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We discuss the existence and nonexistence of solutions to the steady-state Gierer-Meinhardt system $$ \begin{cases} \displaystyle -\Delta u=\frac{u^p}{v^q}+\lambda \rho(x) \,, u>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle -\Delta v=\frac{u^m}{v^s} \,, v>0 &\quad\mbox{ in }\mathbb{R}^N\setminus K,\\[0.1in] \displaystyle \;\;\; \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 &\quad\mbox{ on }\partial K,\\[0.1in] \displaystyle \;\;\; u(x), v(x)\to 0 &\quad\mbox{ as }|x|\to \infty, \end{cases} $$ where $K\subset \mathbb{R}^N$ $(N\geq 2)$ is a compact set, $\rho\in C^{0,\gamma}_{loc}(\overline{\mathbb{R}^N\setminus K})$, $\gamma\in (0,1)$, is a nonnegative function and $p,q,m,s, \lambda>0$. Combining fixed point arguments with suitable barrier functions, we construct solutions with a prescribed asymptotic growth at infinity. Our approach can be extended to many other classes of semilinear elliptic systems with various sign of exponents.

We consider the Lane-Emden system-$\Delta$u = |v| p-1 v,-$\Delta$v = |u| q-1 u in R d. When p $\ge$ q $\ge$ 1, it is known that there exists a positive radial stable solution (u, v) $\in$ C 2 (R d) if and only if d $\ge$ 11 and (p, q) lies on or above the so-called Joseph-Lundgren curve introduced in [5]. In this paper, we prove that for d $\le$ 10, there is no positive stable solution (or merely stable outside a compact set and (p, q) does not lie on the critical Sobolev hyperbola), while for d $\ge$ 11, the Joseph-Lundgren curve is indeed the dividing line for the existence of such solutions, if one assumes in addition that they are asymptotically homogeneous (see Definition 1 below). Most of our results are optimal improvements of previous works in the litterature.

We study the inequality $ -\Delta u - \frac{\mu}{|x|^2} u \geq (|x|^{-\alpha} * u^p)u^q$ in an unbounded cone $\mathcal{C}_\Omega^\rho\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $\Omega$ of the unit sphere $S^{N-1}\subset \mathbb{R}^N,$ $p, q, \rho>0$, $\mu\in \mathbb{R}$ and $0\leq \alpha < N$. In the above, $|x|^{-\alpha} * u^p$ denotes the standard convolution operator in the cone $\mathcal{C}_\Omega^\rho$. We discuss the existence and nonexistence of positive solutions in terms of $N, p, q, \alpha, \mu$ and $\Omega$. Extensions to systems of inequalities are also investigated.

We are concerned with the study of existence and nonexistence of weak solutions to $$ \begin{cases} &\displaystyle \frac{\partial^k u}{\partial t^k}+(-\Delta)^m u\geq (K\ast |u|^p)|u|^q \quad\mbox{ in } \mathbb R^N \times \mathbb R_+,\\[0.1in] &\displaystyle \frac{\partial^i u}{\partial t^i}(x,0) = u_i(x) \,\, \text{ in } \mathbb R^N,\, 0\leq i\leq k-1,\\ \end{cases} $$ where $N,k,m\geq 1$ are positive integers, $p,q>0$ and $u_i\in L^1_{\rm loc}(\mathbb{R}^N)$ for $0\leq i\leq k-1$. We assume that $K$ is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, $K\ast |u|^p$ denotes the standard convolution operation between $K(|x|)$ and $|u|^p$. We obtain necessary conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our analysis emphasizes the role played by the sign of $\displaystyle \frac{\partial^{k-1} u}{\partial t^{k-1}}$.

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

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $$ where $u_0\in L^1_{loc}({\mathbb R}^N)$, $L_{\mathcal{A}}$ denotes a weakly $m$-coercive operator, which includes as prototype the $m$-Laplacian or the generalized mean curvature operator, $p,\,q>0$, while $K\ast u^p$ stands for the standard convolution operator between a weight $K>0$ satisfying suitable conditions at infinity and $u^p$. For problem $(P^-)$ we obtain a Fujita type exponent while for $(P^+)$ we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.

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.

We study the quasilinear elliptic inequality $$ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, \mu \in \mathbb{R}$, $m>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions., Comment: 20 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 are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }\Omega,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s} &&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\in C^2(\Omega)$ ($1\leq k\leq N$) and $\Omega\subset\mathbb{R}^N$ $(N\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\geq 0$, $p\geq s\geq 0$ and $(k-m)(k-s)\neq pq$. In the case where $\Omega$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $\Omega=\mathbb{R}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\leq m

We study the inequality $$ {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq (I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $\alpha>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_\beta$ denotes the Riesz potential of order $\beta\in(0, N)$. We obtain sharp conditions in terms of these parameters for which positive singular solutions exist. We further establish the asymptotic profile of singular solutions to the double inequality $$ a(I_\beta\ast u^p)u^q\geq {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq b(I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $a\geq b>0$ are constants., Comment: 26 pages

We construct positive singular solutions for the problem $-\Delta u=\lambda \exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Differential Equations \textbf{264} (2018), 2684--2707] for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials., Comment: 15 pages

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_\alpha$ stands for the Riesz potential of order $\alpha\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $\alpha$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds., Comment: 18 pages. To appear in Indiana University Mathematics Journal

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q} &&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ where $\Omega\subset\R^N$ $(N\geq 2)$ is either a ball or the whole space, $10$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $(p-1-\alpha)(p-1-\beta)-qm\neq 0$. We first classify all the positive radial solutions in case $\Omega$ is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if $0\leq \alpha

We study the semilinear elliptic equation \begin{equation*} -\Delta u=u^\alpha |\log u|^\beta\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < \alpha < \frac{n+2}{n-2}$ and $-\infty<\beta<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{\alpha-1}} \left(\log \frac{1}{|x|}\right)^{-\frac{\beta}{\alpha-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2}{\alpha-1}\right)^{1-\beta}\left(n-2-\frac{2}{\alpha-1}\right)\right]^{\frac{1}{\alpha-1}}. \end{equation*}, Comment: to appear in Adv. Nonlinear Anal

We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$ with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one in the scalar case, and also a new class of singular solutions provided that the new invariant is nontrivial. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On the technical level, we adopt the method of the moving spheres and the balanced-energy-type monotonicity functionals.

This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in $\mathbb{R}^{N}$ to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory., Comment: 15 pages (There was a gap in a proof in the first version. The main result is unchanged, but the proof has been corrected in the later versions.)

We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\alpha$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach., Comment: 19 pages

We study the behavior near the origin in $\mathbb{R}^n ,n\geq3$, of nonnegative functions \begin{equation}\label{0.1} u\in C^2 (\mathbb{R}^n \backslash \{0\})\cap L^\lambda (\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-\Delta u\leq(|x|^{-\alpha}*u^\lambda )u^\sigma \quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $\alpha\in(0,n),\lambda>0,$ and $\sigma\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin., Comment: 24 pages, 3 figures

We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-\Delta u\leq v^\lambda\] \[0\leq v_t-\Delta v\leq u^\sigma\] in $\Omega\times (0,1)$, where $\lambda$ and $\sigma$ are nonnegative constants and $\Omega$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $\lambda$ and $\sigma$ such that solutions of this system satisfy pointwise bounds in compact subsets of $\Omega$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials., Comment: 31 pages, one figure

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -\Delta u \le (\frac{1}{|x|^\alpha}* v)^\lambda$ $0\le -\Delta v \le (\frac{1}{|x|^\beta}* u)^\sigma$ in $B_2(0)\setminus\{0\} \subset R^n$, $n\ge 3$, where $\lambda,\sigma \ge 0$ and $\alpha,\beta\in (0,n)$. A by-product of our methods used to study these solutions will be results on the behavior near the origin of $L^1(B_1(0))$ solutions $f$ and $g$ of the system $0 \le f(x) \le C(|x|^{2-\alpha} + \int_{|y|<1}\frac{ g(y) dy}{|x-y|^{\alpha-2}} )^\lambda$ $0 \le g(x) \le C(|x|^{2-\beta} + \int_{|y|<1}\frac{ f(y) dy}{|x-y|^{\beta-2}} )^\sigma$ for $0<|x|<1$ where $\lambda,\sigma \ge 0$ and $\alpha, \beta\in (2,n+2)$., Comment: 32 pages, one figure. To be published in Calculus of Variations and PDE. Errors are corrected in the second paragraph of Step 2 on pages 22 and 23. Also corrections are made on page 27, line 9 from the bottom, and on page 29, line 3 from the bottom

We study classical positive solutions of the biharmonic inequality $-\Delta^2 v \geq f(v)$ in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$ and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in any exterior domain of $\mathbb R^n$. Similar results were obtained by Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011-2047] for $-\Delta v\ge f(v)$ using a method which depends only on properties related to the maximum principle. Since the maximum principle does not hold for the biharmonic operator, we adopt a different approach which relies on a new representation formula and an a priori pointwise bound for nonnegative solutions of $-\Delta^2u \ge 0$ in a punctured neighborhood of the origin in $\mathbb{R}^n$., Comment: 36 pages

We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -\Delta u\leq f(v)$ $0\leq -\Delta v\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if $\log^+ f$ or $\log^+g$ grow at most linearly at infinity. In dimension $n\geq 3$ and under the assumption $f(t)=O(t^\lambda)$, $g(t)=O(t^\sigma)$ as $t\to \infty$, ($\lambda, \sigma\geq 0$), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context., Comment: 41 pages, 1 figure

We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions., Comment: 13 pages, 1 figure

We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence region from non-existence. In the critical case $p=p^*$ we show that the existence of a singular solution depends on the rate at which the coefficients stabilize at zero and we provide some optimal conditions in this setting., Comment: 24 pages

We study stable solutions of a fourth order nonlinear elliptic equation, both in entire space and in bounded domains.

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $\rho$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions., Comment: Journal of Differential Equations 2012, 28 pages

We study the semilinear elliptic inequality $-\Delta u\geq\varphi(\delta_K(x))f(u)$ in $R^N\setminus K,$ where $\varphi, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $\delta_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $\varphi$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$., Comment: 18 pages

We study the biharmonic equation $\Delta^2 u =u^{-\alpha}$, $0<\alpha<1$, in a smooth and bounded domain $\Omega\subset\RR^n$, $n\geq 2$, subject to Dirichlet boundary conditions. Under some suitable assumptions on $\o$ related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.