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

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