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

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

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

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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