the domain to a circumscribed and inscribed ball. The proof relies on the method of moving planes and new quantitative versions of the Hopf Lemma and Serrin's corner Lemma.