$$\alpha $$-Dirac-harmonic maps from closed surfaces

Abstract$$\alpha $$ α -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $$\alpha $$ α -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For $$\alpha >1$$ α > 1 , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ α -harmonic maps for $$\alpha >1$$ α > 1 and then letting $$\alpha \rightarrow 1$$ α → 1 . The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $$\alpha $$ α -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By $$\varepsilon $$ ε -regularity and suitable perturbations, we can then show that such a sequence of perturbed $$\alpha $$ α -Dirac-harmonic maps converges to a smooth coupled $$\alpha $$ α -Dirac-harmonic map.

Nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order: Two and three solutions

<abstract><p>In this article, we consider the following nonlocal fractional Kirchhoff-type elliptic systems</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{l} -M_{1}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\eta(x)-\eta(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}}dxdy +\int_{\Omega}\frac{|\eta|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\eta-|\eta|^{\overline{p}(x)}\eta\right)\\ \; \; \; = \lambda F_{\eta}(x, \eta, \xi)+\mu G_{\eta}(x, \eta, \xi), \; \; x \in \Omega, \\ -M_{2}\left(\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|\xi(x)-\xi(y)|^{^{p(x, y)}}}{p(x, y)|x-y|^{N+p(x, y)s(x, y)}}dxdy +\int_{\Omega}\frac{|\xi|^{\overline{p}(x)}}{\overline{p}(x)}dx\right) \left(\Delta_{p(\cdot)}^{s(\cdot)}\xi-|\xi|^{\overline{p}(x)}\xi\right)\\ \; \; \; = \lambda F_{\xi}(x, \eta, \xi)+\mu G_{\xi}(x, \eta, \xi), \; \; x \in \Omega, \\ \; \eta = \xi = 0, \; \; x \in \mathbb{R}^{N}\backslash \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ M_{1}(t), M_{2}(t) $ are the models of Kirchhoff coefficient, $ \Omega $ is a bounded smooth domain in $ \mathbb R^{N} $, $ (-\Delta)_{p(\cdot)}^{s(\cdot)} $ is a fractional Laplace operator, $ \lambda, \mu $ are two real parameters, $ F, G $ are continuous differentiable functions, whose partial derivatives are $ F_{\eta}, F_{\xi}, G_{\eta}, G_{\xi} $. With the help of direct variational methods, we study the existence of solutions for nonlocal fractional $ p(\cdot) $-Kirchhoff systems with variable-order, and obtain at least two and three weak solutions based on Bonanno's and Ricceri's critical points theorem. The outstanding feature is the case that the Palais-Smale condition is not requested. The major difficulties and innovations are nonlocal Kirchhoff functions with the presence of the Laplace operator involving two variable parameters.</p></abstract>

On variational nonlinear equations with monotone operators

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

A Critical Point Theorem for Perturbed Functionals and Low Perturbations of Differential and Nonlocal Systems

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

Multiple solutions for some symmetric supercritical problems

The aim of this paper is to investigate the existence of one or more critical points of a family of functionals which generalizes the model problem [Formula: see text] in the Banach space [Formula: see text], where [Formula: see text] is an open bounded domain, [Formula: see text] and the real terms [Formula: see text] and [Formula: see text] are [Formula: see text] Carathéodory functions on [Formula: see text]. We prove that, even if the coefficient [Formula: see text] makes the variational approach more difficult, if it satisfies “good” growth assumptions then at least one critical point exists also when the nonlinear term [Formula: see text] has a suitable supercritical growth. Moreover, if the functional is even, it has infinitely many critical levels. The proof, which exploits the interaction between two different norms on [Formula: see text], is based on a weak version of the Cerami–Palais–Smale condition and a suitable intersection lemma which allow us to use a Mountain Pass Theorem.

Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in {\mathbb{R}^{N}} ({N\geq 2}):${(*)_{m}}$\displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})% ,\end{cases}where {g(\xi)\in C(\mathbb{R},\mathbb{R})}, {m>0} is a given constant and {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem {(*)_{m}}. We develop a new deformation argument under a new version of the Palais–Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347–375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in {\mathbb{R}^{N}}: Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253–276], it enables us to apply minimax argument for {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem\inf\Bigg{\{}\int_{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int_{0}^{\xi}g(% \tau)\,d\tau.

Infinitely many solutions for some nonlinear supercritical problems with break of symmetry

In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem \[\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x) &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}\] where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N\geq 3\), and \(A(x,t,\xi)\), \(g(x,t)\), \(h(x)\) are given functions, with \(A_t = \frac{\partial A}{\partial t}\), \(a = \nabla_{\xi} A\), such that \(A(x,\cdot,\cdot)\) is even and \(g(x,\cdot)\) is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if \(A(x,t,\xi)\) grows fast enough with respect to \(t\), then the nonlinear term related to \(g(x,t)\) may have also a supercritical growth.

A Generalized Palais-Smale Condition in the Fr\'{e}chet space setting

The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr\'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr\'{e}chet spaces and Fr\'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr\'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr\'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr\'{e}chet spaces. The idea of this approach is to represent a Fr\'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr\'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr\'{e}chet setting.

Solutions for a singular elliptic problem involving the p(x)-Laplacian

Here, a singular elliptic problem involving p(x)-Laplacian operator in a bounded domain in RN is considered. Due to this, the existence of critical points for the energy functional which is unbounded below and satisfies the Palais-Smale condition are proved.