Mountain Pass Research Articles

Infinitely many low‐ and high‐energy solutions for double‐phase problems with variable exponent

AbstractThe aim of this paper is the study of double‐phase problems with variable exponent. Using the Clark's theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively.

Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

Interaction regimes of surface water and groundwater in a hyper-arid endorheic watershed on Tibetan Plateau: Insights from multi-proxy data

The interaction between surface water and groundwater is crucial for the management of water resources and the preservation of ecosystems within arid basins, but knowledge on their interaction regimes at the watershed scale remains relatively limited. The present research synthesizes a range of multi-proxy data, including satellite thermal infrared remote sensing, water piezometric level, hydrochemical analyses, and isotopic signatures (2H, 18O and 222Rn), to elucidate the interaction dynamics and patterns between surface water and groundwater within a representative hyper-arid closed basin on Tibetan Plateau. The findings indicate that the water in the basin originates primarily from the glacier/snow melt water and precipitation in the mountainous regions, and would experience multiple but spatially heterogeneous interactions between river and aquifers from the mountain pass to the tail salt lakes after entering the basin. Water interaction in the piedmont alluvial plain is dominated by the pattern of river seepage into aquifers with a water flux of 25.82 m3/s, which drives the development of hierarchical groundwater flow systems in the basin. The interaction pattern converts to groundwater discharge of the local groundwater flow system at the front of alluvial fan plain, which forms the principal spring-fed rivers and the predominant overflow zone in the basin with the groundwater discharge flux of 3.22 m3/s. Most of these discharged groundwaters would infiltrate back into the aquifers in the middle-upper part of the loess plain with the river water leakage flux of 2.89 m3/s. River water and phreatic groundwater present a gradual enrichment of hydrochemcial components and water stable isotopes (2H and 18O) along the flow path due to the intense evaporation in the loess plain. The multiple water interactions between surface water and groundwater lead to the anomalous distribution of fresh water in the middle-lower stream area, which is related to the intermediate groundwater flow system and the local variable groundwater flow system. Water interaction in the lowest salt-marsh plain is in the pattern of regional groundwater flow system discharge into the tail salt lakes under natural condition, but evolves to only discharge into the artificial brine extraction canals rather than the tail salt lakes after decades of brine exploitation. Our findings provide a conceptual and preliminarily quantificational understanding of the interaction regimes between surface water and groundwater in large arid closed basins at the watershed scale.

Existence results for variable-order fractional Kirchhoff equations with variable exponents

This study establishes the existence of two nontrivial weak solutions for a specific class of Kirchhoff-type equations. Utilizing the Mountain Pass Theorem and the Ekeland variational principle, alongside insights from fractional Sobolev spaces with variable orders and exponents, we provide a concise demonstration of our result.

Reaffirming Loyalty and Legitimacy: Representations of Hui Multi-Layered Identity in Bai Lian’s “Mountain Pass”

In the People’s Republic of China (PRC), the Cultural Revolution (1966–1976) left many writers severed from their cultural roots. Starting in the 1980s, literary authors sought to address this disconnection by turning their attention to rural communities. This tendency is exemplified by the emergence of two significant trends: root-seeking literature and new fiction from Tibet. Root-seeking authors focused on local customs, marginalized cultures, and minority groups to reinvigorate Chinese literature and fill the perceived cultural void. Around the same time, new fiction from Tibet featured diverse responses to post-Mao changes, with some idealizing Tibet as a repository of “authentic” traditions, while others criticized its perceived backwardness. Both trends have been interpreted in scholarship as responses, often critical, to state policies. The short story “Mountain Pass” (1985) by Hui writer Bai Lian intersects with these movements temporally and thematically. However, unlike them, Bai Lian’s portrayal of rural communities emphasizes the Hui’s historical role in resisting the Qing empire, pivotal to the emergence of the PRC, while also highlighting the group’s Arab and Persian origins. This three-layered identification with the local, national, and transnational enriches our understanding of the 1980s literary landscape, challenging the notion that this era was solely characterized by resistance to the central state.

Multiple Solutions for a Critical Steklov Kirchhoff Equation

In the present work, we study some existing results related to a new class of Steklov p(x)-Kirchhoff problems with critical exponents. More precisely, we propose and prove some properties of the associated energy functional. In the first existence result, we use the mountain pass theorem to prove that the energy functional admits a critical point, which is a weak solution for such a problem. In the second main result, we use a symmetric version of the mountain pass theorem to prove that the investigated problem has an infinite number of solutions. Finally, in the third existence result, we use a critical point theorem proposed by Kajikiya to prove the existence of a sequence of solutions that tend to zero.

Mountain pass frozen planet orbits in the helium atom model

We seek frozen planet orbits for the helium atom through an application of the mountain pass lemma to the Lagrangian action functional. Our method applies to a wide class of gravitational-like interaction potentials, thus generalising the results in Cieliebak, Frauenfelder, and Volkov [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), 379–455]. We also let the charges of the two electrons tend to zero and perform an asymptotic analysis to prove convergence to a limit trajectory having a collision-reflection singularity between the electrons.

Existence of solutions for a class of asymptotically linear fractional Schrödinger equations

In this paper, we focus on studying a fractional Schrödinger equation of the form {(−Δ)su+V(x)u=f(x,u)in Ω,u>0in Ω,u=0in Rn∖Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases}(-\\Delta )^{s}u+V(x)u = f(x,u) &\ ext{in }\\Omega , \\\\ u>0 &\ ext{in }\\Omega , \\\\ u =0 &\ ext{in }\\mathbb{R}^{n}\\setminus \\Omega , \\end{cases}$$\\end{document} where 0<s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0< s<1$\\end{document}, n>2s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n>2s$\\end{document}, Ω is a smooth bounded domain in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document}, (−Δ)s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\\Delta )^{s}$\\end{document} denotes the fractional Laplacian of order s, f(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(x, t)$\\end{document} is a function in C(Ω‾×R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$C(\\overline{\\Omega}\ imes \\mathbb{R})$\\end{document}, and f(x,t)/t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(x, t)/t $\\end{document} is nondecreasing in t and converges uniformly to an L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{\\infty}$\\end{document} function q(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q(x)$\\end{document} as t approaches infinity. The potential energy V satisfies appropriate assumptions.In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a “mountain pass” approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(x,t)$\\end{document} exhibits superlinearity in t at infinity, represented by q(x)≡+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q(x)\\equiv +\\infty $\\end{document}.

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

On class of bi-nonlocal problems involving fractional p(.,.)-Laplacian operators

This work aims to obtain the existence and multiplicity of solutions for a bi-nonlocal problem involving the p(.,.)-Laplacian operator. Under specific assumptions, the paper uses the Mountain Pass Theorem combined with the Ekeland variational approach to establish the existence of two nontrivial solutions for the problem. The existence of infinitely many solutions is proved using the Symmetric Mountain Pass Theorem and the Ljusternik-Schnirelman Theorem. Finally, we study a p(.,.)-Laplacian problem with non-variational form.

Estimating rare earth elements at various scales with bastnäsite indices for Mountain Pass

Numerous researchers are exploring innovative ways to map rare earth elements (REEs) critical to components in high technology devices precisely, and one approach that has made significant advances in recent years involves imaging spectroscopy. This research focuses on the utilization of this technique to map REEs across the Sulfide Queen mine and Birthday claim area located in Mountain Pass, California, where the highest concentrations of REEs within a carbonatite deposit in the United States have been reported. New spectral indices based on reflectance measurements of eleven representative ore samples under visible illumination in a laboratory setting were developed and applied to data collected by eight airborne and spaceborne imaging spectrometers. The results show that these indices can effectively detect REEs in any rock formation where an adequate amount of rare earth fluorocarbonates have accumulated. Identification can be achieved using airborne and spaceborne hyperspectral sensors with a spatial resolution of up to 30 m. Geochemical analyses confirm that a REE+Y ore grade of 3.25 wt% or greater is sufficient for detection using these indices.

Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}&amp;amp;#x0005E;{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .

Identifying Rare Earth Elements Using a Tripod and Drone-Mounted Hyperspectral Camera: A Case Study of the Mountain Pass Birthday Stock and Sulphide Queen Mine Pit, California

As the 21st century advances, the demand for rare earth elements (REEs) is rising, necessitating more robust exploration methods. Our research group is using hyperspectral remote sensing as a tool for mapping REEs. Unique spectral features of bastnaesite mineral, has proven effective for detection of REE with both spaceborne and airborne data. In our study, we collected hyperspectral data using a Senop hyperspectral camera in field and a SPECIM hyperspectral camera in the laboratory settings. Data gathered from California’s Mountain Pass district revealed bastnaesite-rich zones and provided detailed insights into bastnaesite distribution within rocks. Further analysis identified specific bastnaesite-rich rock grains. Our results indicated higher concentrations of bastnaesite in carbonatite rocks compared to alkaline igneous rocks. Additionally, rocks from the Sulphide Queen mine showed richer bastnaesite concentrations than those from the Birthday shonkinite stock. Results were validated with thin-section studies and geochemical data, confirming the reliability across different hyperspectral data modalities. This study demonstrates the potential of drone-based hyperspectral technology in augmenting conventional mineral mapping methods and aiding the mining industry in making informed decisions about mining REEs efficiently and effectively.

Existence of solutions for critical Neumann problem with superlinear perturbation in the half‐space

AbstractIn this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem where , , , , , is the outward unit normal vector at the boundary , is the usual critical exponent for the Sobolev embedding and is the critical exponent for the Sobolev trace embedding . By establishing an improved Pohozaev identity, we show that problem () has no nontrivial solution if . Applying the mountain pass theorem without the condition and the delicate estimates for the mountain pass level, we obtain the existence of a positive solution for all and the different values of the parameters and . Particularly, for , , , we prove that problem () has a positive solution if and only if . Moreover, the existence of multiple solutions for () is also obtained by dual variational principle for all and suitable .

Detecting rare earth elements using EnMAP hyperspectral satellite data: a case study from Mountain Pass, California

Rare earth elements (REEs) exhibit diagnostic absorption features in the visible-near infrared region, enabling their detection and identification via spectroscopic methods. Satellite-based remote sensing mapping of REEs, however, has not been attainable so far due to the necessity for high-quality hyperspectral data to resolve their narrow absorption features. This research leverages EnMAP hyperspectral satellite data to map REEs in Mountain Pass, California—a mining area known to host bastnaesite-Ce ore in sövite and beforsite carbonatites. By employing a polynomial fitting technique to characterize the diagnostic absorption features of Neodymium (Nd) at ∼740 and ∼800 nm, the surface occurrence of Nd was successfully mapped at a 30m pixel resolution. The relative abundance of Nd was represented using the continuum-removed area of the 800 nm feature. The resulting map, highlighting hundreds of anomalous pixels, was validated through laboratory spectroscopy, surface geology, and high-resolution satellite imagery. This study marks a major advancement in REE exploration, demonstrating for the first time, the possibility of directly detecting Nd in geologic environments using the EnMAP hyperspectral satellite data. This capability can offer a fast and cost-effective method for screening Earth’s surfaces for REE signature, complementing the existing exploration portfolio and facilitating the discovery of new resources.

세종대 풍수학에 대한 인식과 그 실상 - 세종 15년(1433) ‘풍수학’ 설치 과정을 중심으로

This paper aims to explore the actual state of geomancy (pungsu, 風水) in the early 15th century and examine perceptions of geomancy from various perspectives by investigating the process of establishing the “Pungsuhak (風水學)” organization. In the seventh month of the 15th year of King Sejong (世宗)'s reign (1433), a debate arose in the court over the auspicious site of Gyeongbok-gung (景福宮) Palace, initiated by the geomancer Choi Yang-seon (崔揚善). This led to a geomantic debate in the court, reigniting the controversy over the mountain pass next to Heolleung (獻陵), which had been raised but remained unresolved in the 12th year of Sejong's reign (1430). Despite opposition from his officials, King Sejong ordered scholars from the Jiphyeonjeon to study geomantic texts to resolve the issues. He formalized the scholars' research duties by assigning them official titles and establishing the “Pungsuhak” organization. The establishment of “Pungsuhak” faced considerable opposition. The key points of contention were threefold: first, that geomancy should not be discussed in royal lectures; second, that geomancy was not a subject worthy of the king's personal study; and third, that the Jiphyeonjeon should not be engaged in the study of geomancy. King Sejong countered these objections by citing Zhu Xi (朱熹)'s example, arguing that discussing geomancy was not beyond the scope of a Confucian scholar's duties, and he proceeded with the research through the “Pungsuhak” organization. At the time, two other organizations oversaw geomancy: the Seoun-gwan (書雲觀), an administrative body, and the Yin-Yang Studies (陰陽學) of the Ten Schools (十學), an organization responsible for education and examinations. However, when the geomantic debate arose, Sejong chose not to utilize these existing organizations but instead established the “Pungsuhak” organization, entrusting the scholars of Jiphyeonjeon with the study of geomancy. Consequently, the geomancy departments within the Seoun-gwan and Yin-Yang Studies remained in a disadvantaged position.

On a Schrödinger equation involving fractional [formula omitted]-Laplacian with critical growth and Trudinger–Moser nonlinearity

A nonlinear Schrödinger equation of fractional (N/s1,q)-Laplacian is considered with the Rabinowitz potential, critical Sobolev growth and Trudinger–Moser nonlinearity in RN−ΔN/s1s1u+−Δqs2u+V(ɛx)(uNs1−2u+uq−2u)=λfu+uqs2∗−2u.We establish the global existence of nonnegative ground-state solution for suitable parameter values primarily through variational analysis, fractional Trudinger–Moser inequality and mountain pass approach. It is a crucial ingredient to handle three aspects concerning the limiting setting s1p=N, the critical Sobolev growth and Trudinger–Moser nonlinearity.

New results on fractional advection–dispersion equations

In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result.

Transforming Mining Tailings into Green Steel and Rare Earth Elements

As critical components for energy and defense technologies, the U.S. rare earth element (REE) ore reserves are estimated at ~13 million tons, but domestic production is currently limited to one operational mine residing on high-grade ore deposits in Mountain Pass, CA. While low-grade REE deposits are abundant, high-grade American REE deposits are sparse. Meanwhile, hazardous REE-containing tailings pile up at the economic and environmental cost, and it is estimated that REE value locked within e.g., red mud, is $796 per ton. However, such low-grade deposits or mining tailings are challenging to access because REEs are usually locked within a mineral matrix, typically iron oxide, that renders the ore intractable to mechanical or chemical separation. Here, we propose a process for recovering REEs from low-grade deposits and tailings by electro-metallurgically removing iron oxide, hydro-metallurgically removing aluminum salt, and subsequent separating insoluble rare earth oxides. Our strategy produces not only refinable REEs concentrates but also emissions-free iron metal and supplementary cementitious materials as co-products for decarbonizing steel and cement industries, respectively. By addressing the economic and environmental challenges associated with low-grade REE deposits and mining tailings, our electrochemical approach offers a sustainable pathway for unlocking the potential of domestic rare earth resources.

Fractional Musielak spaces: study of nonlocal elliptic problem with Choquard-logarithmic nonlinearity

In this study, we thoroughly examine a specific type of complex mathematical problems involving a fractional ψ κ , y ( ⋅ ) -Laplacian operator and a Choquard-logarithmic nonlinearity, combined with a real parameter, known as non-homogeneous elliptic problems. Our approach involves setting-specific conditions for the Choquard nonlinearities and the continuous function ψ κ , y . This allows us to successfully identify multiple solutions to these problems. Our analysis is conducted in the area of fractional Musielak spaces. We rely heavily on the mountain pass theorem and Ekeland's variational principle, as well as the Hardy–Littlewood–Sobolev inequality, which is essential for supporting the theoretical basis of our research.