Semilinear Elliptic Equations Research Articles

Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}

In this paper, we investigate variational problems in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document} with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.

An MP-Newton method for finding orthogonal-max type critical points and its application for calculating multiple solutions of semilinear elliptic equations: part I—method

An MP-Newton method for finding orthogonal-max type critical points and its application for calculating multiple solutions of semilinear elliptic equations: part I—method

A Sub-supersolution Method for Integro-differential Semilinear Elliptic Equations and Some Applications

A Sub-supersolution Method for Integro-differential Semilinear Elliptic Equations and Some Applications

Uniqueness results for inverse source problems for semilinear elliptic equations

We study inverse source problems associated to semilinear elliptic equations of the form Δux+ax,u=Fx on a bounded domain Ω⊂Rn , n⩾2 . We show that it is possible to use nonlinearity to recover both the source F and the nonlinearity a(x,u) simultaneously and uniquely for a class of nonlinearities. This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, includes a class of polynomials, which we characterize, and exponential nonlinearities.For general nonlinearities a(x,u) , we recover the source F(x) and the Taylor coefficients ∂uka(x,u) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683–719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553–80) by removing the assumption that u≡0 is a solution. To prove our results, we consider linearizations around possibly large solutions.Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.

Singular solutions of semilinear elliptic equations with supercritical growth on Riemannian manifolds

In this paper, we shall discuss singular solutions of semilinear elliptic equations with general supercritical growth on spherically symmetric Riemannian manifolds. More precisely, we shall prove the existence, uniqueness and asymptotic behavior of the singular radial solution, and also show that regular radial solutions converges to the singular solution. In particular, we shall provide these properties on spherically symmetric Riemannian manifolds including the hyperbolic space as well as the sphere.

Nonlinear Elliptic Equations on the Sierpiński Carpet

This paper deals with a semilinear elliptic equation for a properly defined Laplace operator on the Sierpiński carpet. We investigate the existence of solutions, under Dirichlet boundary conditions, for the problem∆u+a(x)u=f(x,u).This work is motivated by previos studies established for such problems on the Sierpiński gasket by many authors: Kenneth J. Falconer, Jiaxin Hu, G.Bonanno and others (more details can be found in Introduction). The approach used in this study is based on variational methods, more specifically, the the mountain pass theorem of Ambrosetti and Rabinowit, and Sobolev type inequality, which was of principal use for achieving other results.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .

Adaptive Multi-level Algorithm for a Class of Nonlinear Problems

Abstract In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

Cauchy problem for a semilinear elliptic equation with contaminated coefficients

In this paper, we assume that Ω is a bounded domain of RN with smooth boundary. Let A:D(A)→L2(Ω) be a self-adjoint operator defined on a dense subspace D(A)⊂L2(Ω) with an orthonormal basis of eigenfunctions in L2(Ω). For a constant Y>0, giving the function F:Ω×[0,Y]×R→R, we consider the problem of finding a function u:Ω×[0,Y]→R such that −τ(y)Au(x,y)+uyy(x,y)=F(x,y,u(x,y)),x∈Ω,0<y<Y,u(x,0)=f(x),uy(x,0)=g(x),x∈Ω, where f, g∈L2(Ω) are given. In many practical cases, the function τ:[0,Y]→(0,∞) is contaminated with noise and it can only be approximated by an experimentally observable function μ:[0,Y]→(0,∞) such that sup0⩽y⩽Y∑m=02dmτdym(y)−dmμdym(y)⩽δfor a δ>0. Similarly, for ɛ>0, the initial data f,g are often perturbed by fɛ,gɛ as follows. ‖f−fɛ‖L2(Ω)+‖g−gɛ‖L2(Ω)⩽ɛ.The Cauchy problem with these contaminated data is nonlinear and ill-posed. Due to the usefulness in physics and other fields, we consider a regularization for the problem.

A free boundary problem for discontinuous semilinear elliptic equations in convex ring

In this paper, we consider the following free boundary problem: ( P ) { Δ u = λ ϕ ( x ) ∑ i = 1 n H ( u − μ i ) in Ω = Ω 2 ∖ Ω ¯ 1 , u = 0 on ∂ Ω 2 , u = M on ∂ Ω 1 . The domain Ω is a convex ring where Ω 1 and Ω 2 are bounded convex domains in R N , N ≥ 2 , H is the Heaviside step function, λ , M , μ i ( i = 1 , . . , n ) are given positive real parameters and ϕ is a given function. We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which shows that there are more solutions.

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

In an open, bounded Lipschitz polygon Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^2$$\\end{document}, we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. The boundary conditions on each edge of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1(\\Omega )$$\\end{document}, quantized tensor formats and certain deep neural networks.

On oscillating radial solutions for non-autonomous semilinear elliptic equations

&lt;abstract&gt;&lt;p&gt;We consider semilinear elliptic equations of the form $ \Delta u + f(|x|, u) = 0 $ on $ {\mathbb{R}}^{N} $ with $ f(|x|, u) = q(|x|)g(u) $. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $ f $, there exists an oscillating radial solution converging to a zero of the function $ g $. We also study the oscillating and limiting behavior of this solution.&lt;/p&gt;&lt;/abstract&gt;

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Let $\Omega \subset \mathbb {R}^N$ ( $N\geq 3$ ) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$ , $0\leq k \leq N-1$ . We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$ . Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$ , where $\mu \in {\mathbb {R}}$ . We study boundary value problems ( $P_\pm$ ) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$ , where $p&gt;1$ , $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$ . Different critical exponents for the existence of a solution to ( $P_\pm$ ) appear according to concentration of $\nu$ . The solvability for problem ( $P_+$ ) was proved in [3, 29] in subcritical ranges for $p$ , namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$ , we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ( $P_+$ ) in supercritical ranges for $p$ , i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ( $P_-$ ) under a smallness assumption on $\nu$ .

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

On semilinear elliptic equation with negative exponent arising from a closed MEMS model

Semilinear optimal control with Dirac measures

Abstract The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

A class of elliptic problems driven by a mixed local-nonlocal operator

In this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.