Semilinear Elliptic Equations Research Articles

Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

<abstract> This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained. </abstract>

О некоторых емкостных характеристиках некомпактных римановых многообразий

We introduce the notion of an L-massive subset in a noncompact Riemannian manifold and study properties of such subsets. It is proved that a semilinear elliptic equation on a noncompact Riemannian manifold has a nontrivial bounded solution if and only if there exists an L-massive subset in this manifold. A similar assertion is also proved for solutions of semilinear equations with finite energy integral.

Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

&lt;abstract&gt;&lt;p&gt;Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in &lt;italic&gt;conformal dimensions&lt;/italic&gt;, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.&lt;/p&gt;&lt;/abstract&gt;

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

On the number of critical points of solutions of semilinear elliptic equations

&lt;p style='text-indent:20px;'&gt;In this survey we discuss old and new results on the number of critical points of solutions of the problem&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE0.1"&gt; $ \begin{equation} \begin{cases} -\Delta u = f(u)&amp;amp;in\ \Omega\\ u = 0&amp;amp;on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $ &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ \Omega\subset \mathbb{R}^N $&lt;/tex-math&gt;&lt;/inline-formula&gt; with &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;$ N\ge2 $&lt;/tex-math&gt;&lt;/inline-formula&gt; is a smooth bounded domain. Both cases where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;$ u $&lt;/tex-math&gt;&lt;/inline-formula&gt; is a positive or nodal solution will be considered.&lt;/p&gt;

Large positive solutions to an elliptic system of competitive type with nonhomogeneous terms

<abstract> In this paper, we study the elliptic system of competitive type with nonhomogeneous terms $ \Delta u = u^pv^q+h_1(x) $, $ \Delta v = u^rv^s+h_2(x) $ in $ \Omega $ with two types of boundary conditions: (Ⅰ) $ u = v = +\infty $ and (SF) $ u = +\infty $, $ v = f $ on $ \partial\Omega $, where $ f &gt; 0 $, $ (p-1)(s-1)-qr &gt; 0 $, and $ \Omega \subset \mathbb{R^N} $ is a smooth bounded domain. The nonhomogeneous terms $ h_1(x) $ and $ h_2(x) $ may be unbounded near the boundary and may change sign in $ \Omega $. First, for a single semilinear elliptic equation with a singular weight and nonhomogeneous term, boundary asymptotic behaviour of large positive solutions is established. Using this asymptotic behaviour, we show existence of large positive solutions for this elliptic system with the boundary condition (SF), existence of maximal solution, boundary asymptotic behaviour and uniqueness of large positive solutions for this elliptic system with (Ⅰ). </abstract>

Lipschitz stable determination of small conductivity inclusions in a semilinear equation from boundary data

We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [ 4 ]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.

Quasi-regular Dirichlet forms and the obstacle problem for elliptic equations with measure data

We consider the obstacle problem with irregular barriers for semilinear elliptic equations involving measure data and an operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well as its repre

Recent progress on stable and finite Morse index solutions of semilinear elliptic equations

&lt;p style='text-indent:20px;'&gt;We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.&lt;/p&gt;

Singular solutions to some semilinear elliptic equations: an approach of Born–Infeld approximation

Singular solutions to some semilinear elliptic equations: an approach of Born–Infeld approximation

具有Hardy项的半线性椭圆方程正径向对称解的存在性

具有Hardy项的半线性椭圆方程正径向对称解的存在性

Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

Abstract We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau–de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche’s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient a posteriori analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

Positive solutions for a class of elliptic equations

This paper is concerned with a class of semilinear elliptic equations with a potential function−Δu=λ|x|−αu−|x|σupinΩ∖{0}, where λ,σ∈R, α>0, p>1, and Ω⊂RN(N≥3) is a bounded smooth domain with 0∈Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function |x|−α has strong singularity at the origin. When the potential function |x|−α has weak singularity at the origin, we obtain some qualitative properties of positive solutions.

Local and parallel multigrid method for semilinear elliptic equations

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory.

A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation \Delta u+ a(x,u)=0 in a domain in \mathbb{R}^n by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of a(x,z) at z = 0 under general assumptions on a(x,z) . The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem by Ferreira–Kenig–Sjöstrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations \Delta u+ a(x,u) = 0 also proved by Krupchyk–Uhlmann. The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are longstanding open problems, and the solution to the Calderón problem with partial data is known only in special cases when n \geq 3 .

Gradient Lagrangian systems and semilinear PDE

We survey some results about multiplicity of certain classes of entire solutions to semilinear elliptic equations or systems of the form $-\Delta u = F_{u}(x, u)$, $x\in\mathbb{R}^{N+1}$, including the Allen Cahn or the stationary Nonlinear Schr\"odinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional $V(u) = \tfrac 12\|\nabla u\|_{H^{1}(\mathbb{R}^{N})}^{2}-\tfrac 1{p+1}\| u\|_{L^{p+1}(\mathbb{R}^{N})}^{p+1}$ in relation to the value of the exponent $p+1\in (2, 2^{*}_{N})$.

On the semilinear fractional elliptic equations with singular weight functions

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $\end{document} where \begin{document}$ \alpha\in (0,1] $\end{document} , \begin{document}$ 1 2\alpha\right), $\end{document} the potential \begin{document}$ V_{\lambda }(x) = \lambda a(x)-b(x) $\end{document} and the parameter \begin{document}$ \lambda >0. $\end{document} Under some suitable assumptions on \begin{document}$ a,b $\end{document} and the weight functions \begin{document}$ f,g $\end{document} , we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$ \lambda $\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$ f, g $\end{document} are integrable or bounded on whole space \begin{document}$ \mathbb{R}^{N}. $\end{document}

Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators

<p style='text-indent:20px;'>In this paper, we consider the uniformly elliptic nonlocal operators <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A_{\alpha} u(x) = C_{n,\alpha} \rm{P.V.} \int_{\mathbb{R}^n} \frac{a(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha}} dy, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> is positively uniform bounded satisfying a cylindrical condition. We first establish the narrow region principle in the bounded domain. Then using the sliding method, we obtain the monotonicity of solutions for the semi-linear equation involving <inline-formula><tex-math id="M2">\begin{document}$ A_{\alpha} $\end{document}</tex-math></inline-formula> in both the bounded domain and the whole space. In addition, we establish the maximum principle in the unbounded domain and get the non-existence of solutions in the upper half space <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^n_+ $\end{document}</tex-math></inline-formula>.