A log–exp elliptic equation in the plane

Abstract

<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>

A Choquard‐type equation with a singular absorption nonlinearity in two dimensions

“…where C > 0 is independent of 𝜖. Rest of the proof follows exactly as the proof of Proposition 1 in Figueiredo et al 1 This completes the proof.…”

“…From the geometry of J 𝜖,𝜆 and using variational arguments, we prove the existence of a nontrivial solution, u 𝜖 , to ( 𝜖,𝜆 ). Next from a priori estimates and a key point pointwise gradient estimate (in the spirit of works, [1][2][3] see Section 4) independent of 𝜖, we show that u 𝜖 converges as 𝜖 → 0 + to a nontrivial solution of ( 𝜆 ). For reader's convenience, we now present a brief introduction to existence and multiplicity results for the equations involving singular and Choquard nonlinearities.…”

A Choquard type equation with a singular absorption nonlinearity in two dimension

“…where C > 0 is independent of ǫ. Rest of the proof follows exactly as the proof of Proposition 1 in [10]. This completes the proof.…”

“…From the geometry of J ǫ,λ and using variational arguments, we prove the existence of a nontrivial solution, u ǫ , to (P ǫ,λ ). Next from a priori estimates and a key point pointwise gradient estimate (in the spirit of works [10,17,25], see Section 4) independent of ǫ, we show that u ǫ converges as ǫ → 0 + to a nontrivial solution of (P λ ).…”

“…When n = 2, problems with nonlinearities of exponential growth have been considered. In this direction, the case of g of type (b) and h to be of exponential growth was studied in [10]. In this paper, the authors showed the existence of a weak solution under suitable assumptions on h and for small values of λ. Lastly the case of g of type (a) and h to be of exponential growth was handled in [25].…”