Semilinear Elliptic Equations Research Articles

On the positive solutions for some diffusive logistic equations

In this paper, we study the positive solutions of the diffusive logistic equations. It is shown that the asymptotic profiles of the positive solutions display differences between nonlocal dispersal equation and semilinear elliptic equation. When the degeneracy domain becomes small, we find that the positive solution of semilinear elliptic equation admits a blow-up profile. However, when the non-degeneracy domain becomes large, the positive solution of nonlocal equation exhibits a blow-up profile.

Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶1,1 domains

Abstract This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 n\leq 9 , for C 1 , 1 C^{1\smash{,}1} domains. We consider equations − L ⁢ u = f ⁢ ( u ) -Lu=f(u) in a bounded C 1 , 1 C^{1\smash{,}1} domain Ω ⊂ R n \Omega\subset\mathbb{R}^{n} , with u = 0 u=0 on ∂ Ω \partial\Omega , where 𝐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 f\in C^{1} is nonnegative, nondecreasing, and convex. The stability of 𝑢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation − L − f ′ ⁢ ( u ) -L-f^{\prime}(u) . Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), 2, 187–252] proved the Hölder continuity in C 3 C^{3} domains.

Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in ℝn with n ≥ 10

Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in ℝn with n ≥ 10

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.

Sobolev compact embeddings in unbounded domains and its applications to elliptic equations

We give a necessary and sufficient condition for the compact embedding of the Sobolev space W0m,p(Ω) for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.

Bouligand–Newton type methods for non-smooth ill-posed problems

Abstract We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily G\^{a}teaux differentiable. Modifications are proposed with the non-existing Fr'{e}chet derivatives replaced by a family of bounded linear operators satisfying suitable properties. These bounded linear operators can be constructed by the Bouligand subderivatives which are defined as limits of Fr'{e}chet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is G\^{a}teaux differentiable which
introduces challenges for convergence analysis of the corresponding Bouligand-Newton type methods. In this paper we will show that, under the discrepancy principle, these Bouligand-Newton type methods are iterative regularization methods of optimal order. Numerical results for an inverse problem arising from a non-smooth semi-linear elliptic equation are presented to test the performance of the methods.

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper, we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The PSc sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.

A bifurcation diagram of solutions to semilinear elliptic equations with general supercritical growth

We study the global bifurcation diagram of the positive solutions to the problem{Δu+λf(u)=0inB,u=0on∂B, where B is the unit ball in RN with N≥3. Under general supercritical growth conditions on f(u), we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f(u), and we exhibit that a bifurcation curve for Δu+λf(u)=0 has the same qualitative property as a classical Gel'fand problem Δu+λeu=0 for N≥3 except N=10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.

On Dancer’s conjecture for stable solutions with sign-changing nonlinearity

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in R n \mathbb {R}^{n} with sign-changing nonlinearity f f . Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of f f at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer.

Uniqueness of continuation for semilinear elliptic equations

Uniqueness of continuation for semilinear elliptic equations

Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

Kazdan–Warner problem on compact Riemann surfaces with boundary

In this article, we prove that on any compact Riemann surface (M,∂M,g) with non-empty smooth boundary ∂M and a Riemannian metric g, (i) any K∈C∞(M) is the Gaussian curvature function of some Riemannian metric on M; (ii) any σ∈C∞(∂M) is the geodesic curvature of some Riemannian metric on M. These geometric results are obtained analytically by solving a semi-linear elliptic equation −Δgu=Ke2u on M with oblique boundary condition ∂u∂ν=σeu . One essential tool is the existence results of Brezis–Merle type equations −Δgu+Au=Ke2uinM and ∂u∂ν+κu=σeuon∂M with given functions K,σ and some constants A,κ . In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

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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