Solutions Of Semilinear Elliptic Equations Research Articles

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

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

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.

Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Let Nge 3, R>rho >0 and A_{rho }:={xin mathbb {R}^N; rho<|x|<R}. Let U^{pm }_{n,rho }, nge 1, be a radial solution with n nodal domains of ΔU+|x|α|U|p-1U=0inAρ,U=0on∂Aρ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\Delta U+|x|^{\\alpha }|U|^{p-1}U=0 &{} \ ext {in}\\ A_{\\rho },\\\\ U=0 &{} \ ext {on}\\ \\partial A_{\\rho }. \\end{array}\\right. } \\end{aligned}$$\\end{document}We show that if p=frac{N+2+2alpha }{N-2}, alpha >-2 and Nge 3, then U^{pm }_{n,rho } is nondegenerate for small rho >0 and the Morse index {{textsf{m}}}(U^{pm }_{n,rho }) satisfies m(Un,ρ±)=n(N+2ℓ-1)(N+ℓ-1)!(N-1)!ℓ!for smallρ>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {{\ extsf{m}}}(U^{\\pm }_{n,\\rho }) =n\\frac{(N+2\\ell -1)(N+\\ell -1)!}{(N-1)!\\ell !} \\quad \ ext {for small}\\ \\rho >0, \\end{aligned}$$\\end{document}where ell =[frac{alpha }{2}]+1. Using Jacobi elliptic functions, we show that if (p,alpha )=(3,N-4) and Nge 3, then the Morse index of a positive and negative solutions {{textsf{m}}}(U^{pm }_{1,rho }) is completely determined by the ratio rho /Rin (0,1). Upper and lower bounds for {{textsf{m}}}(U^{pm }_{n,rho }), nge 1, are also obtained when (p,alpha )=(3,N-4) and Nge 3.

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

Multiplicity of positive radial solutions for semilinear elliptic equation with locally concave-convex variable exponent

Multiplicity of positive radial solutions for semilinear elliptic equation with locally concave-convex variable exponent

Reaction-diffusion fronts in funnel-shaped domains

We consider bistable reaction-diffusion equations in funnel-shaped domains of RN made up of straight parts and conical parts with positive opening angles. We study the large time dynamics of entire solutions emanating from a planar front in the straight part of such a domain and moving into the conical part. We show a dichotomy between blocking and spreading, by proving especially some new Liouville type results on stable solutions of semilinear elliptic equations in the whole space RN. We also show that any spreading solution is a transition front having a global mean speed, which is the unique speed of planar fronts, and that it converges at large time in the conical part of the domain to a well-formed front whose position is approximated by expanding spheres. Moreover, we provide sufficient conditions on the size R of the straight part of the domain and on the opening angle α of the conical part, under which the solution emanating from a planar front is blocked or spreads completely in the conical part. We finally show the openness of the set of parameters (R,α) for which the propagation is complete.

Singular solutions for semilinear elliptic equations with general supercritical growth

A positive radial singular solution for Delta u+f(u)=0 with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem Delta u+lambda f(u)=0 on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., u^plog u, exp (u^p) and exp (cdots exp (u)cdots ) as well as u^p and e^u. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.

Monotonicity of solutions for fractional p-equations with a gradient term

Abstract In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional p p -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933].

Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds

This paper aims at deriving apriori bounds on the gradient of positve solutions to a class of semilinear elliptic equation, with applications focusing on establishing a liouville type property for the bounded solutions of the Allen-Cahn equation on a complete noncompact Riemannian manifold with nonnegative Ricci curvature.

Nodal solutions of semi-linear elliptic equations with dependence on the gradient

In the present paper we prove the existence of a nontrivial nodal solution of a semi-linear elliptic equation depending on the gradient. Our approach makes use of the theory of invariant sets of descending flow to localize the solution.

The domain derivative for semilinear elliptic inverse obstacle problems

&lt;p style='text-indent:20px;'&gt;We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.&lt;/p&gt;

Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

&lt;abstract&gt;&lt;p&gt;This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.&lt;/p&gt;&lt;/abstract&gt;

Sign-changing solutions for semilinear elliptic equation with variable exponent

This paper is devoted to study a class of semilinear elliptic equation with variable exponent. By means of perturbation technique, the method of invariant sets for the descending flow and necessary estimates, the existence of infinitely many sign-changing solutions to this problem is obtained.

Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland and Matano states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain \(\Omega \) further satisfies \(\Omega \cap \{\vert x\vert \le R\}= O(R^2)\), when \(R\rightarrow +\infty \). If the domain is a straight cylinder, an additional natural assumption is needed. These results can be seen as an extension to more general domains of some results on De Giorgi’s conjecture. As an application, we establish asymptotic symmetries for stable solutions when the domain satisfies a geometric property asymptotically.

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;

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;

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

A monotonicity result under symmetry and Morse index constraints in the plane

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad &amp; \text{ in } \Omega, \\ u= 0 &amp; \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations

Symmetry plays a basic role in variational problems (settled, e.g., in $${\mathbb {R}}^{n}$$ or in a more general manifold), for example, to deal with the lack of compactness which naturally appears when the problem is invariant under the action of a noncompact group. In $${\mathbb {R}}^n$$ , a compactness result for invariant functions with respect to a subgroup G of $$\mathrm {O}(n)$$ has been proved under the condition that the G action on $${\mathbb {R}}^n$$ is compatible, see Willem (Minimax theorem. Progress in nonlinear differential equations and their applications, vol 24, Birkhauser Boston Inc., Boston, 1996). As a first result, we generalize this and show here that the compactness is recovered for particular subgroups of the isometry group of a Riemannian manifold. We investigate also isometric action on Hadamard manifold (M, g) proving that a large class of subgroups of $$\mathrm {Iso}(M,g)$$ is compatible. As an application, we get a compactness result for “invariant” functions which allows us to prove the existence of nonradial solutions for a classical scalar equation and for a nonlocal fractional equation on $${\mathbb {R}}^n$$ for $$n=3$$ and $$n=5$$ , improving some results known in the literature. Finally, we prove the existence of nonradial invariant functions such that a compactness result holds for some symmetric spaces of noncompact type.