Description

• Title:
  Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents
• Creator:
  Li, Shujie;
  Liu, Zhaoli
• Description:
  We study multiplicity of solutions of the following elliptic problems in which critical and supercritical Sobolev exponents are involved: $$ \alignat 2 -\Delta u& =g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \\ -\div(|\nabla u|^{p-2}\nabla u)&=g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \endalignat $$ where $\Omega$ is a smooth bounded domain in ${\mathbb R}^N$, $p> 1$, $\lambda$ is a parameter, and $\lambda h(x, u)$ is regarded as a perturbation term of the problems. Except oddness with respect to $u$ in some cases, we do not assume any condition on $h$. For the first problem, we get a result on existence of three nontrivial solutions for $|\lambda|$ small in the case where $g$ is superlinear and $\limsup_{|t| \to\infty}g(x, t)/|t|^{2^*-1}$ is suitably small. We also prove that the first problem has $2k$ distinct solutions for $|\lambda|$ small when $g$ and $h$ are odd and there are $k$ eigenvalues between $\lim_{t\to0}g(x, t)/t$ and $\lim_{|t|\to\infty}g(x, t)/t$. For the second problem, we prove that it has more and more distinct solutions as $\lambda$ tends to 0 assuming that $g$ and $h$ are odd and $g$ is superlinear and $\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2006.12.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 28, No 2 (December 2006); 235 - 261;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2006.029/8876
• Subject:
  nonlinear elliptic equation;
  Sobolev exponent;
  multiple solutions;
  35J20;
  35J25;
  35J60
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2006.029
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other