Topological Methods in Nonlinear Analysis

Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents

Shujie Li and Zhaoli Liu

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Abstract

We study multiplicity of solutions of the following elliptic problems in which critical and supercritical Sobolev exponents are involved: $$ \begin{alignat}{2} -\Delta u& =g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \\ -{\rm div}(|\nabla u|^{p-2}\nabla u)&=g(x, u)+\lambda h(x, u) &\quad& \text{in } \Omega \text{ and } u=0 \text{ on } \partial\Omega, \end{alignat} $$ 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 $\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 28, Number 2 (2006), 235-261.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463144817

Mathematical Reviews number (MathSciNet)
MR2289687

Zentralblatt MATH identifier
1142.35039

Citation

Li, Shujie; Liu, Zhaoli. Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents. Topol. Methods Nonlinear Anal. 28 (2006), no. 2, 235--261. https://projecteuclid.org/euclid.tmna/1463144817


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