Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.     

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$\{\begin{array}{cc}\hfill & ?{\epsilon }^2\mathrm{\Delta }u+V(x)u+{\lambda }_1|u{|}^{2}u+{\lambda }_2(K?|u{|}^2)u=0\text{ in }{\mathbb{R}}^3,\hfill \\ \hfill & u>0,u\in H^1({\mathbb{R}}^3),\hfill \end{array}$

where ε is a small positive parameter, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ? denotes the convolution, $K(x)=\frac{1?3{\cos }^2?\theta }{|x{|}^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under certain assumptions on $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}^2$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local minima of V as $\epsilon \to 0$. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.