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Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ such that $u^*(x)\to\infty$ as $|x|\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ and $u_0\not\equiv u^*$, while the problem has no nonnegative local-in-time solutions if $u_0\ge u^*$ and $u_0\not\equiv u^*$.

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote the Riesz potentials of order $\alpha$ and $\beta$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $\alpha$, $\beta$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed., Comment: 15 pages

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equation \[ \begin{cases} \partial_tu=\Delta u+f(u) & \textrm{in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N \end{cases} \] under mild assumptions on $f$. A relationship between a growth of $f$ and an integrability of $u_0$ is studied in detail. Our existence theorem gives a sharp integrability condition on $u_0$ in a critical and subcritical cases, and it can be applied to a regularly or rapidly varying function $f$. In a doubly critical case existence and nonexistence of a nonnegative solution can be determined by special treatment. When $f(u)=u^{1+2/N}[\log(u+e)]^{\beta}$, a complete classification of existence and nonexistence of a nonnegative solution is obtained. We also show that the same characterization as in Laister et. al. [11] is still valid in the closure of the space of bounded uniformly continuous functions in the space $L^r_{\rm ul}(\mathbb{R}^N)$. Main technical tools are a monotone iterative method, $L^p$-$L^q$ estimates, Jensen's inequality and differential inequalities., Comment: 31 pages

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $\Psi>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

In this paper, we consider the following minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = \alpha_1, \ \| u_2 \|_{L^2}^2 = \alpha_2 \right\}, \] where $\alpha_1,\alpha_2 > 0$ and $E(u)$ is defined by \[ E(u) := \int_{\mathbf{R}^N} \left\{\frac{1}{2} \sum_{i=1}^2 \left( |\nabla u_1|^2 + V_i (x) |u_i|^2 \right) - \sum_{i=1}^2 \frac{\mu_i}{2p_i+2} |u_i|^{2p_i+2} - \frac{\beta}{p_3+1} |u_1|^{p_3+1} |u_2|^{p_3+1} \right\} \mathrm{d} x. \] Here $N \geq 1$, $ \mu_1,\mu_2,\beta > 0$ and $V_i(x)$ $(i=1,2)$ are given functions. For $V_i(x)$, we consider two cases: (i) both of $V_1$ and $V_2$ are bounded, (ii) one of $V_1$ and $V_2$ is bounded. Under some assumptions on $V_i$ and $p_j$, we discuss the compactness of any minimizing sequence., Comment: 31 pages

We investigate radial solutions for the problem \[ \begin{cases} \displaystyle -\Delta U=\frac{\lambda+\delta|\nabla U|^2}{1-U},\; U>0 & \textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset \mathbb{R}^N$ $(N\geq 2)$ denotes the open unit ball and $\lambda, \delta>0$ are real numbers. Two classes of solutions are considered in this work: (i) {\it regular solutions}, which satisfy $00$ is also discussed., Comment: 14 pages

We study the initial value problem $$ \begin{cases} r^{-(\gamma-1)}\left(r^{\alpha}|u'|^{\beta-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 00 & \textrm{for}\ 0\alpha>\beta\geq 1$ and $f\in C[0,\bar u)\cap C^2(0,\bar u)$, $f(0)=0$, $f(u)>0$ on $(0, \bar u)$ and $f$ satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution $u^*$ to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions $u(\,\cdot\,,a)$ (with $u(0,a)=a$) as $a\to 0$. In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated., Comment: 29 pages

We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a new supersolution which plays a crucial role. Our method does not rely on a change of variables, and hence it can be applied to a wide class of nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$ or $e^u$, a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen's inequality and $L^p$-$L^q$ type estimates., Comment: 25 pages

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge 3$. We show that there exists a sequence $\{\varepsilon_n^*\}_{n=1}^{\infty}$ ($\varepsilon_1^*>\varepsilon_2^*>\cdots\rightarrow 0$) such that this problem has infinitely many singular solutions $\{(\varepsilon_n^*,U_n^*)\}_{n=1}^{\infty}\subset\mathbb{R}\times (C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist of infinitely many smooth curves in the $(\varepsilon,U(0))$-plane. It is shown that each curve blows up at $\varepsilon_n^*$ and if $p_{\rm{S}}0$ such that the problem has no nonconstant regular solution if $\varepsilon>\bar{\varepsilon}$. The main technical tool is the intersection number between the regular and singular solutions., Comment: 37 pages, 1 figure

We study radial solutions of the semilinear elliptic equation $\Delta u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\Delta u+u^p=0$ and $\Delta u+e^u=0$., Comment: 21 pages

Let $\mathbb{S}^N\subset\mathbb{R}^{N+1}$, $N\ge 3$, be the unit sphere, and let $S_{\Theta}\subset\mathbb{S}^N$ be a geodesic ball with geodesic radius $\Theta\in(0,\pi)$. We study the bifurcation diagram $\{(\Theta,\left\|U\right\|_{\infty})\}\subset\mathbb{R}^2$ of the radial solutions of the Emden-Fowler equation on $S_{\Theta}$ $\Delta_{\mathbb{S}^N}U+U^p=0$ in $S_{\Theta}$, $U=0$ on $\partial S_{\Theta}$, $U>0$ in $S_{\Theta}$, where $p>1$. Among other things, we prove the following: For each $p>p_{\rm S}:=(N-2)/(N+2)$, there exists $\underline{\Theta}\in(0,\pi)$ such that the problem has a radial solution for $\Theta\in(\underline{\Theta},\pi)$ and has no radial solution for $\Theta\in(0,\underline{\Theta})$. Moreover, this solution is unique in the space of radial functions if $\Theta$ is close to $\pi$. If $p_{\rm S}

It is known that every positive solution of a one-dimensional Gel'fand problem can be written explicitly. In this paper we obtain exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We also study asymptotic behaviors of eigenvalues and eigenfunctions as the $L^{\infty}$-norm of the solution goes to the infinity., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Journal of Mathematical Physics, 60 (2019), 021506 and may be found at https://doi.org/10.1063/1.5021825

We study the existence and nonexistence of a Cauchy problem of the semilinear heat equation $\partial_tu=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^N\times(0,T)$, $u(x,0)=\phi(x)$ in $\mathbb{R}^N$, in $L^1(\mathbb{R}^N)$. Here, $N \ge 1$, $p=1+2/N$ and $\phi\in L^1( \mathbb{R}^N)$ is a possibly sign-changing initial function. Since $N(p-1)/2=1$, the $L^1$ space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every $\phi\in L^1(\mathbb{R}^N)$. Let $X_q:=\{ \phi\in L^1_{\rm{loc}}(\mathbb{R}^N)\ |\ \int_{\mathbb{R}^N}|\phi| \left[\log (e+|\phi|)\right]^qdx<\infty \} (\subset L^1(\mathbb{R}^N))$. In this paper we construct a local-in-time mild solution in $L^1(\mathbb{R}^N)$ for $\phi\in X_q$ if $q\ge N/2$. We show that, for each $0\le q

The aim of this paper is to deal with the $k$-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem \begin{equation*} (1)\;\;\;\begin{cases} S_k(D^2u)= \lambda \frac{|x|^{\mu-2}}{(1+|x|^2)^{\frac{\mu}{2}}} (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation*} where $B$ denotes the unit ball in $\mathbb{R}^n,n>2k$ ($k\in\mathbb{N}$), $\lambda>0$ is an additional parameter, $q>k$ and $\mu\geq 2$. In this setting, through a transformation recently introduced by two of the authors that reduces problem (1) to a non-autonomous two-dimensional generalized Lotka-Volterra system, we prove the existence and multiplicity of solutions for the above problem combining dynamical-systems tools, the intersection number between a regular and a singular solution and the super and subsolution method., Comment: Paper submitted with date 2017-05-23

We construct countably infinitely many nonradial singular solutions of the problem \[ \Delta u+e^u=0\ \ \textrm{in}\ \ \mathbb{R}^N\backslash\{0\},\ \ 4\le N\le 10 \] of the form \[ u(r,\sigma)=-2\log r+\log 2(N-2)+v(\sigma), \] where $v(\sigma)$ depends only on $\sigma\in \mathbb{S}^{N-1}$. To this end we construct countably infinitely many solutions of \[ \Delta_{\mathbb{S}^{N-1}}v+2(N-2)(e^v-1)=0,\ \ 4\le N\le 10, \] using ODE techniques., Comment: 15 pages