We consider a class of elliptic equation with natural growth. We obtain a region of the natural growth term with precise lower boundary less than zero.

We prove in this paper a periodic existence theorem for neutral differential equations in finite dimension with high frequency terms. This study completes previous works about applications of averaging methods to periodic problems.

We consider nonlinear Neumann problems driven by the $p$-Laplacian differential operator with a Caratheodory nonlinearity. Under hypotheses which allow resonance with respect to the principal eigenvalue $\lambda_{0}=0$ at $\pm\infty$, we prove existence and multiplicity results. Our approach is variational and uses critical point theory and Morse theory (critical groups).

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb{R}^N$: \begin{gather*} -\Delta u=g(u) \quad \text{in }\mathbb{R}^N,\\ u\in H^1(\mathbb R^N). \end{gather*} We give an extension of the existence results due to H. Berestycki, T. Gallouët and O. Kavian [Equations de Champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math. 297, 307–310].

We take a mountain pass approach in $H^1(\mathbb{R}^N)$ and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity.

Global solvability and dynamical behaviour of the modified viscous Cahn-Hilliard equation is studied in the Sobolev space $H^1({\mathbb{R}}^N)$. For $\nu \in [0,1]$ we construct $H^1({\mathbb{R}}^N)$ global attractors and show their upper semicontinuity at $\nu = 0$.

In this paper, some existence results for a differential equation of fractional order with anti-periodic boundary conditions are presented. The main tool of study is Leray-Schauder degree theory.

In this paper we prove the existence of solutions for a two point boundary value problem for a second order differential inclusion governed by a maximal monotone operator with a mixed semicontinuous perturbation.

Let $(M,g)$ be a smooth connected compact Riemannian manifold of finite dimension $n\geq 2$\ with a smooth boundary $\partial M$. We consider the problem $$ \begin{cases} -\varepsilon ^{2}\Delta _{g}u+u=|u|^{p-2}u,\quad u> 0 &\text{ on }M,\\ \displaystyle \frac{\partial u}{\partial \nu }=0 & \text{on }\partial M, \end{cases} $$ where $\nu $ is an exterior normal to $\partial M$.

The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.

Let $M$ be a possibly noncompact manifold. We prove, generically in the $C^k$-topology ($2\leq k\leq \infty$), that semi-Riemannian metrics of a given index on $M$ do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [Genericity of nondegenerate critical points and Morse geodesic functionals, Indiana Univ. Math. J. 58 (2009), 1797–1830] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold $\mathcal P\subset M\times M$ that satisfies an admissibility condition. Such condition holds, for example, when $\mathcal P$ is transversal to the diagonal $\Delta\subset M\times M$. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are $C^k$-generic are given.

We obtain nontrivial solutions of a class of $p$-superlinear anisotropic $p$-Laplacian systems using Morse theory.

We prove an odd mapping theorem of Borsuk type for locally compact perturbations of Fredholm maps of index zero between Banach spaces. We extend this result to a more general class of perturbations of Fredholm maps, defined in terms of measure of noncompactness.

Let $M^n$, $N^n$, $n\geq 2$, be compact connected manifolds. We first observe that mappings of zero degree have high dimensional critical sets and show that the only possible degree is zero for maps $f\colon M\to N$, under the assumption on the index $[\pi_1(N):{\rm Im}(f_*)]$ to be infinite. By contrast with the described situation one shows, after some estimates on the algebraic $\varphi$-category of some pairs of finite groups, that a critical set of smaller dimension keeps the degree away from zero.