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We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a k -star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the k -star graph to obtain a continuous solution for this coupled system. First, we showed the existence and uniqueness of the solution using the variational formulation approach. Then, we reformulated it into a minimization problem over a function space to conclude the uniqueness of the solution. For the approximation of the continuous problem, note that the upwind scheme for the flux condition at the free boundary leads to a parameter uniform first-order approximation. To obtain a higher-order uniform accuracy, we utilized a three-point scheme for first-order derivatives and a five-point approximation at the point of discontinuity. These approximations typically did not yield an M-matrix or strict diagonally dominant structure of the stiffness matrix. Hence, we provided a suitable transformation that could lead to a sufficient condition for preserving the strict diagonally dominant structure of the stiffness matrix. We performed a comprehensive convergence analysis to demonstrate the almost second-order uniform accuracy on each edge of the k -star graph. Numerical experiments highly validate the theory on the k -star graph. [ABSTRACT FROM AUTHOR]

A study was conducted on the existence of solutions for a class of nonlinear Caputo type higher-order fractional Langevin equations with mixed boundary conditions on a star graph with $ k+1 $ nodes and $ k $ edges. By applying a variable transformation, a system of fractional differential equations with mixed boundary conditions and different domains was converted into an equivalent system with identical boundary conditions and domains. Subsequently, the existence and uniqueness of solutions were verified using Krasnoselskii's fixed point theorem and Banach's contraction principle. In addition, the stability results of different types of solutions for the system were further discussed. Finally, two examples are illustrated to reinforce the main study outcomes.

A variational solution to the transient heat flow measure above a closed conductive region of arbitrary perimeter aspect, held at constant temperature, and which is embedded in an otherwise insulating boundary plane, is presented. Upon developing the general variational formulation, a full range solution for the circular disk conductor is considered as an example when implementing a two term trial function that comprises a general form based on known physical solutions to the problem at long and short times. The particular combination of the Lagrange density for the total transient diffusive flux and of the form of the trial functions are evidently effective in dealing with the difficulties often experienced when dealing with mixed boundary value heat transfer problems of the parabolic type which have a non-periodic time variable.

A source identification problem for a parabolic equation with mixed boundary condition is studied. Stability estimates for the solution of identification problem with mixed boundary conditions are obtained. Numerical algorithm for solving this inverse problem are proposed. Stability estimates for difference schemes are established. The numerical result in test example is presented.

This paper is concerned with the study of two kinds of new double phase problems with mixed boundary conditions and multivalued convection terms, which are, exactly, a double phase inclusion problem with Dirichlet–Neumann–Dirichlet–Neumann boundary conditions (DNDN, for short) and a double phase inclusion problem with Dirichlet–Neumann–Neumann–Neumann boundary conditions (DNNN, for short), respectively. On the one hand, we examine the nonemptiness, boundedness and closedness of solution sets to (DNDN) and (DNNN), respectively, by employing the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings which is formulated by the sum of a multivalued maximal monotone operator and a multivalued bounded pseudomonotone mapping. On the other hand, we explore a significant result on asymptotic behavior of solution set to (DNNN) which reveals that the solution set of (DNDN) can be approached by the solution set of (DNNN) in the sense of Kuratowski, when a parameter tends to infinity. [ABSTRACT FROM AUTHOR]

Although peridynamics is widely used to investigate mechanical responses in materials, the ability of peridynamics to capture the main features of realistic stress states remains unknown. Here, we present a procedure that combines analytic investigation and numerical simulation to capture the elastic field in the mixed boundary condition. By using the displacement potential function, the mixed boundary condition elasticity problem is reduced to a single partial differential equation which can be analytically solved through Fourier analysis. To validate the peridynamic model, we conduct a numerical uniaxial tensile test using peridynamics, which is further compared with the analytic solution through a convergence study. We find that, when the parameters are carefully calibrated, the numerical predicted stress distribution agrees very well with the one obtained from the theoretical calculation. [ABSTRACT FROM AUTHOR]

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the pp-Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Abstract In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms D a + β H C ( H C D a + α + p ( t ) ) x ( t ) + q ( t ) x ( t ) = 0 , 0 < a < t < b , $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\beta } \bigl({}_{H}^{C}D_{a + }^{\alpha }+ p(t)\bigr)x(t) + q(t)x(t) = 0,\quad 0 < a < t < b, \end{aligned} $$ and D a + η H C ϕ p [ ( H C D a + γ + u ( t ) ) x ( t ) ] + v ( t ) ϕ p ( x ( t ) ) = 0 , 0 < a < t < b , $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a + }^{\gamma }+ u(t)\bigr)x(t)\bigr] + v(t){\phi _{p}}\bigl(x(t)\bigr) = 0,\quad 0 < a < t < b, \end{aligned} $$ subject to mixed boundary conditions, respectively, where p ( t ) $p(t)$ , q ( t ) $q(t)$ , u ( t ) $u(t)$ , v ( t ) $v(t)$ are real-valued functions and 0 < β < 1 < α < 2 $0 < \beta < 1 < \alpha < 2$ , 1 < γ $1 < \gamma $ , η < 2 $\eta < 2$ , ϕ p ( s ) = | s | p − 2 s ${\phi _{p}}(s) = |s{|^{p - 2}}s$ , p > 1 $p > 1$ . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to investigate two double phase problems and two distributed optimal control problems driven by the double phase problems. First, we prove the existence and uniqueness of weak solution to a double phase problem with Dirichlet–Neumann–Dirichlet–Neumann boundary conditions (DNDN) and a double phase problem with Dirichlet–Neumann–Neumann–Neumann boundary conditions (DNNN), respectively. Then, a comparison principle and a monotonicity property for the solutions of double phase problems are obtained. After that, we establish a convergence result that the solution of DNDN can be approached by the solution of DNNN. Moreover, two distributed optimal control problems driven by DNDN and DNNN, respectively, are studied, and a theorem concerning the nonemptiness, closedness and boundedness of the set of optimal solutions to distributed optimal control problems is examined. Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values, when a parameter tends to infinity. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with the study of a nonlinear complementarity problem (NCP, for short) with a nonlinear and nonhomogeneous partial differential operator (called double phase differential operator), a convection term (i.e., a reaction depending on the gradient), a generalized multivalued boundary condition, and two nonlocal terms which appear in the domain and boundary, respectively. First, we formulate NCP to a nonlinear bilateral obstacle variational problem with feedback effect. Then, a regularized approximation problem corresponding to NCP is introduced via applying the Moreau–Yosida approximating method. By employing a surjectivity theorem to multivalued pseudomonotone operators and a limiting procedure for solutions of approximating problems, we obtain the properties of solution set to NCP, including the nonemptiness and compactness. Finally, under further assumptions, we examine several extended versions of existence theorem to NCP. [ABSTRACT FROM AUTHOR]

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms D a + β H C (H C D a + α + p (t)) x (t) + q (t) x (t) = 0 , 0 < a < t < b , and D a + η H C ϕ p [ (H C D a + γ + u (t)) x (t) ] + v (t) ϕ p (x (t)) = 0 , 0 < a < t < b , subject to mixed boundary conditions, respectively, where p (t) , q (t) , u (t) , v (t) are real-valued functions and 0 < β < 1 < α < 2 , 1 < γ , η < 2 , ϕ p (s) = | s | p − 2 s , p > 1 . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic. [ABSTRACT FROM AUTHOR]

We study the problem of constructing the asymptotics of solutions to a boundary value problem for the hyperbolic equation with holomorphic coefficients depending on the time parameter t in the space of functions of exponential growth. In addition, sufficient conditions are established for the convergence of asymptotic series contained in the asymptotics of solutions of the boundary value problem in a neighborhood of infinity. [ABSTRACT FROM AUTHOR]

The unusual properties of quasicrystals (QCs) have attracted tremendous attention from researchers. In this paper, a semi-analytical solution is presented for the static response of a functionally graded (FG) multilayered two-dimensional (2D) decagonal QC rectangular plate with mixed boundary conditions. Based on the elastic theory of FG 2D QCs, the state-space method is used to derive the state equations composed of partial differential along the thickness direction. Besides, the Fourier series expansion and the differential quadrature technique are utilized to simulate the simply supported boundary conditions and the mixed boundary conditions, respectively. Then, the propagator matrix which connects the field variables at the upper interface to those at the lower interface of any homogeneous layer can be derived based on the state equations. Combined with the interface continuity condition, the static response can be obtained by imposing the sinusoidal load on the top surfaces of laminates. Finally, the numerical examples are presented to verify the effectiveness of this method, and the results are very useful for the design and understanding of the characterization of FG QC materials in their applications to multilayered systems. [ABSTRACT FROM AUTHOR]

In this paper, we present a finite difference scheme for a linear complementarity problem with a mixed boundary condition arising from pricing a Russian option with a finite time horizon. An implicit Euler method for the temporal discretization and second-order difference schemes on a piecewise uniform mesh for the spatial discretization are used to solve the linear complementarity problem with a mixed boundary condition. It is shown that the transformed discrete operator satisfies a maximum principle, which is used to derive the error estimate. It is proved that the scheme is first- and second-order convergent with respect to the temporal and spatial variables, respectively. Numerical experiments verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This article presents a close form of solution to the magnetohydrodynamics free convection between two parallel vertical plates filled with nanoparticles with induced magnetic field effect. The surface of the channel is maintained at constant heat flux or constant temperature.The exact and the numerical solution is obtained through the method of undetermined coefficient and RKF45 in maple software respectively while the analytic solution is obtained through perturbation. The governing equations include the significant effects of the thermophoretic and Brownian motion parameters due to the presence of nanofluid. The role of the active parameters on the velocity, temperature, induced current density, concentration and induced magnetic field is illustrated using graphs. The results of the study showed that Brownian motion parameter Buoyancy ratio and the heat source parameter plays a supportive role on the velocity whereas other active parameters found to have decreasing effect on the velocity profile. Regarding the temperature distribution, the heat source parameter Brownian motion parameter and the thermophoretic parameter augment enhance the nanofluid temperature. The skin friction decreases with Hartmann number and magnetic Prandtl number augment but increases with Buoyancy ratio and heat source parameter

A definition of the Laplacian on Cartesian products with mixed boundary conditions using quadratic forms is proposed. Its consistency with the standard definition for homogeneous and certain mixed boundary conditions is proved and, as a consequence, tensor representations of the corresponding Sobolev spaces of first order are derived. Moreover, a criterion for the domain to belong to the Sobolev space of second order is proved. [ABSTRACT FROM AUTHOR]

In this article, we have presented a parametric finite difference method, a numerical technique for the solution of two point boundary value problems in ordinary differential equations with mixed boundary conditions. We have tested proposed method for the numerical solution of a model problem. The numerical results obtained for the model problem with constructed exact solution depends on the choice of parameters. The computed result of a model problem suggests that proposed method is efficient.

Abstract In this paper, we consider a class of fractional Langevin equations with integral and anti-periodic boundary conditions. By using some fixed point theorems and the Leray–Schauder degree theory, several new existence results of solutions are obtained.

Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces Hσ with σ ∈ [ 0, 1). We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have "smoother" interior data. Numerical tests validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Stationary incompressible Newtonian fluid flow governed by external force and external pressure is considered in a thin rough pipe. The transversal size of the pipe is assumed to be of the order ε , i.e., cross-sectional area is about ε 2 , and the wavelength in longitudinal direction is modeled by a small parameter μ . Under general assumption ε , μ → 0 , the Poiseuille law is obtained. Depending on ε , μ -relation ( ε ≪ μ , ε / μ ∼ constant , ε ≫ μ ), different cell problems describing the local behavior of the fluid are deduced and analyzed. Error estimates are presented. [ABSTRACT FROM AUTHOR]

This article presents a close form of solution to the magnetohydrodynamics free convection between two parallel vertical plates filled with nanoparticles with induced magnetic field effect. The surface of the channel is maintained at constant heat flux or constant temperature.The exact and the numerical solution is obtained through the method of undetermined coefficient and RKF45 in maple software respectively while the analytic solution is obtained through perturbation. The governing equations include the significant effects of the thermophoretic and Brownian motion parameters due to the presence of nanofluid. The role of the active parameters on the velocity, temperature, induced current density, concentration and induced magnetic field is illustrated using graphs. The results of the study showed that Brownian motion parameter (N b) , Buoyancy ratio (B r) , and the heat source parameter (S) plays a supportive role on the velocity whereas other active parameters found to have decreasing effect on the velocity profile. Regarding the temperature distribution, the heat source parameter (S) , Brownian motion parameter (N b) , and the thermophoretic parameter (N t) augment enhance the nanofluid temperature. The skin friction decreases with Hartmann number (H a) and magnetic Prandtl number (P m) augment but increases with Buoyancy ratio (B r) and heat source parameter (S). [ABSTRACT FROM AUTHOR]

In recent decades, research utilizing numerical schemes dealing with fluid and nanoparticle interaction has been relatively intensive. It is known that CuO nanofluid with a volume fraction of 0.1 and a special thermal boundary condition with heat supplied to part of the wall increases the average Nusselt number for different aspect ratios ranges and for high Rayleigh numbers. Due to its simplicity, stability, accuracy, efficiency, and ease of parallelization, we use the thermal single relaxation time Bhatnagar-Gross-Krook (SRT BGK) mesoscopic approach D2Q9 scheme lattice Boltzmann method in order to solve the coupled Navier–Stokes equations. Convection of CuO nanofluid in a square enclosure with a moderate Rayleigh number of 105 and with new boundary conditions is highlighted. After a successful validation with a simple partial Dirichlet boundary condition, this paper extends the study to deal with linear and sinusoidal thermal boundary conditions applied to part of the wall.

In this paper, we investigate (1+2)-dimensional Black-Scholes partial differential equations(PDE) with mixed boundary conditions. The main idea of our method is to transform the given PDE into the relatively simple ordinary differential equations(ODE) using double Mellin transforms. By using inverse double Mellin transforms, we derive the analytic representation of the solutions for the (1+2)-dimensional Black-Scholes equation with a mixed boundary condition. Moreover, we apply our method to European maximum-quanto lookback options and derive the pricing formula of this options. [ABSTRACT FROM AUTHOR]

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In this paper, we study the discrete eigenvalue problems for the p -Schrödinger equations under the mixed boundary conditions defined on networks as follows: − Δ p , ω ϕ (x) + q (x) | ϕ (x) | p − 2 ϕ (x) = λ | ϕ (x) | p − 2 ϕ (x) , x ∈ S , μ (z) ∂ ϕ ∂ p n (z) + σ (z) | ϕ (z) | p − 2 ϕ (z) = 0 , z ∈ ∂ S , where p > 1 , q is a real-valued function on a network S , and μ , σ are nonnegative functions on the boundary ∂ S of S , with μ (z) + σ (z) > 0 , z ∈ ∂ S. Next, we use the above result to provide the existence of a positive solution to the discrete Poisson equation − Δ p , ω u (x) + q (x) | u (x) | p − 2 u (x) = f (x) , x ∈ S , μ (z) ∂ u ∂ p n (z) + σ (z) | u (z) | p − 2 u (z) = 0 , z ∈ ∂ S. • Discussing p -Schrödinger equation under the mixed boundary conditions. • Existence and uniqueness of the eigenvalue for the p -Schrödinger equation. • A lower bound of the first eigenvalue for the p -Schrödinger equation. • Existence and uniqueness of solution to the p -Poisson equation. [ABSTRACT FROM AUTHOR]

We study the stationary problem of the drift–diffusion model with a mixed boundary condition. For this problem, the existence of solutions was established in general settings, while the uniqueness was investigated only in some special cases which do not entirely cover situations that semiconductor devices are used in integrated circuits. In this paper, we prove the uniqueness in a physically relevant situation. The key to the proof is to derive two-sided uniform estimates for the densities of the electron and hole. We establish a new technique to show the lower bound. This together with the Moser iteration method leads to the upper bound. [ABSTRACT FROM AUTHOR]

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and must fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup. [ABSTRACT FROM AUTHOR]

We study the solvability of an inhomogeneous mixed boundary value problem for a stationary reaction-convection-diffusion model. Such models are often used in science and engineering for the description and analysis of various processes of heat and mass transfer. We focus on the issues of solvability of the boundary value problem in different functional spaces and on the stability of the solution and its continuous dependence on the input data in natural metrics. The peculiarity of the problem consists in the inhomogeneity and irregularity of the mixed boundary data. These boundary data, in general, cannot be continued inside the domain so that the continuation is sufficiently smooth and can be used in the known way to transform the problem to a problem with homogeneous boundary data. To prove the solvability of the problem, we use the Lax-Milgram theorem. Estimates for the norms of the solution follow from the same theorem. The cases where the solution operator is completely continuous are also established. The properties of the solution of the direct problem found in this study will be used later to solve inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, a brand-new wavelet-homotopy Galerkin technique is developed to solve nonlinear ordinary or partial differential equations. Before this investigation, few studies have been done for handling nonlinear problems with non-uniform boundary conditions by means of the wavelet Galerkin technique, especially in the field of fluid mechanics and heat transfer. The lid-driven cavity flow and heat transfer are illustrated as a typical example to verify the validity and correctness of this proposed technique. The cavity is subject to the upper and lower walls' motions in the same or opposite directions. The inclined angle of the square cavity is from 0 to π/2. Four different modes including uniform, linear, exponential, and sinusoidal heating are considered on the top and bottom walls, respectively, while the left and right walls are thermally isolated and stationary. A parametric analysis of heating distribution between upper and lower walls including the amplitude ratio from 0 to 1 and the phase deviation from 0 to 2π is conducted. The governing equations are non-dimensionalized in terms of the stream function-vorticity formulation and the temperature distribution function and then solved analytically subject to various boundary conditions. Comparisons with previous publications are given, showing high efficiency and great feasibility of the proposed technique. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the fractional Hardy inequality ... The second aim of the paper is to study the mixed Dirichlet-Neumann boundary problem associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the fractional Laplacian, that is, ... with N and D open sets in ℝd\Ω such that N ∩ D = 0 and N ∪ D =ℝd\Ω, d>2s, λ>0 and 0s*-1, 2s*=2d/d-2s. We emphasize that the nonlinear term can be critical. The operators (-Δ)s, fractional laplacian, and Ns, nonlocal Neumann condition, are defined below in (1.5) and (1.6) respectively. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of weak solutions to initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains ⋃ t ∈ ( 0 , T ) Ω ( t ) × { t } of spatial-temporal space R N × R . In the case of the linear equation, each boundary condition is given on any open subset of the boundary surface Σ = ⋃ t ∈ ( 0 , T ) ∂ Ω ( t ) × { t } under a condition that the boundary portion for Dirichlet condition Σ 0 ⊂ Σ is nonempty at any time t . Due to this, it is difficult to reduce the problem to the one on a cylindrical domain by diffeomorphism of the spatial domains Ω ( t ) . By a transformation of the unknown function and the penalty method, we connect the problem to a monotone operator equation for functions defined on the non-cylindrical domain. We are also concerned with a semilinear problem when the boundary portion for Dirichlet condition is cylindrical. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the following problem ... In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (Pλ) for suitable ranges of λ and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data. [ABSTRACT FROM AUTHOR]

We study a semilinear parametric elliptic equation with superdiffusive reaction and mixed boundary conditions. Using variational methods, together with suitable truncation techniques, we prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions. [ABSTRACT FROM AUTHOR]