Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents
Keywords
nonlinear elliptic equation, Sobolev exponent, multiple solutionsAbstract
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 $\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.Downloads
Published
2006-12-01
How to Cite
1.
LI, Shujie and LIU, Zhaoli. Multiplicity of solutions for some elliptic equations involving critical and supercritical Sobolev exponents. Topological Methods in Nonlinear Analysis. Online. 1 December 2006. Vol. 28, no. 2, pp. 235 - 261. [Accessed 25 April 2024].
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