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Let $\Omega$ be a bounded domain in $\mathbb{R}^{N+1}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, u=0\,\,\text{on}\,\, \partial\Omega,\,\,\frac{\partial u}{\partial n} =c\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a nontrivial solution, then $\Omega$ is a ball, which is exactly the affirmative answer to the Berenstein conjecture. Similarly, we show that, if $\Omega$ has a Lipschitz connected boundary and the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, \frac{\partial u}{\partial n}=0\,\,\text{on}\,\, \partial\Omega,\,\,u =c\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a nontrivial solution, then $\Omega$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture.

In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the corresponding functional spaces, namely the Musielak-Orlicz Sobolev spaces, hence we investigate their regularity properties and the extension of the classical Sobolev embedding results to the new context. Then, we focus on the regularity properties of our operators, and prove that these operators are bounded, continuous, strictly monotone, coercive and satisfy the (S_+)-property. Further, we discuss suitable problems driven by such operators. In particular, we deal with Dirichlet problems in which the nonlinearity is gradient dependent. Under very general assumptions, we establish the existence of a nontrivial solution for such problems. Also, we give additional conditions on the nonlinearity which guarantee the uniqueness of solution. Lastly, we produce some local regularity results (namely, Caccioppoli-type inequality, Sobolev-Poincar\'e-type inequalities and higher integrability) for minimizers of the integral functionals corresponding to the operators., Comment: 39 pages. arXiv admin note: text overlap with arXiv:2103.08928 by other authors

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha}\left(\Omega\right) \cap H^1\left(\Omega\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space., Comment: 26 pages

In this paper, we prove the existence of $k$ families of smooth unbounded domains $\Omega_s\subset\mathbb{R}^{N+1}$ with $N\geq1$, where \begin{equation} \Omega_s=\left\{(x,t)\in \mathbb{R}^N\times \mathbb{R}:\vert x\vert<1+s\cos \left(\frac{2\pi}{T(s)}t\right)+s w_s\left(\frac{2\pi}{T(s)}t\right)\right\},\nonumber \end{equation} such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, \partial_\nu u=0,\,\,u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a bounded sign-changing solution with exactly $k+1$ nodal domains. These results can be regarded as counterexamples to the Schiffer conjecture on unbounded domain. These results also indicate that there exist non-spherical unbounded regions without Pompeiu property. Our construction shows that the condition "$\partial\Omega$ is homeomorphic to the unit sphere" is necessary for Williams conjecture to hold. In addition, these conclusions may have potential applications in remote sensing or CT., Comment: We find that the proof of our main Theorem 1.1 and Theorem 1.2 are wrong, where the essential reason is that the linearized operator in 23 page is not order 1 elliptic. Thus we wish to withdraw all versions of this paper

In this paper, we consider the following overdetermined eigenvalue problem on an unbounded domain $\Omega\subset\mathbb{R}^{N+1}$ with $N\geq1$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u=\lambda u\,\, &\text{in}\,\, \Omega,\\ u=0 &\text{on}\,\, \partial \Omega,\\ \partial_\nu u=\text{const} &\text{on}\,\, \partial \Omega. \end{array} \right.\nonumber \end{equation} Let $\lambda_k$ be the $k$-th eigenvalue of the zero-Dirichlet Laplacian on the unit ball for any $k\in \mathbb{N^+}$ with $k\geq 3$. We can construct $k$ smooth families of nontrivial unbounded domains $\Omega$, bifurcating from the straight cylinder, which admit a nonsymmetric solution with changing the sign by $k-1$ times to the overdetermined problem. While the existence of such domains for $k=1,2$ has been well-known, to the best of our knowledge this is the first construction for any positive integer $k\geq 3$. Due to the complexity of studying high eigenvalue problem, our proof involves some novel analytic ingredients. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain., Comment: 33 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2304.05550

A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor critical graph if $G-V'$ contains a $P_{\geq k}$-factor for any $V'\subseteq V(G)$ with $|V'|=l$. A graph $G$ is called a $(P_{\geq k},m)$-factor deleted graph if $G-E'$ has a $P_{\geq k}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\geq 3}$-factor. In this paper, we give some sufficient conditions for a graph to be a $(P_{\geq 3},l)$-factor critical graph or a $(P_{\geq 3},m)$-factor deleted graph. In this paper, we demonstrate that (i) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its sun toughness $s(G)>\frac{l+1}{3}$ and $\kappa(G)\geq l+2$. (ii) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its degree sum $\sigma_3(G)\geq n+2l$ and $\kappa(G)\geq l+1$. (iii) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if its sun toughness $s(G)\geq \frac{m+1}{m+2}$ and $\kappa(G)\geq 2m+1$. (iv) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if its degree sum $\sigma_3(G)\geq n+2m$ and $\kappa(G)\geq 2m+1$.

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a sign-changing solution. The domains bifurcate from the straight cylinder $B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal., Comment: 32 pages, 3 figures

We obtain a continuous family of nontrivial domains $\Omega_s\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_s, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_s \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result obtained by Ruiz, here we obtain a family domains $\Omega_s$ by using Crandall-Rabinowitz bifurcation theorem instead of a sequence of domains., Comment: We find that some essential mistakes in this paper. The proof of Proposition 4.1 in page 12-13 is wrong because we used the implicit function theorem incorrectly. Thus we wish to withdraw this article. We also did not submit this work to any journal

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Furthermore, we find that there is at least one inflection point on the wave profile between successive crests and troughs and the free surface is strictly concave at any crest and strictly convex at any trough. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the fibering method by using the Nehari manifold we are going to prove the existence of at least two weak solutions for such problems when the parameter is sufficiently small.

A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in $\mathcal{F}$. In particular, $H$ is called an $\{P_2,P_3\}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called an $\mathcal{S}_n$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A spanning subgraph of a graph $G$ is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component is isomorphic to a path of order at least $k$, where $k\geq2$. A graph $G$ is called a $\mathcal{F}$-factor covered graph if there is a $\mathcal{F}$-factor of $G$ including $e$ for any $e\in E(G)$. In this paper, we give a minimum degree condition for a $K_{1,r}$-free graph to have an $\mathcal{S}_n$-factor and a $\mathcal{P}_{\geq 3}$-factor, respectively. Further, we obtain sufficient conditions for $K_{1,r}$-free graphs to be $\mathcal{P}_{\geq 2}$-factor, $\mathcal{P}_{\geq 3}$-factor or $\{P_2,P_3\}$-factor covered graphs. In addition, examples show that our results are sharp., Comment: 13 pages

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=\lambda^N a(x)f(-u)\,\, &\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial \Omega. \end{array} \right.\nonumber \end{eqnarray} We establish global bifurcation results for the problem. We find intervals of $\lambda$ for the existence, multiplicity and nonexistence of strictly convex solutions for this problem., Comment: arXiv admin note: substantial text overlap with arXiv:1207.6669

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda u+h(x,u,\lambda)\,\,\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,\Omega. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(a\lambda_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of $a=0$., Comment: 20 pages

In this paper, we consider the following eigenvalue problem {{l} (|u"|^{p-2}u")"=\lambda m(x)|u|^{p-2}u, x\in (0,1), u(0)=u(1)=u"(0)=u"(1)=0, where $1

In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(\mu_k^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^\nu)^+$ and $(\mathcal{C}_{k}^\nu)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^\nu$ from $(\mu_k^\nu, 0)$, where $\mu_k^\nu$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $\nu\in{+,-}$. As the applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight., Comment: arXiv admin note: substantial text overlap with arXiv:1203.3262