Existence of groundstates for Choquard type equations with Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.

Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\triangle u + u + \phi u = u^{5}+\lambda g(u), &amp;\hbox{in}\ \ \Omega, \\\ &amp; -\triangle \phi = u^{2}, &amp; \hbox{in}\ \ \Omega, \\\ &amp; u, \phi = 0, &amp; \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> and the nonlinear growth of <inline-formula><tex-math id="M4">\begin{document}$ u^{5} $\end{document}</tex-math></inline-formula> reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.</p>

Sign Changing Solutions for Coupled Critical Elliptic Equations

In this paper, we consider the coupled elliptic system with a Sobolev critical exponent. We show the existence of a sign changing solution for problem P for the coupling parameter −μ1μ2<β<0. We also construct multiple sign changing solutions for the symmetric case.