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In this paper, we analyze the following nonlinear elliptic problem A (u) = ρ (u) | ∇ φ | 2 in Ω , div (ρ (u) ∇ φ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω. where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique. [ABSTRACT FROM AUTHOR]

This work is devoted to establish the existence of multiple positive radial solutions to the following equations associated with certain boundary conditions. The Six Functionals Fixed Point Theorem serves as the basis for the technique. To illustrate the key findings, an application is given. [ABSTRACT FROM AUTHOR]

This work is devoted to establish the existence of multiple positive radial solutions to the following equations associated with certain boundary conditions. The Six Functionals Fixed Point Theorem serves as the basis for the technique. To illustrate the key findings, an application is given. [ABSTRACT FROM AUTHOR]

Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new 'Nodal Discrete Duality' (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

Abstract We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold ( M , g ) $(M, g)$ Δ u + a u ( ln u ) p + b u ln u = 0 , $$ \Delta u +au(\ln{u})^{p}+bu\ln{u}=0, $$ where a ≠ 0 $a\ne 0$ , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 $p=\frac{k_{1}}{2k_{2}+1}\ge 2$ , here k 1 $k_{1}$ and k 2 $k_{2}$ are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold (M , g) Δ u + a u (ln u) p + b u ln u = 0 , where a ≠ 0 , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 , here k 1 and k 2 are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution. [ABSTRACT FROM AUTHOR]

In this paper we give conditions for the existence of, and describe the asymtotic behavior of, radial positive solutions of the nonlinear elliptic equation of Emden-Fowler type with convection term... [ABSTRACT FROM AUTHOR]

The Miura ori is a very classical origami pattern used in numerous applications in engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then an H 2 -conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that nontrivial shapes can be achieved using periodic Miura tessellations. [ABSTRACT FROM AUTHOR]

The paper deals with the following elliptic equation with a critical potential div |∇u|p-2∇u + γ |x|-p |u|p-2 u = 0, x ∈ ℝN - {0}, where N > p > 2 and γ > 0. We prove the existence of positive solutions in the radial case. The study relies heavily on comparing γ with ((N - p)/p)p. [ABSTRACT FROM AUTHOR]

We prove by using the topological degree theory the existence of at least one weak solution for the nonlinear elliptic equation ... with homogeneous Dirichlet boundary condition in Musielak-Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method. The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by \alignat2&-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^p = f - {\rm div}(c(x)|u|^{\alpha}) &\quad&in \Omega, & u = 0 &\quad&on \partial\Omega, where $\Omega$ is a bounded open set of $\mathbb{R}^N$ ($N\geq2$) with $1

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by ... where is a bounded open set of RN (N > 2) with 1 < p < N and f ∈ L1(), under some growth conditions on the function b(·) and d(·), where c(·) is assumed to be in LN/(p-1)(). We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result of the entropy solution to some nonlinear elliptic Dirichlet problem defined on Orlicz–Sobolev spaces and whose simplest model is (1) - div A (| ∇ u |) ∇ u | ∇ u | 2 + α u = β (u) A (| ∇ u |) + f in Ω u = 0 on ∂ Ω , where A is a N -function, α > 0 is a real number, β is a real continuous function and f ∈ E A β (Ω) , with Ω is a bounded open in R N ( N ≥ 2 ), such that A β (s) = ∫ 0 s e ∫ 0 v β (η) d η - 1 d v . [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on Γ-convergence. The relevance and applicability of this approach are then shown through several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters is solved by finite difference method. The main aim of this paper is to investigate the conditions under those all eigenvalues of corresponding difference eigenvalue problem are positive. For this purpose, we investigate the spectrum structure of one-dimensional difference eigenvalue problem with integral condition. In particular, conditions of the existence and some properties of negative eigenvalue are analyzed in details.

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also study the power type elliptic equation in higher dimension case related to mean field equation and derive such estimate. [ABSTRACT FROM AUTHOR]

We prove convergence for the nonoverlapping Robin--Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear Steklov--Poincaré operators based on the p-structure and the Lp-generalization of the Lions--Magenes spaces. This framework allows the reformulation of the Robin--Robin method into a Peaceman--Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions. [ABSTRACT FROM AUTHOR]

We consider the nonlinear elliptic equation Δ u + V (x) u + f (x , u (x)) = 0 on D \ { 0 } , where D is a bounded domain containing 0 in R n , n ≥ 2 , and V and f are Borel measurable functions. Under general conditions on the functions V and f, we prove the existence of positive singular solutions globally comparable to the Dirichlet Green's function of the Laplacian with pole at the origin. Our result applies to various types of semilinear equations, in particular to Δ u + W (x) u p = 0 for all real exponent p which was extensively studied for the range p > 1 . Moreover for this equation with sign-unchanging function W our condition is the optimal one. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form ▵ z j + (N - 2) 2 r 0 2 N - 2 | x | 2 N - 2 z j + φ (| x |) g j (z j + 1) = 0 , R 1 < | x | < R 2 , where j ∈ { 1 , 2 , 3 , · · · , ℓ } , z 1 = z ℓ + 1 , ▵ z = div (▿ z) , N > 2 , 0 < r 0 < π / 2 , φ = ∏ i = 1 n φ i , each φ i : (r 0 , + ∞) → (0 , + ∞) is continuous, r N - 1 φ is integrable, and g j : [ 0 , + ∞) → R is continuous, by an application of various fixed point theorems in a Banach space. Further, we also establish uniqueness of the solution for the addressed system by using Rus's theorem in a complete metric space. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of $W_0^{1,1}(\Omega)$ solutions to the following elliptic equation with principal part having noncoercivity and singular quadratic term \begin{equation*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{\nabla u}{(1+|u|)^{\gamma}}\right)+\frac{|\nabla u|^2}{u^{\theta}}=f,&x\in\Omega,\\ u=0,&x\in\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N(N\geq3)$, $\gamma>0$, $\frac{N}{N-1}\leq\theta

The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Our main result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ* such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ*); (ii) there is at least one solution if λ = λ*; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ*). The proof combines variational methods, analytic tools, and monotonicity arguments.

In this paper, we study the existence of a non-trivial weak solution to the following singular elliptic equations with subcritical nonlinearities:

We consider the equation ε2s(−Δ)su+V(x)u−f(u)=0inℝN,where s ∈ (0, 1), p∈1,N+2sN−2s, N > 2s, f(u)=|u|p−1u, V∈L∞(ℝN) is such that infℝNV>0 and ε > 0 is small. We study the existence and nonexistence of solutions concentrating at a local minimum point of V as ε → 0 without using any symmetry assumption on V. First, we prove that certain type of positive solutions exhibiting peaks does not exist. Then, we study the existence of sign‐changing solutions under a suitable configuration of positive and negative peaks. To guarantee the existence, we cannot neglect the interaction between peaks. In particular, by using a minimization argument, we found solutions exhibiting peaks at the vertices of a 2ℓ‐regular polygon, such that two adjacent peaks have alternate sign. [ABSTRACT FROM AUTHOR]

We employ the photography method to estimate the number of solutions to a nonlinear elliptic problem on a Riemannian orbifold in function of the Lusternik–Schnirelmann category of its submanifold of points with largest local group. [ABSTRACT FROM AUTHOR]

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data.

Summary: The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular. [ABSTRACT FROM AUTHOR]

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they can be adapted to model inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we show that the method used to investigate Stuart vortices on a fixed sphere provides insight into the dynamics of the large-scale zonal flows on a rotating sphere that model the background flow of polar vortices. Our approach takes advantage of the fact that while a sphere is spinning around its polar axis, every point on the sphere has the same angular velocity but its tangential velocity is proportional to the distance from the polar axis of rotation, so that points move fastest at the Equator and slower as we go towards the poles, both of which remain fixed. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\] where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq. [ABSTRACT FROM AUTHOR]

We consider the nonlinear elliptic equation with quadratic convection - Δ u + g (u) | ∇ u | 2 = λ f (u) in a smooth bounded domain Ω ⊂ R N ( N ≥ 3 ) with zero Dirichlet boundary condition. Here, λ is a positive parameter, f : [ 0 , ∞) : (0 ∞) is a strictly increasing function of class C 1 , and g is a continuous positive decreasing function in (0 , ∞) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function h (t) = f (t) e - ∫ 0 t g (s) d s , nor that the functions g h / h ′ or h ′ ′ h / h ′ 2 admit a limit at infinity. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction u acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable class of the weak solutions, we provide asymptotic analysis of the above mentioned optimal control problem posed in a family of perturbed domains and give the characterization of the limiting behavior of its optimal solutions. [ABSTRACT FROM AUTHOR]

Abstract We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space (M,g,e−fdv) $(M, g,e^{-f}\,dv)$: Δfu+aulogu+bu=0, $$\Delta_{f} u+au\log u+bu=0, $$ where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on |∇f| $|\nabla f|$. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.

We study an irregular double obstacle problem with Orlicz growth over a nonsmooth bounded domain. We establish a global Calderón–Zygmund estimate by proving that the gradient of the solution to such a nonlinear elliptic problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles. We also investigate minimal regularity requirements on the relevant nonlinear operator for the desired regularity estimate. [ABSTRACT FROM AUTHOR]

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation Δ V (u p) + λ u = 0 , p ≥ 1 , on a Riemannian manifold (M, g) with k-Bakry–Émery Ricci curvature bounded from below. We consider both the case where M is a compact manifold with or without boundary and the case where M is a complete manifold. [ABSTRACT FROM AUTHOR]

We study a free boundary problem modeling solid tumor growth with ECM and MDE interactions. The production rate of MDE by tumor cells is a nonlinear function depending on nutrients and MDE. This nonlinear term is more complicated. For this model, we can not give the explicit expressions or integral expressions for MDE terms while we only get the existence and uniqueness. At first, we show the existence and the uniqueness of radially symmetric stationary solutions. Then the existence of symmetry-breaking solutions bifurcating from the radially symmetric stationary solutions is obtained. [ABSTRACT FROM AUTHOR]

The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction–diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected. [ABSTRACT FROM AUTHOR]

In this article we study a nonlinear elliptic equation by using the maximum principle and cutoff functions, We establish related gradient estimates, the Liouville theorem, and the Harnack inequality.

In this paper, we prove the existence of solutions for the nonlinear p(.) degenerate problems involving nonlinear operators of the form - div a(x, ∇u) = f(x, u, ∇u) where a and fare Caratheodoryfunctions satisfying some nonstandard growth conditions. [ABSTRACT FROM AUTHOR]

We consider the semilinear elliptic problem: where is an open bounded subset, Under assumptions small we show that there exists a solution to this problem. The interesting point of the problem under consideration is that it has a strong singularity. [ABSTRACT FROM AUTHOR]

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation (-ΔΩ)su = |u|4/N-2s u + ε ƒ(x) in Ω, under zero Dirichlet boundary conditions in a bounded domain Ω in RN, N > 4s, s ∈ (0,1], with f ∈ L∞(Ω), ƒ ≥ 0 and ƒ ≠ 0. Here, ε > 0 is a small parameter, and (−∆Ω)s represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as ε → 0 when it is assumed that Ω and ƒ have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method. [ABSTRACT FROM AUTHOR]