In this note, we complete the study of [3], where we got existence results for the quasilinear elliptic equation \begin{equation*} -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In [3], we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper, we remove this obstruction. To face the problem, we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t > 0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.

In this paper, by using the variational methods, we study the existence and concentration of positive solutions for the following fractional logarithmic Schrödinger equation $$ \epsilon^{2s}(-\Delta )^{s} u+V(x)u=u\,\text{log}\,u^{2}, \,\, x\in\mathbb{R}^{N}, $$ where $\epsilon>0$ is a parameter, $N > 2s$, $s\in(0,1)$ and $(-\Delta )^{s}$ is the fractional Laplacian, the potential $V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a continuous function satisfying a local assumption. We generalize the result obtained by Alves and Ji [3] for the case $s=1$ to the fractional logarithmic Schrödinger equation.

We consider the following system $$ \begin{cases} \displaystyle - \Big (a_1+b_1\int_{\mathbb{R}^3} |\nabla u|^2 dx + d\int_{\mathbb{R}^3} |\nabla v|^2 dx \Big ) \Delta u + u \\ \qquad\qquad\qquad \qquad\qquad\qquad = |u|^{p-2}u +\frac{\alpha\lambda}{p}|v|^\beta |u|^{\alpha-2}u , \\[8pt] \displaystyle - \Big (a_2+b_2\int_{\mathbb{R}^3} {|\nabla v|^2} dx + d\int_{\mathbb{R}^3} |\nabla u|^2 dx \Big ) \Delta v + v \\ \qquad\qquad\qquad \qquad\qquad\qquad = |v|^{p-2}v +\frac{\beta\lambda}{p} |u|^\alpha |v|^{\beta-2}v , \end{cases} $$ where $a_1 > 0$, $a_2 > 0$; $b_1\geq 0$, $b_2\geq 0$, $d\geq 0$ and $b_1 + b_2 + d \neq 0$; $\alpha > 1$, $\beta > 1$ with $\alpha+\beta=p\in (2, 6)$ and $\lambda \geq 0$. By introducing a variant variational identity and a constraint set, we are able to prove that for any positive integer $m$, the system admits a non-radially symmetric sign-changing solution $(u,v)\in H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$. Moreover, $u\neq 0$, $v\neq 0$ and both $u$ and $v$ change their sign exactly $m-$times.

The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.

In this paper, we deal with the multiple solutions for the degenerate Kirchhoff system with critical nonlinearity on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle -M \Big (\int_{\Omega}|\nabla_Hu|^{2}d\xi\Big )\Delta_{H} u + \phi_{1}|u|^{q-2} u=\lambda|u|^{2} u+F_{u}(\xi, u, v) & \text { in } \Omega, \\[8pt] \displaystyle -M\Big (\int_{\Omega}|\nabla_Hv|^{2}d\xi\Big )\Delta_{H} v + \phi_{2}|v|^{q-2} v=\lambda|v|^{2} v+F_{v}(\xi, u, v)& \text { in } \Omega, \\ \displaystyle -\Delta_{H} \phi_{1}=|u|^q, \quad-\Delta_{H} \phi_{2}=|v|^q & \text { in } \Omega, \\ \displaystyle \phi_{1}=\phi_{2}=u = v =0& \text { on } \partial \Omega,\end{cases} \end{equation*} where $\Delta_{H}$ is the Kohn-Laplacian, $1 < q < 2$, ${\lambda}$ is a positive real parameter, and $F=F(\xi,u,v),F_{u}=\frac{\partial F}{\partial u}$, $F_{v}=\frac{\partial F}{\partial v}$. Under some suitable assumptions on the Kirchhoff function $M$ and $F$, together with the symmetric mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in the degenerate cases on the Heisenberg group. The result of this paper extends or else completes recent papers and is new in several directions for the critical Kirchhoff-Poisson systems on the Heisenberg group.

In this paper, we consider the nonlinear Chern-Simons-Schrödinger equation \begin{equation*} -\Delta u+ \Big ( \frac{h_u^2(|x|)}{|x|^2} +\int^{\infty}_{|x|}\frac{h_u(s)}{s}u^2(s)ds \Big ) u =-a|u|^{p-2}u+f(u),\,\, x\in\mathbb R^2, \end{equation*} where $a > 0$, $p\in (2,3)$ and \begin{equation*} h_u(s)=\int^s_0\frac{\tau}{2}u^2(\tau)d\tau =\frac{1}{4\pi}\int_{B_s}u^2(x)dx \end{equation*} is the so-called Chern-Simons term, $f$ has sub-critical exponential growth $\lim_{|t|\to\infty}\frac{|f(t)|}{e^{\alpha t^2}}=0$ for every $\alpha > 0$. Under some wild assumptions on $f$, we prove the existence of a ground state solution, a mountain-pass type solution and infinitely many geometrically distinct of solutions for the above equation.

In this paper, we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s \to 1$.

In this paper, we study the traveling wave solutions for a three species competition system with two weak aboriginal competitors and one strong alien competitor. We are concerned with the existence of traveling waves such that these two co-existence aboriginal competitors are wiped out by the invading alien strong competitor. First, we derive the existence of wave profiles based on an application of Schauder's fixed point theorem with the help of constructing suitable generalized upper-lower solutions to capture the unstable wave tail limit. Then a new method for deriving the stable wave tail limit is introduced. Finally, the minimal invading speed is characterized.