Raffaella Servadei
Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity
(76K, LaTeX)
ABSTRACT. In this paper we discuss the existence of infinitely many solutions for a nonlocal, nonlinear equation with homogeneous Dirichlet boundary data.
Our model problem is the following one
$$ \left\{
egin{array}{ll}
(-\Delta)^s u-\lambda u=|u|^{q-2}u+h & {\mbox{ in }} \Omega\
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,,
\end{array}
ight.
$$
where $s\in (0,1)$ is a fixed parameter, $(-\Delta)^s$ is the fractional Laplace operator,
which (up to normalization factors) may be defined as
$$-(-\Delta)^s u(x)=
\int_{\RR^n}rac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in \RR^n\,,$$
while $\lambda$ is a real parameter, the exponent~$q\in (2, 2^*)$, with $2^*=2n/(n-2s)$, $n>2s$, the function~$h$ belongs to the space $L^2(\Omega)$ and, finally,
the set $\Omega$ is an open, bounded subset of $\RR^n$ with
Lipschitz boundary. Here the solution is sought to satisfy $u = 0$ on $\RR^n\setminus \Omega$ and not simply on $\partial \Omega$,
consistently with the non-local character of the fractional Laplace operator.
Adapting the classical variational techniques used in order to study the standard Laplace equation with subcritical growth nonlinearities to the nonlocal framework, along the present paper we prove that this problem admits infinitely many weak solutions~$u_k$, with the property that their Sobolev norm goes to infinity as $k o +\infty$\,, provided the exponent $q<2^*-2s/(n-2s)$\,.
In this sense, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.