Supercritical Elliptic Equations Research Articles

A supercritical elliptic equation in the annulus

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation -\Delta u + u = a(x)|u|^{p-2}u in an annulus A \subset \mathbb{R}^N ( N\ge3 ). Here p>2 is allowed to be supercritical and a(x) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution u we construct. In the case where a equals a positive constant, we detect conditions, only depending on the exponent p and on the inner radius of the annulus, that ensure that the solution is nonradial.

Clustered solutions for supercritical elliptic equations on Riemannian manifolds

AbstractLet{(M,g)}be a smooth compact Riemannian manifold of dimension{n\geq 5}. We are concerned with the following elliptic problem:-\Delta_{g}u+a(x)u=u^{\frac{n+2}{n-2}+\varepsilon},\quad u>0\text{ in }M,where{\Delta_{g}=\mathrm{div}_{g}(\nabla)}is the Laplace–Beltrami operator onM,{a(x)}is a{C^{2}}function onMsuch that the operator{-\Delta_{g}+a}is coercive, and{\varepsilon>0}is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has ak-peaks solution for positive integer{k\geq 2}, which blow up and concentrate at one point inM.

Singular extremal solutions for supercritical elliptic equations in a ball

We study positive singular solutions to the Dirichlet problem for the semilinear elliptic equation Δu+λf(u)=0 in the unit ball on RN. We assume that f has the form f(u)=up+g(u), where p>(N+2)/(N−2) and g(u) is a lower order term. We first show the uniqueness of the singular solution to the problem, and then study the existence of the singular extremal solution. In particular, we show a necessary and sufficient condition for the existence of the singular extremal solution in terms of the weak eigenvalue of the linearized problem.

Supercritical Elliptic Equation in Hyperbolic Space

Supercritical Elliptic Equation in Hyperbolic Space

Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

Given a smooth Riemannian manifold (M,g) we investigate the existence of positive solutions to the equation−ε2Δgu+u=up−1on M which concentrate at some submanifold of M as ε→0, for supercritical nonlinearities. We obtain a positive answer for some manifolds, which include warped products. Using one of the projections of the warped product or some harmonic morphism, we reduce this problem to a problem of the form−ε2divh(c(x)∇hu)+a(x)u=b(x)up−1, with the same exponent p, on a Riemannian manifold (M,h) of smaller dimension, so that p turns out to be subcritical for this last problem. Then, applying Lyapunov–Schmidt reduction, we establish existence of a solution to the last problem which concentrates at a point as ε→0.

Existence, Stability and Oscillation Properties of Slow-Decay Positive Solutions of Supercritical Elliptic Equations with Hardy Potential

AbstractWe prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potentialand study the stability and oscillation properties of these solutions. We also show that if the equation on ℝN has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝN a family of the exterior Dirichlet problems in ℝN \ K admits a continuum of stable slow-decay infinite-energy solutions.

Structure of the positive solutions for supercritical elliptic equations in a ball

Let B⊂RN (N⩾3) be a unit ball. We consider the bifurcation diagram of the positive solutions to the supercritical elliptic equation in B{Δu+λf(u)=0inB,u=0on∂B,u>0inB, where f(u)=up+g(u)(p>pS:=(N+2)/(N−2)) and g(u) is a lower order term which satisfies certain assumptions. We show that if pS<p<pJL, then the branch has infinitely many turning points around some λ⁎>0. Here,pJL:={1+4N−4−2N−1(N⩾11),∞(2⩽N⩽10). We give an example such that the bifurcation diagram can be classified by using our theorem and the characterization of the extremal solution by Brezis and Vázquez. The main tool is the intersection number between regular and singular solutions.

Addendum to: Supercritical Elliptic Equations

Addendum to: Supercritical Elliptic Equations

Supercritical Elliptic Equations

Abstract For an elliptic model equation with supercritical power non-linearity we give a complete description of radial solutions and discuss self-similar blow-up solutions.

Partial regularity of stable solutions to the supercritical equations and its applications

We prove the partial regularity of stable solutions of supercritical elliptic equations. As an application, we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric.

Radial Solutions of a Supercritical Elliptic Equation with Hardy Potential

Abstract Various properties of radial solutions of the supercritical elliptic equation with Hardy Potential are studied, where Ω = int{x ∈ ℝN | α ≤ |x| &lt; b}, which is a ball if 0 = α &lt; b &lt; +∞, an annulus if 0 &lt; α &lt; b &lt; +∞, an exterior domain if 0 &lt; α &lt; b = +∞, and the whole space ℝN if α = 0, b = +∞. We assume p is supercritical, that is, p &gt; 2∗ with 2∗ = being the critical Sobolev exponent, and N ≥ 3.

Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains

We construct solutions of the semilinear elliptic problem$\Delta u+|u|^{p-1}u+$ε1/2 f = 0 in Ω u=ε1/2 g on $\partial$Ωin a bounded smoothdomain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical andclose enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutionshave multiple blow up at finitely many points which are thecritical points of a function whose definition involves Green'sfunction. As applications, we will give some existence results, in particular,when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

Super-position of spikes for a slightly super-critical elliptic equation in $R^N$

We prove existence of radial positive solutions for the equation $ -\Delta u + V(y) u =u^{\frac{N+2}{N-2}+\varepsilon} \quad$ in $\quad \mathbb R^N$ where the potential $V$ is a radial smooth function with $V(0) In particular, we show that the solutions have the shape of a super-position of spikes blowing-up at the origin as $\varepsilon \to 0$, with different rates of concentration.

Positive solutions of slightly supercritical elliptic equations in symmetric domains

This paper deals with existence and multiplicity of solutions for problem P(ε,Ω) below, which concentrate and blow-up at a finite number of points as ε→0. We give sufficient conditions on Ω which guarantee that the following property holds: there exists k―̄(Ω) such that, for each k⩾ k―̄(Ω), problem P(ε,Ω), for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. Exploiting the properties of the Green and Robin functions, we also prove that the blow up points approach the boundary of Ω as k→∞. Moreover we present some examples which show that P(ε,Ω) may have k-spike solutions of this type also when Ω is a contractible domain, not necessarily close to domains with nontrivial topology and, for ε>0 small and k large enough, even when it is very close to star-shaped domains.

On the Existence of Positive Solutions of Slightly Supercritical Elliptic Equations

Abstract In this paper we deal with existence and multiplicity of solutions for problem P(ε, Ω) below, in bounded domains Ω which do not satisfy some symmetry prop erties that play an important role in the proof of previous results on this subject. For ε &gt; 0 small and k large enough, we construct solutions blowing up, as ε → 0, at exactly k points which approach the boundary of Ω as k tends to infinity. We also present some examples showing that solutions of this type also exist in some contractible domains which may be even very close to starshaped domains.

Positive solutions for slightly super-critical elliptic equations in contractible domains

We give examples of bounded domains Ω, even contractible, having the following property: there exists k ̄ (Ω) such that, for every integer k⩾ k ̄ (Ω) , problem P(ε,Ω) below, for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. We also prove that the blow-up points tend to some points of ∂Ω as k→∞. To cite this article: R. Molle, D. Passaseo, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 459–462.

Best constants for supercritical Sobolev inequalities and application to some quasilinear supercritical elliptic equations of scalar curvature type

Let Σ be the set of O( k)× O( m)-invariant functions on the unit sphere S n , k+ m= n+1, k≥ m≥2. We investigate the optimal value of the constant B in the inequality ‖u‖ p≤B·‖∇u‖ q+A·‖u‖ q, where u∈ H 1 q ( S n )∩ Σ and 1/ p=1/ q−1/ k. We then give an application to the existence of positive solutions to scalar curvature type equations with critical supercritical Sobolev growth.

Positive solutions of super-critical elliptic equations and asymptotics

This paper is devoted to the study of positive solutions of semilinear elliptic equations . Asymp- totic behavior of ground states and uniqueness of singular ground states are proved via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has one positive solution with fast decay and infinitely many positive solutions with slow decay. The asymptotics of the singular sequence of fast decay solutions when p approaches to is also discussed.