In this paper, we study the feedback optimal control for a class of evolution hemivariational inequalities with delay. First, we obtain the existence of feasible pairs by applying the Cesari property, the Filippov theorem, the properties of Clarke subdifferential and a fixed point theorem for multivalued maps. Next, the results of optimal feedback control pairs and time optimal control for delay evolution hemivariational inequalities are presented under sufficient conditions. Finally, an example is included to illustrate our main results.

In this paper, we analyse a linear quadratic multistage game of extraction of a common renewable resource --- a fishery --- by many players with inherent state dependent constraints for exploitation and an infinite time horizon. To the best of our knowledge, such games have never been studied. We analyse the social optimum and Nash equilibrium for the feedback information structure and compare the results obtained in both cases. For the Nash equilibria, we obtain a value function that is contrary to intuitions from standard linear quadratic games. In our game, we face a situation in which the social optimum results in sustainability, while the Nash equilibrium leads to the depletion of the fishery in a finite time for realistic levels of the initial biomass of fish. Therefore, we also study an introduction of a tax in order to enforce socially optimal behaviour of the players. Besides, this game constitutes a counterexample to simplifications of techniques often used in computation of Nash equilibria and/or optimal control problems.

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: \begin{equation*} \begin{cases} -{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) &\text{ in }\Omega , \\ u=0 &\text{ on }\partial \Omega . \end{cases} \end{equation*} We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Computers and Mathematics with Applications, 2015) and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

We consider the following Schrödinger-Poisson system: $$ \begin{cases} -\triangle u+V(|x|)u+\phi u= Q(|x|) f(u) &\hbox{in } \mathbb{R}^3,\\ -\triangle \phi=u^{2} & \hbox{in } \mathbb{R}^3, \end{cases} $$ with more general radial potentials $V,Q$ and discontinuous nonlinearity $f$. The Lagrange functional may be locally Lipschitz. Using nonsmooth critical point theorem, we obtain the multiplicity results of radial solutions, we also show concentration properties of the solutions. This is in contrast with some recent papers concerning similar problems by using the classical Sobolev embedding theorems.

The Benjamin-Bona-Mahony equation with a distribution force on torus is studied in low regularity spaces. The global well-posedness and the existence of a global attractor in $\dot{H}^{s,p}(\mathbb{T})$ are proved.

We study the following class of elliptic equations: \begin{equation*} -\bigg(a+b\int_{{\mathbb R}^3}|\nabla u|^2\,dx\bigg)\Delta u+\lambda V(x)u=f(u), \quad x\in{\mathbb R}^3, \end{equation*} where $\lambda,a,b>0$, $V\in \mathcal{C}({\mathbb R}^3,{\mathbb R})$ and $V^{-1}(0)$ has nonempty interior. First, we obtain one ground state sign-changing solution $u_{b,\lambda}$ applying the non-Nehari manifold method. We show that the energy of $u_{b,\lambda}$ is strictly larger than twice that of the ground state solutions of Nehari-type. Next we establish the convergence property of $u_{b,\lambda}$ as $b\searrow0$. Finally, the concentration of $u_{b,\lambda}$ is explored on the set $V^{-1}(0)$ as $\lambda\rightarrow\infty$.

We discuss the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities ((DMQVIs), for short) in Hilbert spaces. This gives us an outlook to the convergence analysis of approximating sequences of solutions for (DMQVIs). Using these concepts we point out the relation between metric characterizations and well-posedness of (DMQVIs). We also prove that the solution set of (DMQVIs) is compact, if problem (DMQVIs) is well-posed in the generalized sense.

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0(\Omega)$ to the equation \[ -\Delta u+h u=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \quad \textrm{in } \Omega, \] where $2^*_\sigma:={2(N-\sigma)}/({N-2})$, $\sigma\in (0,2)$, and $\rho_\Gamma$ is the distance function to $\Gamma$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\Delta+h$ is positive at a point of the curve.

We investigate the classification of closed curves and eight curves of saddle points defined on non-orientable closed surfaces, up to an ambient homeomorphism. The classification obtained here is applied to Morse-Bott foliations on non-orientable closed surfaces in order to define a complete topological invariant.

In this paper, we consider the following semilinear elliptic systems: $$ \begin{cases} -\Delta u+V(x)u=F_{u}(x, u, v)-\Gamma(x)|u|^{q-2}u & \mbox{in }\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v)-\Gamma(x)|v|^{q-2}v & \mbox{in }\mathbb{R}^{N},\\ \end{cases} $$ where $q\in[2,2^{*})$, $V=V_{{\rm per}}+V_{{\rm loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{{\rm per}}$ and a localized potential $V_{{\rm loc}}$ and $\Gamma\in L^{\infty}(\mathbb{R}^{N})$ is periodic and $\Gamma(x)\geq0$ for almost every $x\in\mathbb{R}^{N}$. Under some appropriate assumptions on $F$, we investigate the existence and nonexistence of ground state solutions for the above system. Recent results from the literature are improved and extended.

We prove an $h$-principle for Poisson structures on closed manifolds. Equivalently, we prove an $h$-principle for symplectic foliations (singular) on closed manifolds. On open manifolds however the singularities could be avoided and it is a known result by Fernandes and Frejlich [Int. Math. Res. Not. IMRN (2012), no. 7, 1505-1518].

A spectral sequence analysis of a filtered Novikov complex $(\mathcal{N}_{\ast}(f),\Delta)$ over $\mathbb{Z}((t))$ is developed with the goal of obtaining results relating the algebraic and dynamical settings. Specifically, the unfolding of a spectral sequence of $(\mathcal{N}_{\ast}(f),\Delta)$ and the cancellation of its modules is associated to a one parameter family of circle-valued Morse functions on a~surface and the dynamical cancellations of its critical points. The data of a spectral sequence computed for $(\mathcal{N}_{\ast}(f),\Delta)$ is encoded in a family of matrices $\Delta^r$ produced by the Spectral Sequence Sweeping Algorithm (SSSA), which has as its initial input the differential $\Delta$. As one ``turns the pages'' of the spectral sequence, differentials which are isomorphisms produce cancellation of pairs of modules. Corresponding to these cancellations, a family of circle-valued Morse functions $f^r$ is obtained by successively removing the corresponding pairs of critical points of $f$. We also keep track of all dynamical information on the birth and death of connecting orbits between consecutive critical points, as well as periodic orbits that arise within a family of negative gradient flows associated to $f^r$.

We derive a conditional variational principle of the saturated set for systems with the non-uniform structure. Our result applies to a broad class of systems including $\beta$-shifts, $S$-gap shifts and their subshift factors.