<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ u = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula>. We replace the singular function <inline-formula><tex-math id="M7">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> by a function <inline-formula><tex-math id="M8">\begin{document}$ g_\epsilon(u) $\end{document}</tex-math></inline-formula> which pointwisely converges to -<inline-formula><tex-math id="M9">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. When the parameter <inline-formula><tex-math id="M11">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is small enough, the corresponding energy functional to the perturbed equation <inline-formula><tex-math id="M12">\begin{document}$ -\Delta u + g_\epsilon(u) = \lambda f(u) $\end{document}</tex-math></inline-formula> has a critical point <inline-formula><tex-math id="M13">\begin{document}$ u_\epsilon $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ H_0^1(\Omega) $\end{document}</tex-math></inline-formula>, which converges to a nontrivial nonnegative solution of the original problem as <inline-formula><tex-math id="M15">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>
We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, βΞπ’ = (log π’ + π(π’))π {π’>0} in Ξ© β β 2 with Dirichlet boundary condition. The function π has exponential growth, which can be subcritical or critical with respect to the Trudinger-Moser inequality. We study the energy functional πΌ π corresponding to the perturbed equation βΞπ’ + π π (π’) = π(π’), where π π is well defined at 0 and approximates β log π’. We show that πΌ π has a critical point π’ π in π» 1 0 (Ξ©), which converges to a legitimate nontrivial nonnegative solution of the original problem as π β 0. We also investigate the problem with π(π’) replaced by ππ(π’), when the parameter π > 0 is sufficiently large.
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 $\epsilon \rightarrow 0$ .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citationsβcitations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright Β© 2024 scite LLC. All rights reserved.
Made with π for researchers
Part of the Research Solutions Family.