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

Shujie Li, Zhaoli Liu

DOI: http://dx.doi.org/10.12775/TMNA.2006.029

Abstract


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
$\lim_{|t| \to\infty}g(x, t)/|t|^{p^*-1}=0$.

Keywords


nonlinear elliptic equation; Sobolev exponent; multiple solutions

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