We consider the semilinear elliptic eigenvalue problem $-\Delta u+k\left(\left|x\right|\right)u^p=\lambda u$, $u>0$ in $B_R$, $u=0$ on $\partial B_R$, where $p>1$ is a constant, , and $\lambda >0$ is a parameter. We investigate the global structure of the branch of $\left(\lambda ,u_{\lambda }\right)$ of bifurcation diagram from a point of view of $L^2$-theory. To do this, we establish a precise asymptotic formula for $\lambda =\lambda \left(\alpha \right)$ as $\alpha \to \infty$, where $\alpha :={\Vert u_{\lambda }\Vert }_2$.

We characterise the set of fixed points of a class of holomorphic maps on complex manifolds with a prescribed homology. Our main tool is the Lefschetz number and the action of maps on the first homology group.

We define classes of mappings of monotone type with respect to a given direct sum decomposition of the underlying Hilbert space $H$. The new classes are extensions of classes of mappings of monotone type familiar in the study of partial differential equations, for example, the class $\left(S_{+}\right)$ and the class of pseudomonotone mappings. We then construct an extension of the Leray-Schauder degree for mappings involving the above classes. As shown by (semi-abstract) examples, this extension of the degree should be useful in the study of semilinear equations, when the linear part has an infinite-dimensional kernel.

We study the best constant involving the $L^2$ norm of the $p$-derivative solution of Wente's problem in ${ℝ}^{2p}$. We prove that this best constant is achieved by the choice of some function $u$. We give also explicitly the expression of this constant in the special case $p=2$.

We will investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation: $\partial u/\partial t=ℒu+V\left(w\right)u^{p-1}$ in $\Omega \times \left(0,T\right)$, , $u\left(w,0\right)=u_0\left(w\right)\ge 0$ in $\Omega$, $u\left(w,t\right)=0$ on $\partial \Omega \times \left(0,T\right)$ where $ℒ$ is the subelliptic $p$-Laplacian and $V\in L_{loc}^1\left(\Omega \right)$.

The exact controllability of a semilinear wave equation in a bounded open domain of $R^n$, with controls on a part of the boundary and in the interior, is shown. Feedback laws are established.

We present some results of existence for the following problem: $-\Delta u=a\left(x\right)g\left(u\right)+u{\left|u\right|}^{2\ast -2}$, $x\in {ℝ}^N\left(N\ge 3\right)$, $u\in D^{1,2}\left({ℝ}^N\right)$, where the function $a$ is a sign-changing function with a singularity at the origin and $g$ has growth up to the Sobolev critical exponent $2^{\ast }=2N/\left(N-2\right)$.

We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dynamical system corresponding to a differential equation of second order in time variable and fourth order in spatial variable with nonlinearity involving the gradients.

By means of Minimax theory, we study the existence of one nontrivial solution and multiple nontrivial solutions for a fourth-order semilinear elliptic problem with Navier boundary conditions.

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between $C^{\ast }$-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if $A$ and $B$ are semisimple Banach algebras and $\Phi :A\to B$ is a linear map onto $B$ that preserves the spectrum of elements, then $\Phi$ is a Jordan isomorphism if either $A$ or $B$ is a $C^{\ast }$-algebra of real rank zero. We also generalize a theorem of Russo.