Solutions Of Semilinear Elliptic Equations Research Articles

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 \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] 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.

A Morse index formula for minimax type saddle points by a Ljusternik–Schnirelman minimax algorithm and its application in computation of multiple solutions of semilinear elliptic equation

As soon as a saddle point is found, people will pay attention to its Morse index. The instability is an important character to a saddle point. For nondegenerate saddle points, the Morse indices can be used to measure their instability and classify them. In this paper, a formula on the Morse index of minimax type saddle point by a Ljusternik–Schnirelman minimax algorithm is established. For nondegenerate minimax type saddle points, the formula clearly answers the question what their Morse indices are. Also, a numerical example on the Hénon equation is presented to illustrate the application of the formula for calculating the Morse indices of numerical solutions in the computation of multiple solutions of the semilinear elliptic equation.

Least energy solutions to semi-linear elliptic problems on metric graphs

We consider positive solutions of semi-linear elliptic equations −ε2Δu+u=up on compact metric graphs, where 1<p<∞. For each ε>0, there exists a least energy positive solution uε. We focus on the asymptotic behavior of uε and show that uε has exactly one local maximum point xε and concentrates like a peak for sufficiently small ε. Moreover, we prove that the location of xε is determined by the length of edges of graphs. These results are shown for the more general super-linear term f(u) instead of up.

Radial solutions of semilinear elliptic equations with prescribed asymptotic behavior

AbstractWe consider semilinear elliptic equations of the form on . Given a suitable positive root z of g, we discuss the existence of positive radial solutions u with .

On supercritical problems involving the Laplace operator

AbstractWe discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

Boundary estimates for superharmonic functions and solutions of semilinear elliptic equations with source

In a certain Lipschitz domain $$\Omega \subset {\mathbb {R}}^n$$ , we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality $$-\Delta u\le cu^p$$ , where $$c>0$$ and $$1<p<(n+\alpha )/(n+\alpha -2)$$ with constant $$\alpha $$ regarding the lower bound estimate of the Green function on $$\Omega $$ . An argument combined estimates for certain Green potentials and iteration methods enables us to prove it. Results are applicable to positive solutions of semilinear elliptic equations like $$-\Delta u=a(x)u^p$$ with a(x) being nonnegative and bounded on $$\Omega $$ . Also, we present an a priori estimate and a removability theorem for positive solutions having isolated singularities at a boundary point. The former extends one given by Bidaut-Veron and Vivier (Rev Mat Iberoam 16:477–513, 2000) in the case where $$\Omega $$ has a smooth boundary and $$a(x)\equiv 1$$ .

Renormalized solutions of semilinear elliptic equations with general measure data

In the paper we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia’s definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.

Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data

We study existence and stability of solutions of (E 1) −∆u + µ |x| 2 u + g(u) = ν in Ω, u = 0 on ∂Ω, where Ω is a bounded, smooth domain of R N , N ≥ 2, containing the origin, µ ≥ − (N −2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and ν is a Radon measure on Ω. We show that the situation differs according ν is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.

Symmetry and monotonicity of positive solutions of elliptic equations with mixed boundary conditions in a super-spherical cone

In this paper, we study symmetry and monotonicity properties for positive solutions of semilinear elliptic equations with mixed boundary conditions. We establish a version of the maximum principle for mixed boundary conditions in a narrow domain, and prove the positive solution in super-spherical cone is axially symmetric and monotone in some direction by the methods of moving planes.

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Let Ω⊂RN (N≥3) be a bounded C2 domain and δ(x)=dist(x,∂Ω). Put Lμ=Δ+μδ2 with μ>0. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to−Lμu=up+τin Ω,u=νon ∂Ω, where μ>0, p>0, τ and ν are measures on Ω and ∂Ω respectively. We then establish existence results for the system{−Lμu=ϵvp+τin Ω,−Lμv=ϵup˜+τ˜in Ω,u=ν,v=ν˜on ∂Ω, where ϵ=±1, p>0, p˜>0, τ and τ˜ are measures on Ω, ν and ν˜ are measures on ∂Ω. We also deal with elliptic systems where the nonlinearities are more general.

Liouville theorems for radial solutions of semilinear elliptic equations

ABSTRACTIn this work, we obtain some new Liouville's theorems for positive, radially symmetric solutions of the equation where f is a continuous function in which is positive in . Our methods adapt to cover more general problems, where the nonlinearity is multiplied by some radially symmetric weights and/or the Laplacian is replaced by the p-Laplacian, 1<p<N. Some results for related elliptic systems are also obtained.

Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and $\delta_\Omega(x)=\mathrm{dist}(x,\partial\Omega)$. Assuming that $0\leq \mu<C_H(\Omega)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\infty<\mu<1/4$, even if $C_H(\Omega)<1/4$. We recall that $C_H(\Omega)\leq 1/4$ and, in general, strict inequality holds. The key to our study is the fact that, if $\mu<1/4$ then in smooth domains there exist local $L_\mu$-superharmonic functions in a neighborhood of $\partial\Omega$ (even if $C_H(\Omega)<1/4$). Using this fact we extend the notion of normalized boundary trace to arbitrary domains, provided that $\mu<1/4$. Further we study the b.v.p. with normalized boundary trace $\nu$ in the space of positive finite measures on $\partial\Omega$. We show that existence depends on two critical values of the exponent $q$ and discuss the question of uniqueness. Part of the paper is devoted to the study of the linear operator: properties of local $L_\mu$ subharmonic and superharmonic functions and the related notion of moderate solutions.

Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity

We consider the semilinear elliptic equation \begin{document} $Δ u + K(|x|)e^u = 0$\end{document} in \begin{document} $\mathbf{R}^N$\end{document} for \begin{document} $N > 2$\end{document} , and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For \begin{document} $N = 10+4\ell$\end{document} with \begin{document} $\ell>-2$\end{document} , the equation changes its nature drastically according to the sign of the derivative of \begin{document} $r^{-\ell}K(r)$\end{document} when \begin{document} $r^{-\ell}K(r)$\end{document} is monotonic in \begin{document} $r$\end{document} and \begin{document} $r^{-\ell} K(r)\to1$\end{document} as \begin{document} $r\to∞$\end{document} .

Symmetry of large solutions for semilinear elliptic equations in a ball

In this work we consider the boundary blow-up problem{Δu=f(u)in Bu=+∞on ∂B where B stands for the unit ball of RN and f is a locally Lipschitz function which is positive for large values and verifies the Keller–Osserman condition. Under an additional hypothesis on the asymptotic behavior of f we show that all solutions of the above problem are radially symmetric and radially increasing. Our condition is sharp enough to generalize several results in previous literature.

Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the linear problem $-\Delta h -\frac{\mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $\mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

Weak solutions of semilinear elliptic equation involving Dirac mass

In this paper, we study the elliptic problem with Dirac mass(1){−Δu=Vup+kδ0inRN,lim|x|→+∞⁡u(x)=0, where N>2, p>0, k>0, δ0 is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in RN∖{0}, with non-empty support and satisfying0≤V(x)≤σ1|x|a0(1+|x|a∞−a0), with a0<N, a0<a∞ and σ1>0. We obtain two positive solutions of (1) with additional conditions for parameters on a∞,a0, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.

Sign-Changing Solutions for Semilinear Elliptic Equations with Dependence on the Gradient Via the Nehari Method

In this paper, we consider a class of elliptic equation with dependence on the gradient. The existence of sign-changing solutions is established via the Nehari method and iterative technique.

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display) where , or , with is the fractional Laplacian defined in the principle value sense, is a bounded open set in with , is a bounded Radon measure supported in and is defined in the distribution sense, i.e. here denotes the unit inward normal vector at . In this paper, we prove that (0.1) with admits a unique weak solution when g is a continuous nondecreasing function satisfying Our interest then is to analyse the properties of weak solution when with , including the asymptotic behaviour near and the limit of weak solutions as . Furthermore, we show the optimality of the critical value in a certain sense, by proving the non-existence of weak solutions when . The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when and is a bounded nonnegative Radon measure supported in . We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying -pagination