Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples

Bifurcation of sign-changing solutions for an overdetermined boundary problem in bounded domains

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: 17 pages, 1 figur

Bifurcation domains from any high eigenvalue for an overdetermined elliptic problem

Let $\lambda_k$ be the $k$-th eigenvalue of the zero-Dirichlet Laplacian on the unit ball for $k\in \mathbb{N^+}$. We prove the existence of $k$ 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 with changing the sign by $k-1$ times. This nonsymmetric sign-changing solution can be seen as the perturbation of the eigenfunction corresponding to $\lambda_k$ with $k\geq2$. The main contribution of the paper is to provide some counterexamples to the Berenstein conjecture on unbounded domain.Comment: 26pp. arXiv admin note: substantial text overlap with arXiv:2304.0555

The global bifurcation of periodic internal waves with point vortex and capillary effect

In this paper, we first construct two-dimensional periodic interface waves with point vortex and capillary effect and then obtain the global structure of the set of solutions. This is done using the local and global bifurcation argument. Especially, we establish a global continuation theorem by using the degree for $C^1$ Fredholm mappings. As far as we know, the global result is new and general, which is promising to deal with other new phenomena in water waves.Comment: 19p

Geochronological, geochemical and Nd-Hf-Os isotopic fingerprinting of an early Neoproterozoic arc-back-arc system in South China and its accretionary assembly along the margin of Rodinia

This research was jointly funded by the NSFC project (40825009 and 40830319), Closure of Eastern Paleotethys Ocean and assembly of South China continents (41190073) of Major NSFC Program (41190070) Reconstruction of East Asian blocks in Pangea and the State Key Laboratory of Continental Dynamics, Northwest University (BJ081331). PAC acknowledges support from the University of St Andrews and NERC (NE/J021822/1).U-Pb geochronology along with elemental and Nd-Hf-Os isotopic data from the earliest Neoproterozoic metabasic rocks within the Cathaysia Block of the South China Block (SCB) constrain the tectonic setting and paleogeography of the block within the Rodinia supercontinent. The metabasic rocks give zircon U-Pb ages of 969-984 Ma, epsilon(Hf)(t) values of +1.8 to +15.3 and Hf model ages of 0.92-1.44 Ga. They are subalkaline basalts that can be geochemically classified into four groups. Group 1 has low Nb contents (1.24-4.33 ppm), highly positive epsilon(Nd)(t) values (+4.3 to +5.2), and REE and multi-elemental patterns similar to fore-arc MORB-type basalt. Group 2 has Nb contents ranging from 3.13 ppm to 6.48 ppm, epsilon(Nd)(t) of +3.1 to +6.2, low Re and Os contents and high initial Os isotopic ratios, and displays an E-MORB geochemical signature. Group 3 has Nb = 7.18-29.87 ppm, Nb/La = 0.60-1.40, Nb/U = 5.0-37, Ce/Pb = 1.1-6.6, epsilon(Nd)( t) = +2.9 to +7.0, Re-187/Os-188 = 5.87-8.87 and gamma Os (t) = 178-772, geochemically resembling to the Pickle Nb-enriched basalt. Group 4 has strong LREE/HREE and HREE fractionation and high epsilon(Nd)(t) values (+2.3 to +5.6), and is characterized by similar element patterns to arc volcanic rocks. Serpentinites coeval to Group 4 show Os-187/Os-188 of 0.1143-0.1442 and gamma Os (t) of -7.8 to +0.1. Groups 1 and 2 are interpreted to originate from the N-MORB and E-MORB-like sources with the addition of an arc-like component, genetically linked to fore- and back-arc settings, respectively. Groups 3 and 4 show inputs of newly subduction-derived melt and fluid in the wedge source. These geochronological and geochemical signatures fingerprint the development of an earliest Neoproterozoic (similar to 970 Ma) arc-back-arc system along the Wuyi-Yunkai domain of the Cathaysia Block. Regional relationships indicate that the Wuyi-Yunkai arc-back-arc system was one of a series of separate convergent margin settings, which included the Shuangxiwu (similar to 970-880 Ma) and Jiangnan (similar to 870-820 Ma) systems that developed in the SCB. The formation and closure of these arc-back-arc systems resulted in the northwestwardly episodic amalgamation of various pieces of the Yangtze and Cathaysia to finally form the SCB. These signatures require the SCB to occupy an exterior accretionary orogen along the periphery of Rodinia during 990-820 Ma, rather than to have formed through Mesoproterozoic Sibao orogenesis within the interior of Rodinia. (c) 2013 Elsevier B.V. All rights reserved.Peer reviewe