1 results
Singular elliptic equations with nonlinearities of subcritical and critical growth
- Part of
- Journal: Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 3 / August 2022
- Published online by Cambridge University Press: 27 July 2022, pp. 652-690
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View abstract
In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$
in $\Omega$
with $u=0$
on $\partial \Omega$
, where $0<\beta <1$
and $\Omega$
is a smooth bounded domain in $\mathbb {R}^{N}$
, $N\geq 2$
. We are able to solve this problem provided $f$
has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$
and $N\geq 3$
. In this case, we are able to obtain a solution for large values of $\lambda$
. We replace the singular function $u^{-\beta }$
by a function $g_\epsilon (u)$
which pointwisely converges to $u^{-\beta }$
as $\epsilon \rightarrow 0$
. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$
has a critical point $u_\epsilon$
in $H_0^{1}(\Omega )$
, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$
.
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