In this paper, we study the Einstein-scalar field Lichnerowicz equations on CR manifolds. We prove some existence results involving the equations.

This paper considers the following forager-exploiter model \begin{eqnarray*} \left \{ \begin{array}{lll} u_t=u_{xx}-\chi_1(uw_{x})_{x} + \mu_1 u(1-u-a_1v), \quad & x\in \Omega,\quad t > 0,\\ v_t=v_{xx}-\chi_2(vu_{x})_{x} + \mu_2 v(1-v-a_2u), \quad & x\in \Omega,\quad t > 0,\\ w_t=w_{xx}-\lambda(u + v)w-\mu w + r, \quad & x\in \Omega,\quad t > 0 \end{array} \right. \end{eqnarray*} in a smooth bounded domain $\Omega\subset \mathbb{R}$ with homogeneous Neumann boundary conditions, where the parameters $\chi_{1},\chi_{2},\mu_{1},\mu_{2},a_{1},a_{2},\lambda,\mu$, and $r$ are some positive constants. It is proved that for all appropriately regular initial data, the corresponding initial-boundary value problem has a global bounded classical solution. Moreover, the large time behavior of solution will be investigated. When $0 < a_{1},a_{2} < 1$, it is asserted that forager and exploiter will approach spatially homogeneous distributions under some explicit conditions; if $0 < a_{1} < 1,a_{2}\geq1$, the exploiter will become extinct.

We continue the study of positive singular solutions of PDEs arising from double phase functionals started in [6]. In particular, we consider the case $p < q < 2$, and we relax the assumption on the capacity of the singular set using an intrinsic notion of capacity.

In this article, we consider the following modified quasilinear critical Kirchhoff-Schrödinger problem involving Stein-Weiss type nonlinearity: $$ \mathcal K(u)= \lambda f(x) |u(x)|^{q-2} u(x)+ \Big ( \int_{\mathbb R^N}\frac{|u(y)|^{2p_{\beta,\mu}^{*}}} {|x-y|^{\mu}|y|^{\beta}}dy \Big ) \frac {|u(x)|^{2p_{\beta,\mu}^{*}-2} u(x)}{|x|^\beta} \; \text{ in }\; \mathbb R^N, $$ where $\lambda > 0$ is a parameter, $N\geq 3$, $$ \mathcal K(u) = \Big ( a+b \int_{\mathbb R^N}|\nabla u|^{p}dx \Big ) \Delta_{p} u - a u \Delta_{p}(u^2) $$ with $a > 0$, $b\geq 0$, $\beta\geq0,$ $0 < \mu < N$, $0 < 2\beta+ \mu < N$, $2\leq q < 2 p^*$. Here, $p^*=\frac{Np}{N-p}$ is the Sobolev critical exponent and $ p_{\beta,\mu}^{*}:= \frac p2\frac{(2N-2\beta-\mu)}{N-p} $ is the critical exponent with respect to the doubly weighted Hardy-Littlewood-Sobolev inequality (also called Stein-Weiss type inequality). Then by establishing a concentration-compactness argument for our problem, we show the existence of infinitely many nontrivial solutions to the equations with respect to the parameter $\lambda$ by using Krasnoselskii's genus theory, symmetric mountain pass theorem and $\mathbb Z_2$- symmetric version of mountain pass theorem for different ranges of $q$. We further show that these solutions belong to $L^\infty(\mathbb R^N)$.

In this paper, we study the existence of solutions to the Kirchhoff equation \begin{equation*} - \Big ( {a + b\int_{{\mathbb{R}}^{3}} {|\nabla u{|^2}dx} } \Big ) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~{\rm in}~{\mathbb{R}}^{3}, \end{equation*} having prescribed mass $$ \int_{{\mathbb{R}}^{3}} {|u{|^2}dx} = c, $$ where $a,b > 0$, $\mu \in {\mathbb{R}}$, $2 < q < p < 6$. When $(p,q)$ belongs to a certain domain in ${{\mathbb{R}}^{2}}$, we prove the existence and nonexistence of normalized solutions by using constraint minimization and concentration compactness principle, our main results may be illustrated by the red areas and green areas shown in Figure 1. In particular, our results are closely related to the values of $\mu$ and prescribed mass $c$, and partially extend the results of Li et al. [10] and Soave [20].

In this paper, we establish the existence of positive solutions to the semilinear problem for bi-harmonic operator $$ (-\Delta)^2u+u=|u|^{p-1}u, \ \ \text{ in }\ R^n. $$ One key tool is to use the Schwartz symmetrization method, in particular the Riesz inequality to this problem. The other tool is the generalized Ni-Strauss Lemma to radially symmetric functions in higher order Sobolev spaces.