In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha> 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.

We show that for a given finite graph $G$ without loop edges and isolated vertices, there exists an embedding of a closed orientable surface in $\mathbb{R}^3$ such that the Reeb graph of the associated height function has the structure of $G$. In particular, this gives a positive answer to the corresponding question posed by Masumoto and Saeki in 2011. We also give a criterion for a given surface to admit such a realization of a given graph, and study the problem in the class of Morse functions and in the class of round Morse-Bott functions. In the case of realization up to homeomorphism, the height function can be chosen Morse-Bott; we estimate from below the number of its critical circles and the number of its isolated critical points in terms of the graph structure.

In this paper, we deal with the following Kirchhoff-type equation:\begin{equation*}-\bigg(1+\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\bigg)\Delta u+\frac{A}{|x|^{\alpha}}u=f(u),\quad x\in\mathbb{R}^{3},\end{equation*}where $A> 0$ is a real parameter and $\alpha\in(0,1)\cup ({4}/{3},2)$. Remark that $f(u)=|u|^{2_{\alpha}^{*}-2}u +\lambda|u|^{q-2}u +|u|^{4}u$, where $\lambda> 0$, $q\in(2_{\alpha}^{*},6)$, $2_{\alpha}^{*}=2+{4\alpha}/({4-\alpha})$ is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent. We point out that the nonlinearity $f$ is the almost ''optimal'' choice. First, for $\alpha\in({4}/{3},2)$, applying the generalized version of Lions-type theorem and the Nehari manifold, we show the existence of nonnegative Nehari-type ground sate solution for above equation. Second, for $\alpha\in(0,1)$, using the generalized version of Lions-type theorem and the Pohozaev manifold, we establish the existence of nonnegative Pohozaev-type ground state solution for above equation. Based on our new generalized version of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. 66 (2015)].

Under appropriate assumptions on the $N(\Omega)$-fucntion, locally uniform Morrey estimate is presented in the Musielak-Orlicz-Sobolev space. The assumptions include a new increasing condition on the $x$-derivative of the Young complementary function of the $N(\Omega)$-fucntion. The conclusion applies to several important nonlinear examples frequently appeared in mathematical literature.

We are concerned with of existence of solutions to the semilinear elliptic problem$$ \begin{cases} - \Delta u=\lambda_{k}u+u^3 &\text{in } \Omega, \\ u= 0 &\text{on }\partial \Omega, \end{cases}$$in a bounded domain $\Omega \subset \mathbb{R}^{4}$. Here $\lambda_k$ is an eigenvalue of the $-\Delta$ in $H_0^1(\Omega)$. We prove that this problem has a nontrivial solution.

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\mathbb Z)$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence in particular does not coincide with $\mathcal{W}$. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within $\mathcal{D}$ or in general, is necessarily a constant. An example of a metric, strictly ergodic, distal flow is constructed which admits a non-strictly ergodic $2$-fold minimal self-joining. It then follows that the enveloping group of this flow is not strictly ergodic (as a $T$-flow). Finally we show that the distal, strictly ergodic Heisenberg nil-flow is relatively disjoint over its largest equicontinuous factor from the universal Weyl flow $|\mathcal{W}|$.

In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\[\begin{cases}-\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} |\nabla u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, & x\in\mathbb{R}^3,\\u\in H^1\big(\mathbb{R}^3\big), \quad u> 0, & x\in\mathbb{R}^3,\end{cases}\]where $\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.

In this paper, we use Morse theory to study existence of nontrivial solutions to the following Schrödinger system with linear and nonlinear couplings which arises from Bose-Einstein condensates:$$\begin{cases}-\Delta u+\lambda_{1} u+\kappa v=\mu_{1} u^{3}+\beta uv^{2}& \text{in } \Omega,\\-\Delta v+\lambda_{2} v+\kappa u=\mu_{2} v^{3}+\beta vu^{2}& \text{in } \Omega,\\u=v=0 & \text{on } \partial\Omega,\end{cases}$$where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ ($N=2,3$), $\lambda_{1},\lambda_{2},\mu_{1},\mu_{2} \in \mathbb{R} \setminus \{ 0 \}$, $\beta, \kappa \in \mathbb{R}$. In two cases of $\kappa=0$ and $\kappa\neq 0$, by transferring an eigenvalue problem into an algebraic problem, we compute the Morse index and critical groups of the trivial solution. Furthermore, even when the trivial solution is degenerate, we show a local linking structure of energy functional at zero within a suitable parameter range and then get critical groups of the trivial solution. As an application, we use Morse theory to get an existence theorem on existence of nontrivial solutions under some conditions.

In this article, we establish the existence of positive and multiple sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity \begin{equation}\tag{P$_{p,\Omega}^{s}$}\begin{cases} (-\Delta)_{p}^s u =|u|^{p_s^*-2} u& \text{in }\Omega, \\ u =0 & \text{on }\Omega^{c}, \end{cases}\end{equation}in a bounded domain $\Omega\subset \mathbb{R}^{N}$ for $s\in (0,1)$, $p\in (1,\infty)$, and the fractional critical Sobolev exponent $p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions. We study Struwe's type global compactness results for the Palais-Smale sequence in the presence of symmetries.

In this paper we characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem in context of (sub)Riemannian manifolds. We also present some spectral gap and convexity results.

In a 1961 paper by E. Rakotch it was shown that a uniformly locally contractive mapping has a fixed point. In the present paper we show that for such a mapping, the fixed point problem is well posed and that inexact iterates of such a mapping converge to its unique fixed point, uniformly on bounded sets. Using the porosity notion, we also show that most uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.

We consider a class of singular weighted anisotropic $p$-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.

A method which is a combination of the Halpern method and proximal point method (PPA) is introduced in this paper. It is proved that the sequence of iterates generated by our method converges strongly to a point which is a common solution to some monotone inclusion problem and fixed point problem in CAT$(0)$ spaces under some appropriate conditions.

The main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems$$-\Delta_{\Phi}u + V'(u)=0\quad \text{in }\mathbb{R}^2,$$where$$\Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u),$$$\Phi\colon \mathbb{R}\rightarrow [0,+\infty)$ is an $N$-function and the potential $V$ satisfies some technical condition and we have as an example $ V(t)=\Phi(|t^2-1|)$.

In this article, we analyse an $\alpha$-th order, $1 < \alpha \leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.

In this work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.