A new kind of blowing-up solutions for the Brezis-Nirenberg problem
type: Item
instance of: scholarly article
title: A new kind of blowing-up solutions for the Brezis-Nirenberg problem
zbMATH Open document ID: 1314.35050
author: Giusi Vaira
published in: Calculus of Variations and Partial Differential Equations
publication date: 2015-01-23T00:00:00Z
review text: In this paper, the author is interested in the existence of solutions for the Brezis-Nirenberg problem \[ \begin{cases} -\Delta u = |u|^{p-1}u+\varepsilon\lambda u \quad & \text{ in }\Omega, \\ u=0 & \text{ on }\partial\Omega, \end{cases} \eqno (\mathcal {BN}) \] where \(\varepsilon\) is a small and positive parameter, \(\lambda\in\{-1,1\}, p=\frac{N+2}{N-2}\) and \(\Omega\) is a bounded open domain in \(\mathbb R^N, N>6\), with nontrivial topology and smooth boundary such that the problem \[ \begin{cases} -\Delta u_0 = |u_0|^{p-1}u_0 \quad & \text{ in }\Omega, \\ u_0=0 &\text{ on }\partial\Omega, \\ u_0>0 & \text{ in }\Omega,\end{cases} \eqno(1.1) \] has a positive solution \(u_0\) which is non-degenerate. The author gives an improvement about the existence of positive and sign-changing solutions for \((\mathcal{BN})\) and shows that the result obtained can be extended to the problem \[ \begin{cases} -\Delta u = |u|^{p-1-\lambda\varepsilon} u \quad & \text{ in }\Omega, \\ u=0 & \text{ on } \partial\Omega. \end{cases} \eqno(\mathcal {AC}) \] The following results are obtained. \(\quad\) {Theorem 1.1.} Let \(N>6\) and \(\lambda =1\). (i) Assume that \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a sign-changing solution which blows-up negatively at \(\overline{\xi}_0\). (ii) Assume that \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a positive solution which blows-up positively at \(\overline{\xi}_0\). (iii) Assume that \(\overline{\xi}_0\in\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\map…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 6390176
zbMATH Keywords: Brezis-Nirenberg problem
zbMATH Keywords: supercritical case
zbMATH Keywords: blowing-up solution
MaRDI profile type: Publication
full work available at URL: s00526-014-0716-y
OpenAlex ID: W2083287007
cites work: On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain
cites work: On a variational problem with lack of compactness: The topological effect of the critical points at infinity
cites work: On the existence and the profile of nodal solutions of elliptic equations involving critical growth
cites work: Nonexistence results of sign-changing solutions to a supercritical nonlinear problem
cites work: A NONEXISTENCE RESULT OF SINGLE PEAKED SOLUTIONS TO A SUPERCRITICAL NONLINEAR PROBLEM
cites work: Blow-up and nonexistence of sign changing solutions to the Brezis-Nirenberg problem in dimension three
cites work: Blow-up and symmetry of sign-changing solutions to some critical elliptic equations
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain
cites work: Some existence results for superlinear elliptic boundary value problems involving critical exponents
cites work: ``Bubble-tower'' radial solutions in the slightly supercritical Brezis-Nirenberg problem.
cites work: The Brezis-Nirenberg problem near criticality in dimension 3
cites work: Two-bubble solutions in the super-critical Bahri-Coron's problem
cites work: MULTI-BUBBLE SOLUTIONS FOR SLIGHTLY SUPER-CRITICAL ELLIPTIC PROBLEMS IN DOMAINS WITH SYMMETRIES
cites work: Multi-peak solutions for super-critical elliptic problems in domains with small holes
cites work: Concentration estimates and multiple solutions to elliptic problems at critical growth
cites work: A MULTIPLICITY RESULT FOR ELLIPTIC EQUATIONS AT CRITICAL GROWTH IN LOW DIMENSION
cites work: Elliptic equations with critical Sobolev exponents in dimension 3
cites work: Bubble towers for supercritical semilinear elliptic equations
cites work: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent
cites work: On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth
cites work: Multispike solutions for a nonlinear elliptic problem involving critical Sobolev exponent
cites work: DOUBLE BLOW-UP SOLUTIONS FOR A BREZIS–NIRENBERG TYPE PROBLEM
cites work: Multiplicity of solutions to the supercritical Bahri--Coron's problem in pierced domains
cites work: Q5516356
cites work: The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent
cites work: Proof of two conjectures of H. Brezis and L. A. Peletier
cites work: Sign-Changing Blow-Up for Scalar Curvature Type Equations
cites work: Generic properties of nonlinear boundary value problems
DOI: 10.1007/S00526-014-0716-Y
description: scientific article; zbMATH DE number 6390176