We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation $\ddot{x} + q(t) g(x) = 0$, where $g(x)$ has superlinear growth at infinity and $q(t)$ changes sign.

In this paper the existence of a positive measurable solution of the Hammerstein equation of the first kind with a singular nonlinear term at the origin is presented.

We study a semilinear elliptic equation with an asymptotic linear nonlinearity. Exact multiplicity of solutions are obtained under various conditions on the nonlinearity and the spectrum set. Our method combines a bifurcation approach and Leray-Schauder degree theory.

The existence and Hausdorff dimension of the global attractor for discretization of a damped wave equation with the periodic nonlinearity under the periodic boundary conditions are studied for any space dimension. The obtained Hausdorff dimension is independent of the mesh sizes and the space dimension and remains small for large damping, which conforms to the physics.

We study the degenerate elliptic equation $$ -{\rm div}(a(x)\nabla u)+b(x)u= K(x)\vert u\vert ^{p-2}u+g(x)\quad \text{in } \mathbb R^{N}, $$ where $N\geq 2$ and $2< p< 2^{*}$. We assume that $a\not\equiv 0$ is a continuous, bounded and nonnegative function, while $b$ and $K$ are positive and essentially bounded in $\mathbb R^{N}$. Under some assumptions on $a,b$ and $K$, which control the location of zeros of $a$ and the behaviour of $a,b$ and $K$ at infinity we prove that if the perturbation $g$ is sufficiently small then the above problem has at least two distinct solutions in an appropriate weighted Sobolev space. The proof relies essentially on the Ekeland Variational Principle [Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–473] and on the Mountain Pass Theorem without the Palais-Smale condition, established in Brezis-Nirenberg [Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437–477], combined with a weighted variant of the Brezis-Lieb Lemma [A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490], in order to overcome the lack of compactness.

The paper is concerned with a global bifurcation result for the equation $$ -\text{div} (A(|\nabla u|) \nabla u) = g(x,u,\lambda) $$ in a general domain $\Omega$ with non necessarily radial solutions. Using a variational inequality formulation together with calculations of the Leray-Schauder degrees for mappings in Orlicz-Sobolev spaces, we show a global behavior (the Rabinowitz alternative) of the bifurcating branches.

We consider the perturbed elliptic Sine-Gordon ODE with two positive parameters $\mu$ and $\lambda$, and show the existence of solutions which have $2n$ multiple interior layers for $\lambda \gg 1$. We also determine the location of multiple interior layers as $\lambda \to \infty$.

Of concern is the Cauchy problem $$ \frac{du}{dt} \in Au,\quad u(0) = u_{0},\quad t > 0, $$ where $ u : [0, \infty) \to X$, $X $ is a real Banach space, and $ A : D(A) \subset X \to X $ is nonlinear and multi-valued. It is showed by the method of lines, combined with the Crandall-Liggett theorem that this problem has a limit solution, and that the limit solution is a unique strong one if $ A $ is what is called embeddedly quasi-demi-closed. In the case of linear, single-valued $ A $, further results are given. An application to nonlinear partial differential equations in non-reflexive $ X $ is given.

For a semilinear functional differential inclusion of the form $$ y'(t) \in Ay(t0+f(t, x_t) $$ satisfying a dissipativity condition in a separable Banach space we prove the existence of a periodic solution and a global compact attractor.

The main result of the paper asserts that for every separable measurable space $(T,\mathfrak F,\mu)$, where $\mathfrak F$ is the $\sigma$-algebra of measurable subsets of $T$ and $\mu$ is a nonatomic probability measure on $\mathfrak F$, every Banach space $E$ and every paracompact space $X$, each dispersible closed-valued mapping $F: x \rightsquigarrow L_1(T,E)$ of $X$ into the Banach space $L_1(T,E)$ of all Bochner integrable functions $u: T\to E$, admits a continuous selection. Our work generalizes some results of Gon\v carov and Tol'stonogov.

A number of misprints are listed.