Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components by Luisa Toscano, Speranza Toscano

“…We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. …”
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The Dirichlet Problem for the Equation Δu−k2u=0 in the Exterior of Nonclosed Lipschitz Surfaces by P. A. Krutitskii

“…We study the Dirichlet problem for the equation Δu−k2u=0 in the exterior of nonclosed Lipschitz surfaces in R3. …”
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Existence of Positive Solutions to Singular 𝑝-Laplacian General Dirichlet Boundary Value Problems with Sign Changing Nonlinearity by Qiying Wei, You-Hui Su, Subei Li, Xing-Xue Yan

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The Dirichlet Problem for the 2D Laplace Equation in a Domain with Cracks without Compatibility Conditions at Tips of the Cracks by P. A. Krutitskii

“…We study the Dirichlet problem for the 2D Laplace equation in a domain bounded by smooth closed curves and smooth cracks. …”
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Existence of Weak Solutions for a Sturm–Liouville Type System Using Critical Points Theorems by Saeid Shokooh, Amir Ali Shamlou

“…In this paper, we will study a system of Sturm–Liouville differential equations under the Dirichlet boundary condition. First, by using a three-critical-point theorem, we check the existence of at least three weak solutions for the problem, and then, by utilizing a local minimum theorem, we present sufficient conditions so that the existence of at least one nontrivial weak solution for the problem is guaranteed. …”
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ON SOLVABILITY OF SOME BOUNDARY-VALUE PROBLEMS FOR THE NON-LOCAL POISS by B. Kh. Turmetov

“…The first problem generalizes the well-known Dirichlet, Neumann, and Robin problems for fractional-order boundary operators. …”
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Identifying Copy Number Variants under Selection in Geographically Structured Populations Based on -statistics by Hae-Hiang Song, Hae-Jin Hu, In-Hae Seok, Yeun-Jun Chung

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Decomposition conditions for two-point boundary value problems by Wenying Feng

“…We study the solvability of the equation x″=f(t,x,x′) subject to Dirichlet, Neumann, periodic, and antiperiodic boundary conditions. …”
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Positive solutions of higher order quasilinear elliptic equations by Marcelo Montenegro

“…The higher order quasilinear elliptic equation −Δ(Δp(Δu))=f(x,u) subject to Dirichlet boundary conditions may have unique and regular positive solution. …”
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Existence of Solutions of a Nonlocal Elliptic System via Galerkin Method by Alberto Cabada, Francisco Julio S. A. Corrêa

“…By means of the Galerkin method and by using a suitable version of the Brouwer fixed-point theorem, we establish the existence of at least one positive solution of a nonlocal elliptic N-dimensional system coupled with Dirichlet boundary conditions.…”
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Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation by Xinfu Li, Guang Zhang, Ying Wang

“…By using sub- and upper-solution method, the existence of bounded positive solutions and the existence and uniqueness of positive solitons will be established. To this end, the Dirichlet eigenvalue problem with positive and negative coefficients is considered, and a general sub- and upper-solution theorem is also obtained.…”
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𝑊𝟐,𝟐 A Priori Bounds for a Class of Elliptic Operators by Sara Monsurrò, Maria Salvato, Maria Transirico

“…We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.…”
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Infinitely Many Solutions for a Superlinear Fractional p-Kirchhoff-Type Problem without the (AR) Condition by Xiangsheng Ren, Jiabin Zuo, Zhenhua Qiao, Lisa Zhu

“…We only consider the non-Ambrosetti-Rabinowitz condition to prove our results by using the symmetric mountain pass theorem.…”
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Existence Results for Polyharmonic Boundary Value Problems in the Unit Ball by Sonia Ben Othman, Habib Mâagli, Malek Zribi

“…Our approach is based on estimates for the polyharmonic Green function on B with zero Dirichlet boundary conditions, including a 3G-theorem, which leeds to some useful properties on functions belonging to the Kato class.…”
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Estimates for the Green function and singular solutions for polyharmonic nonlinear equation by Imed Bachar, Habib Màagli, Syrine Masmoudi, Malek Zribi

“…We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of ℝn(n≥2) corresponding to zero Dirichlet boundary conditions. …”
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Positive Solutions for Some Competitive Fractional Systems in Bounded Domains by Imed Bachar, Habib Mâagli, Noureddine Zeddini

“…Using some potential theory tools and the Schauder fixed point theorem, we prove the existence and precise global behavior of positive continuous solutions for the competitive fractional system (−Δ|D)α/2u+p(x)uσvr=0, (−Δ|D)α/2v+q(x)usvβ=0 in a bounded C1,1-domain D in ℝn (n≥3), subject to some Dirichlet conditions, where 0<α<2, σ,β≥1,s,r≥0. …”
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On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces by A. Lastra, S. Malek

“…This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.…”
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On the uniqueness of linear convection–diffusion equations with integral boundary conditions by Lee, Chiun-Chang, Mizuno, Masashi, Moon, Sang-Hyuck

“…This approach has the advantage of transforming the integral boundary conditions into new Dirichlet boundary conditions so that we can obtain refined estimates, and the comparison theorem can be applied to the equations. …”
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Definition and Properties of the Libera Operator on Mixed Norm Spaces by Miroslav Pavlovic

“…This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. …”
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Efficient Variational Approaches for Deformable Registration of Images by Mehmet Ali Akinlar, Muhammet Kurulay, Aydin Secer, Mustafa Bayram

“…Dirichlet, anisotropic, and Huber regularization terms are presented for efficient registration of deformable images. …”
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