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In this present paper, we are first of all interested in some continuity and compactness results for the space ψ-fractional 핊 풜 α , β ; ψ ⁢ ( Λ ) {\mathbb{S}^{\alpha,\beta;\psi}_{\mathcal{A}}(\Lambda)} . In this sense, we investigate the existence of at least two solutions with constant signs using truncation arguments and comparison methods of a new class of fractional differential equations with m ⁢ ( ξ ) {m(\xi)} -Laplacian with double phase. [ABSTRACT FROM AUTHOR]

We study a random matching problem on closed compact 2-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m = m (n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures μ n and ν m is quantitatively well-approximated by (Id , exp (∇ h n) ) # μ n where h n solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge–Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the α -mixing coefficient holds and for a class of discrete-time sub-geometrically ergodic Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure. [ABSTRACT FROM AUTHOR]

In this paper, the existence of solutions, (λ , u) , of the problem - Δ u = λ u - a (x) | u | p - 1 u in Ω , u = 0 on ∂ Ω , is explored for 0 < p < 1 . When p > 1 , it is known that there is an unbounded component of such solutions bifurcating from (σ 1 , 0) , where σ 1 is the smallest eigenvalue of - Δ in Ω under Dirichlet boundary conditions on ∂ Ω . These solutions have u ∈ P , the interior of the positive cone. The continuation argument used when p > 1 to keep u ∈ P fails if 0 < p < 1 . Nevertheless when 0 < p < 1 , we are still able to show that there is a component of solutions bifurcating from (σ 1 , ∞) , unbounded outside of a neighborhood of (σ 1 , ∞) , and having u ⪈ 0 . This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described. [ABSTRACT FROM AUTHOR]

We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts - Δ u + V (x) u + κ 2 u Δ (u 2) = 1 | x | β (∫ R 2 H (u) | x - y | μ | y | β d y) h (u) , x ∈ R 2 , where κ ∈ R \ { 0 } is a parameter, β > 0 , 0 < μ < 2 with 0 < 2 β + μ < 2 and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For κ < 0 : (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that V ∈ C 0 (R 2 , R +) and inf x ∈ R 2 V (x) > 0 , which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when V ≡ 1 . Moreover, if κ > 0 is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when κ → 0 + . It seems that the results presented above are even new for the case κ = 0 . [ABSTRACT FROM AUTHOR]

Let. Let and. Consider the formal second-order differential operator in. We show that the closure of is quasi-m-accretive under certain conditions on the coefficients. [ABSTRACT FROM AUTHOR]

This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of C 1 , α solutions to address the research gap between low-regularity results for bounded and measurable diffusions and the smooth results modeled by the Laplacian. We use the Hopf-Cole transformation to convert the MFG system into a scalar elliptic equation. Then, we apply Morrey space methods to establish the existence and regularity of solutions. The introduction of Morrey space methods offers a novel approach to address regularity issues in the context of MFGs. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the properties of the category, we introduce suitable maps between the expanding domains and the critical levels of the energy functional related to the problem, which allow us to estimate the number of positive solutions by the shape of the domain. [ABSTRACT FROM AUTHOR]

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases, or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus. [ABSTRACT FROM AUTHOR]

In this article, we consider a nonlinear elliptic unilateral equation whose model is −∑i=1N∂iσi(x,u,∇u)+L(x,u,∇u)+N(x,u,∇u)=μ−divϕ(u)inΩ.-\mathop{\sum }\limits_{i=1}^{N}{\partial }^{i}{\sigma }_{i}\left(x,u,\nabla u)+L\left(x,u,\nabla u)+N\left(x,u,\nabla u)=\mu -{\rm{div}}\phi \left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega . We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ=f−divF\mu =f-{\rm{div}}F belongs to L1(Ω)+W−1,p′(Ω){L}^{1}\left(\Omega )+{W}^{-1,{p}^{^{\prime} }}\left(\Omega ). The nonlinear terms L(x,s,∇u)L\left(x,s,\nabla u) satisfy the sign and growth conditions, and N(x,s,∇u)N\left(x,s,\nabla u) verifies only the growth conditions.

Let w 0 be a bounded, C 3 , strictly plurisubharmonic function defined on B 1 ⊂ C n . Then w 0 has a neighborhood in L ∞ (B 1) . Suppose that we have a function ϕ in this neighborhood with 1 - ε ≤ M A (ϕ) ≤ 1 + ε and there exists a function u solving the linearized complex Monge–Amp e ' re equation: d e t (ϕ k l ¯ ) ϕ i j ¯ u i j ¯ = 0 . Then there exist constants α > 0 and C such that | u | C α (B 1 2 (0)) ≤ C , where α > 0 depends on n and C depends on n and | u | L ∞ (B 1 (0)) , as long as ϵ is small depending on n. This partially generalizes Caffarelli–Gutierrez's estimate for linearized real Monge–Amp e ' re equation to the complex version. [ABSTRACT FROM AUTHOR]

In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . Assume that L R is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on L 2 (Ω) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of L R , the authors establish a new atomic characterization of the Hardy space H L R p (Ω) associated with L R . Using this, the authors further show that, for any given p ∈ (n n + δ 0 , 1 ] , H z p (Ω) + L ∞ (Ω) = H L N p (Ω) = H L R p (Ω) ⫋ H L D p (Ω) = H r p (Ω) , where H L D p (Ω) and H L N p (Ω) denote the Hardy spaces on Ω associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, H z p (Ω) and H r p (Ω) respectively denote the "supported type" and the "restricted type" Hardy spaces on Ω , and δ 0 ∈ (0 , 1 ] is the critical index depending on the operators L D , L N , and L R . The authors then obtain the boundedness of the Riesz transform ∇ L R - 1 / 2 on the Lebesgue space L p (Ω) when p ∈ (1 , ∞) [if p > 2 , some extra assumptions are needed] and its boundedness from H L R p (Ω) to L p (Ω) when p ∈ (0 , 1 ] or to H r p (Ω) when p ∈ (n n + 1 , 1 ] . As applications, the authors further obtain the global regularity estimates, in L p (Ω) when p ∈ (0 , p 0) and in H r p (Ω) when p ∈ (n n + 1 , 1 ] , for the inhomogeneous Robin problem of L R on Ω , where p 0 ∈ (2 , ∞) is a constant depending only on n, Ω , and the operator L R . The main novelties of these results are that the range (0 , p 0) of p for the global regularity estimates in the scale of L p (Ω) is sharp and that, in some sense, the space X : = H L R 1 (Ω) is also optimal to guarantee both the boundedness of ∇ L R - 1 / 2 from X to L 1 (Ω) or to H r 1 (Ω) and the global regularity estimate ‖ ∇ u ‖ L n n - 1 (Ω ; R n) ≤ C ‖ f ‖ X for inhomogeneous Robin problems with C being a positive constant independent of both u and f. [ABSTRACT FROM AUTHOR]

In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: (−Δ)su+λu=f(u)+∣u∣2s*−2u,inRN,(Pm)∫RN∣u∣2dx=m2,\hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}\left({P}_{m})\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 00m\gt 0, 2s*≔2NN−2s{2}_{s}^{* }:= \frac{2N}{N-2s}, λ\lambda is an unknown parameter that will appear as a Lagrange multiplier, and ff is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about ff, with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to (Pm)\left({P}_{m}). We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s=1s=1.

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form. [ABSTRACT FROM AUTHOR]

In the framework of Clifford analysis, we study higher order Dirac operators constructed with k-vectors. We find a necessary and sufficient condition to determine whether a function cancels them. [ABSTRACT FROM AUTHOR]

In this paper, we study the unique continuation property for the fractional p-elliptic equations in a semigroup form with variable coefficients. By employing an extension procedure, we derive a monotonicity formula for an extended frequency function. Utilizing this monotonicity together with a blow-up analysis, we establish the unique continuation property. [ABSTRACT FROM AUTHOR]

The main thrust of our current work is to exploit very specific characteristics of a given problem in order to acquire improved compactness for supercritical problems and to prove existence of new types of solutions. To this end, we first introduce an efficient tool in the context of variational methods in order to construct a new type of classical solutions for a large class of supercritical elliptic partial differential equations. The issue of symmetry and symmetry breaking is challenging and fundamental in mathematics and physics. Symmetry breaking is the source of many interesting phenomena namely phase transitions, instabilities, segregation, etc. As a consequence of our results we shall establish the existence of several symmetry breaking solutions when the underlying problem is fully symmetric. Our methodology is variational, and we are not seeking non symmetric solutions which bifurcate from the symmetric one. Instead, we construct many new positive solutions by utilizing a minimax principle for general semilinear elliptic problems restricted to a given convex subset instead of the whole space. As a byproduct of our investigation, several new Sobolev embeddings are established for functions having a mild monotonicity on symmetric monotonic domains. [ABSTRACT FROM AUTHOR]

We revisit the following fractional Schrödinger equation 0.1 ε 2 s (- Δ) s u + V u = u p - 1 , u > 0 , in R N , where ε > 0 is a small parameter, (- Δ) s denotes the fractional Laplacian, s ∈ (0 , 1) , p ∈ (2 , 2 s ∗) , 2 s ∗ = 2 N N - 2 s , N > 2 s , V ∈ C ( R N , [ 0 , + ∞)) is a general potential. Under various assumptions on V(x) at infinity, including V(x) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value p ∗ , such that the above problem admits positive concentration solutions if p ∈ (p ∗ , 2 s ∗) , while it has no positive weak solutions for p ∈ (2 , p ∗) if p ∗ > 2 , where the threshold p ∗ ∈ [ 2 , 2 s ∗) can be characterized explicitly by p ∗ = 2 + 2 s N - 2 s if lim | x | → ∞ (1 + | x | 2 s ) V (x) = 0 , 2 + ω N + 2 s - ω if 0 < inf (1 + | x | ω ) V (x) ≤ sup (1 + | x | ω) V (x) < ∞ for some ω ∈ [ 0 , 2 s ] , 2 if inf V (x) log (e + | x | 2) > 0. Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω ⊂ R n be a bounded Lipschitz domain. Assume that b ∈ L n ∗ (Ω ; R n) and γ is a non-negative function on ∂ Ω satisfying some mild assumptions, where n ∗ : = n when n ≥ 3 and n ∗ ∈ (2 , ∞) when n = 2 . In this article, we establish the unique solvability of the Robin problems - Δ u + div (u b) = f in Ω , ∇ u - u b · ν + γ u = u R on ∂ Ω and - Δ v - b · ∇ v = g in Ω , ∇ v · ν + γ v = v R on ∂ Ω in the Bessel potential space L α p (Ω) , where α ∈ (0 , 2) and p ∈ (1 , ∞) satisfy some restraint conditions, and ν denotes the outward unit normal to the boundary ∂ Ω . The results obtained in this article extend the corresponding results established by Kim and Kwon (Trans Am Math Soc 375:6537–6574, 2022) for the Dirichlet and the Neumann problems to the case of the Robin problem. [ABSTRACT FROM AUTHOR]

We present the application of Lie group theory analysis with ζ -Riemann–Liouville fractional derivative (ζ -RLFD, for short) detailing the construction of infinitesimal prolongation to obtain Lie symmetries. In addition, it addresses the invariance condition without necessarily imposing that the lower limit of the fractional integral is fixed. We find an expression that expands the knowledge regarding the study of exact solutions for fractional differential equations. We apply the Leibniz-type rule for the derivative operator in question to build the prolongation. At last, we calculate the Lie symmetries of the generalized Burgers equation and fractional porous medium equation. [ABSTRACT FROM AUTHOR]

We consider the magnetic Schrödinger operator H = (i ∇ + A) 2 - V with a non-negative potential V supported over a strip which is a local deformation of a straight one, and the magnetic field B : = rot (A) is assumed to be non-zero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of H to be empty. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

Subsolutions and concavity play critical roles in classical solvability, especially a priori estimates, of fully nonlinear elliptic equations. Our first primary goal in this paper is to explore the possibility to weaken the concavity condition. The second is to clarify relations between weak notions of subsolution introduced by Székelyhidi and the author, respectively, in attempt to treat equations on closed manifolds. More precisely, we show that these weak notions of subsolutions are equivalent for equations defined on convex cones of type 1 in the sense defined by Caffarelli, Nirenberg and Spruck.

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.

Let Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x)uqinΩu>0inΩu=0on∂Ω(k−1)π

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4). [ABSTRACT FROM AUTHOR]

The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the $ f^* $ f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We present a canonical series expansion for generalized harmonic functions in the open unit disc in the complex plane that generalizes that recently obtained for the class of $ (p,q) $ (p , q) -harmonic functions. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to discuss the local Hölder continuity of the gradients for weak solutions of the following non-homogenous elliptic p(x)-Laplacian equations of divergence form div A (x) ∇ u (x) · ∇ u (x) p (x) - 2 2 A (x) ∇ u (x) = div | f (x) | p (x) - 2 f (x) in Ω , where Ω ⊂ R n is an open bounded domain for n ≥ 2 , under some proper non-Hölder conditions on the variable exponents p(x) and the coefficients matrix A(x). [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth (- Δ) A s u + V (x) u = λ | u | p - 2 u + f (x , | u | 2) u , x ∈ R N , where (- Δ) A s is the fractional magnetic operator with 0 < s < 1 , N > 2 s , 2 s ∗ = 2 N N - 2 s , λ > 0 , V ∈ C (R N , R) and A ∈ C (R N , R N) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, and f is a continuous function and there exists 2 < q < 2 s ∗ such that | f (x , t) | ≤ C (1 + | t | q - 2 2) for all (x, t), for 2 s ∗ ≤ p < 22 s ∗ - q . For any D > 0 fixed, if λ ∈ (0 , D ] we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior L ∞ -estimate. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence of generalized variational solutions for nonlinear Dirichlet problems driven by anisotropic and Finsler Laplacian competing operators. The main difficulty consists in the lack of ellipticity and monotonicity in the principal part of the equations. This difficulty is overcome by developing a Galerkin-type procedure. [ABSTRACT FROM AUTHOR]

It is established existence and multiplicity of solutions for semilinear elliptic problems defined in the whole space R N considering subcritical nonlinearities with some parameters. Here we emphasize that our nonlinearities can be sign-changing functions. The main difficulty is proving the existence of nontrivial solutions by using the Nehari method, taking into account that the Lagrange multipliers theorem cannot be directly applied in our setting. In fact, we consider the case where the fibering map admits inflection points. In other words, we consider the case where the Nehari set admits degenerate critical points. Hence our main contribution is to consider a huge class of semilinear elliptic problems where the standard Nehari method cannot be applied. Using some fine estimates and recovering some compactness results together with the nonlinear Rayleigh quotient, we prove that our main problem admits at least three nontrivial solutions depending on the parameters. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of positive multi-peak solutions to the following elliptic problem - Δ v + (λ + V (x)) v = v p in Ω , v > 0 in Ω , ∫ Ω v 2 d x = ρ , where Ω is a bounded smooth domain of R N or the whole space R N , the exponent p satisfies 1 < p < N + 2 N - 2 for N ≥ 3 and p > 1 for N = 1 , 2 . For the case of mass subcritical, mass critical, and mass supercritical, we shall deal with the effect of ρ on the existence of the solution concentrating at k different points, which belong to either ∂ Ω or Ω , or R N . [ABSTRACT FROM AUTHOR]

This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of L 2 → L 2 Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result. [ABSTRACT FROM AUTHOR]

In this present paper, we concern the discussion of multiplicity non-trivial solution for a new class of fractional differential equations of the Kirchhoff type in the ψ -fractional space S H , 0 α , β ; ψ (Λ) via critical point result and variational methods. [ABSTRACT FROM AUTHOR]

In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system { − Δ u 1 − λ 1 u 1 = μ 1 u 1 3 + β u 1 u 2 2 + κ (x) u 2 in R 3 , − Δ u 2 − λ 2 u 2 = μ 2 u 2 3 + β u 1 2 u 2 + κ (x) u 1 in R 3 , ∫ R 3 u 1 2 = a 1 2 , ∫ R 3 u 2 2 = a 2 2 , where a 1 , a 2 , μ 1 , μ 2 , κ = κ (x) > 0 , β < 0 , and λ 1 , λ 2 are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive. [ABSTRACT FROM AUTHOR]

This paper is concerned with the elliptic equation - Δ u = λ (a - u) p in a connected, bounded C 2 domain Ω of R N subject to zero Dirichlet boundary conditions, where λ > 0 , p > 0 and a : Ω ¯ → [ 0 , 1 ] vanishes at the boundary with the rate dist (x , ∂ Ω) γ for γ > 0 . When N = 2 and p = 2 , this equation models the closed micro-electromechanical systems devices, where the elastic membrane sticks the curved ground plate on the boundary, but insulating on the boundary. The function a shapes the curved ground plate. Our aim in this paper is to study some qualitative properties of minimal solutions of this equation when λ > 0 , p > 0 and to show how the boundary decaying of a works on the behavior of minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of renormalized solution for nonlinear multivalued elliptic problems β (u) - d i v (a (x , ∇ u) + F (u)) ∋ f in the framework of Orlicz spaces and L 1 -data. [ABSTRACT FROM AUTHOR]

We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings. [ABSTRACT FROM AUTHOR]

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion $ a_0({\boldsymbol {x}})+\sum _{j\in {\mathbb N}}y_ja_j({\boldsymbol {x}}) $ a 0 (x) + ∑ j ∈ N y j a j (x) , where elements of the parameter vector $ {\boldsymbol {y}}=(y_j)_{j\in {\mathbb N}}\in U^\infty $ y = (y j) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on $ U:=[-\frac {1}{2},\frac {1}{2}] $ U := [ − 1 2 , 1 2 ]. Under the assumption $ \|\sum _{j\in {\mathbb N}}\rho _j|a_j|\|_{L_\infty (D)} ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ (D) < ∞ with some positive sequence $ (\rho _j)_{j\in {\mathbb N}}\in \ell _p({\mathbb N}) $ (ρ j) j ∈ N ∈ ℓ p (N) for $ p\in (0,1] $ p ∈ (0 , 1 ] we show that for any $ {\boldsymbol {y}}\in U^\infty $ y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues $ (\lambda _j({\boldsymbol {y}}))_{j\in {\mathbb N}} $ (λ j (y)) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap $ \lambda _2({\boldsymbol {y}})-\lambda _1({\boldsymbol {y}}) $ λ 2 (y) − λ 1 (y) is uniformly positive in $ U^\infty $ U ∞ . From this, we prove the holomorphic extension property of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) to a complex domain in $ {\mathbb C}^\infty $ C ∞ and estimate partial derivatives of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y) with respect to the parameter $ {\boldsymbol {y}} $ y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of $ \lambda _1({\boldsymbol {y}}) $ λ 1 (y). [ABSTRACT FROM AUTHOR]