This paper uses critical point theory and variational methods to investigate the multiple solutions of a boundary value problem for second order impulsive differential equations. The conditions for the existence of multiple solutions are established.

In this note, we investigate a non-monotone and non-coercive dynamic model of poroplasticity with mixed boundary conditions. The existence of the solution to this non-monotone model, where the inelastic constitutive equation is satisfied in the sense of Young measures, is proved using the coercive and monotone approximations.

In this paper, we show the existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness, by using the technique of penalized functionals and an infinite dimensional Morse theory developed by Kryszewski and Szulkin. Two applications are given to Hamiltonian systems and elliptic systems.

Fractional differential equations have been used for constructing many mathematical models in science and engineering. In this paper, we study the approximate controllability results for a class of impulsive fractional differential equations with infinite delay. A new set of sufficient conditions are formulated and proved for achieving the required result. In particular, the results are established under the natural assumptions that the corresponding linear system is approximately controllable. The results are obtained by using the fractional calculus, solution operators and fixed point technique. An example is also provided to illustrate the theory. Further, as a corollary, exact controllability result is discussed without assuming compactness of characteristic solution operators.

In this paper we construct two retracts in a cone by nonnegative functionals of convex and concave types, and an example is given to illustrate that the retracts are nonconvex. Then the nonconvex retracts are used to compute the fixed point index for the completely continuous operator in the domains $D_1\cap D_2$ and $D_1\cup D_2$, where $D_1$ and $D_2$ are bounded open sets in the cone. The computation for fixed point index can be applied to the existence and the more precise location of positive fixed points.

We study the existence of infinitely many sign-changing critical points and nonexistence of critical points to a class of noncoercive functionals.

In the present paper we study singularly perturbed Hamiltonian elliptic systems with nonlocal nonlinearities $$ \begin{cases} \displaystyle -\varepsilon^2\Delta u +V(x)u =\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}\,dy\bigg)|z|^{p-2}u, \\ \displaystyle -\varepsilon^2\Delta v +V(x)v =-\bigg(\int_{\mathbb{R}^N} \frac{|z|^{p}}{|x-y|^{\mu}}\,dy\bigg)|z|^{p-2}v, \end{cases} $$ where $z=(u,v)\in H^1(\mathbb{R}^N,\mathbb{R}^2)$, $V(x)$ is a continuous real function on $\mathbb{R}^N$, $0< \mu< N$ and $2-{\mu}/{N}< p< (2N-\mu)/(N-2)$. Under suitable assumptions on the potential $V(x)$, we can prove the existence of solutions for small parameter $\varepsilon$ by variational methods. Moreover, if $N> 2$ and $2+(2-\mu)/(N-2)< p< (2N-\mu)/(N-2)$ then the solutions $z_\varepsilon\to 0$ as the parameter $\varepsilon\to 0$.

We are concerned here with the classical problem of Poincaré of persistence of periodic solutions under small perturbations. The main contribution of this work is to give the expression of the second order bifurcation function in more general hypotheses than the ones already existing in the literature. We illustrate our main result constructing a second order bifurcation function for the perturbed symmetric Euler top.

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a Caratheodory reaction which is $(p-1)$-sublinear near $\pm\infty$. Using variational tools we show that the problem has at least three nontrivial smooth solutions (one positive, one negative and a third nodal). Our formulation unifies problems driven by the $p$-Laplacian, the $(p,q)$ Laplacian and the $p$-generalized mean curvature operator.

In this paper we study the quasilinear equation \begin{equation} \begin{cases} - {\rm div}(|\nabla u|^{p-2} \nabla u)+V(|x|)|u|^{p-2} u= Q(|x|)f(u), & x\in \mathbb{R}^N, \\ u(x)\rightarrow 0,\quad |x|\rightarrow \infty. \end{cases} \tag{$\rm P$} \end{equation} with singular radial potentials $V,Q$ and bounded measurable function $f$. The approaches used here are based on a compact embedding from the space $W^{1,p}_r(\mathbb{R}^N; V)$ into $L^1 (\mathbb{R}^N; Q)$ and a new multiple critical point theorem for locally Lipschitz continuous functionals.

In this paper an existence result is presented for solution of a parabolic boundary value problem under Dirichlet null boundary conditions for a class of general equations of infinite order with strongly nonlinear perturbation terms.

The paper is devoted to local and global solvability and existence of a global attractor for an exemplary `parabolic' problem containing fractional powers of the minus Laplace operator. We want to compare, which properties of the similar semilinear heat equation are preserved when we replace the pure minus Laplace operator by a fractional power of that operator. Several useful technical tools and estimates are collected in that paper.

This paper deals with variational-hemivariational inequalities involving the $p$-Laplace operator and a nonlinear Neumann boundary condition. Based on an abstract critical point result, which is developed at the beginning of the paper, it is shown the existence of at least three solutions to such inequalities whereby the cases $p> N$ and $p \leq N$ are treated separately. The applicability of these results is emphasized with suitable examples.

We discuss the role of the diffusion coefficient $a(x)$ on the existence of a positive solution for the quasilinear elliptic problem involving critical exponent $$ \begin{cases} - {\rm div}( a(x) |\nabla u|^{p-2} \nabla u) = u^{p^* - 1} + \lambda u^{p-1} & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases} $$ where $\Omega$ is a smooth bounded domain in $\mathbb R^n$, $n \geq 2$, $1 < p < n$, $p^* = np/(n-p)$ is the critical exponent from the viewpoint of Sobolev embedding, $\lambda$ is a real parameter and $a\colon \overline{\Omega} \rightarrow \mathbb R$ is a positive continuous function. We prove that if the function $a(x)$ has an interior global minimum point $x_0$ of order $\sigma$, then the range of values $\lambda$ for which the problem above has a positive solution relies strongly on $\sigma$. We discover in particular that the picture changes drastically from $\sigma > p$ to $\sigma \leq p$. Some sharp answers are also provided.

We study the existence of solutions for the following fractional hybrid boundary value problem $$ \begin{cases} \displaystyle D_{0^+}^{\alpha}\bigg[\frac{x(t)}{f(t,x(t))}\bigg]+g(t,x(t))=0, &0< t< 1,\\ x(0)=x(1)=0, \end{cases} $$ where $1< \alpha\leq 2$ and $D_{0^+}^{\alpha}$ denotes the Riemann-Liouville fractional derivative. The main tool is our study is the technique of measures of noncompactness in the Banach algebras. Some examples are presented to illustrate our results. Finally, we compare the results of paper with the ones obtained by other authors.