Existence of solutions for some degenerate semilinear elliptic equations with measure data

We study the existence of a weak solution for a certain degenerate semilinear elliptic problem.

Planar vortices in a bounded domain with a hole

<p style='text-indent:20px;'>In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1111"> \begin{document}$ \begin{equation} \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &amp;\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &amp;\text{on}\; \partial O_0,\\ \psi = 0,\quad &amp;\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> is a positive constant, <inline-formula><tex-math id="M3">\begin{document}$ \rho_\lambda $\end{document}</tex-math></inline-formula> is a constant, depending on <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Omega = \Omega_0\setminus \bar{O}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ O_0 $\end{document}</tex-math></inline-formula> are two planar bounded simply-connected domains. We show that under the assumption <inline-formula><tex-math id="M8">\begin{document}$ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> small, (1) has a solution <inline-formula><tex-math id="M10">\begin{document}$ \psi_\lambda $\end{document}</tex-math></inline-formula>, whose vorticity set <inline-formula><tex-math id="M11">\begin{document}$ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)&gt;0\} $\end{document}</tex-math></inline-formula> shrinks to the boundary of the hole as <inline-formula><tex-math id="M12">\begin{document}$ \lambda\to +\infty $\end{document}</tex-math></inline-formula>.</p>

An Application of the Browder-Minty Theorem in a Problem of Partial Differential Equations

In this work, we will show existence of weak solution for a semilinear elliptic problem using as main tool the Browder-Minty Theorem. First, we will make a brief introduction about basic theory of the Sobolev Spaces to support our study and provide sufficient tools for the development of our work. Then we will take a quick approach on the Browder-Minty Theorem and use this result to show the existence of at least one weak solution to an elliptic Partial Differential Equations (PDE) problem whose nonlinearity, denoted by f, is a known function. For this, in addition to the already mentioned results, we will also use as study tools: Embedding Sobolev Theorems, Linear Continuous Operators Theory, Poincaré Inequality and Hölder Inequality.

Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.