Let $λ^*>0$ denote the largest possible value of $λ$ such that \begin{align*} \left\{\begin{aligned} Δ^2 u & = \la e^u && \text{in $B $} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a solution, where $B$ is the unit ball in $\R^N$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^*$ this problem possesses a unique {\em weak} solution $u^*$. We prove that $u^*$ is smooth if $N\le 12$ and singular when $N\ge 13$, in which case $ u^*(r) = - 4 \log r + \log (8(N-2)(N-4) / λ^*) + o(1)$ as $r\to 0$. We also consider the problem with general constant Dirichlet boundary conditions.
Consider the problem {ll} Δ^2 u= λe^{u} &\text{in} B u=\frac{\partial u}{\partial n}=0 &\text{on}\partial B, where $B$ is the unit ball in $\R^N$ and $λ$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln(|x|)$, for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Dávila et al. in \cite{DDGM} used a computer assisted proof to show that the extremal solution is singular in dimensions $13\leq N\leq 31$. Here by an improved Hardy-…
By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth.
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and mild solutions, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.
When the spatial dimensions $n$=2, the initial data $u_0\in H^1$ and the Hamiltonian $H(u_0)\leq 1$, we prove that the scattering operator is well-defined in the whole energy space $H^1(\mathbb{R}^2)$ for nonlinear Schrödinger equation with exponential nonlinearity $(e^{λ|u|^2}-1)u$, where $0<λ<4π$.
We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity $$ i\partial_t u+Δu=\frac{u}{|x|^b}(e^{α|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $α=2π(2-b)$. We establish local and global well-posedness in the subcritical and critical regimes.
The mean field equation involving the $N$-Laplace operator and an exponential nonlinearity is considered in dimension $N\geq2$ on bounded domains with homogenoeus Dirichlet boundary condition. By a detailed asymptotic analysis we derive a quantization property in the non-compact case, yielding to the compactness of the solutions set in the so-called non-resonant regime. In such a regime, an existence result is then provided by a variational approach.
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential n…
In the present work we are concerned with the Choquard Logarithmic equation $-Δu + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.
We study the two dimensional non-linear Schrödinger equation with two types of exponential non-linearities. It is well-known by a work of Ruf - Sani, that such models support solitary wave solutions, which are solutions of some constrained minimization problem. We show that these Ruf - Sani solitons, are spectrally unstable as well as unstable by blow-up.
In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation $$ -ε^{2}Δ{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, $$ where $ε>0$ is a positive parameter, $V$ is a saddle-like potential and $f$ has an exponential critical growth.
We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any $N>2$.
We are interested in the qualitative properties of solutions of the Hénon type equations with exponential nonlinearity. First, we classify the stable at infinity solutions of $Δu +|x|^αe^u=0$ in $\mathbb{R}^N$, which gives a complete answer to the problem considered in [Wang-Ye]. Secondly, existence and precise asymptotic behaviors of entire radial solutions to $Δ^2 u=|x|^α e^u$ are obtained. Then we classify the stable and stable at infinity radial solutions to $Δ^2 u=|x|^α e^u$ in any dimension.
In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-Δ)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable solutions provided that $N<10s$. This result is optimal when $s\uparrow 1$.
We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical regime.
This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i\partial_t u+Δu= θ(ωt)\big(e^{4π|u|^2}-1\big)$, where $θ$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1(\mathbb{R}^2)$, the solution $u_ω$ converges, as $|ω|$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U+ΔU= I(θ)\big(e^{4π|U|^2}-1\big)$ with the same initial data, where $I(θ)$ is the average of $θ$.
In the present work we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -Δu + λu + μ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2 , \\ \intR |u(x)|^2 dx = c,\ c> 0 , \end{array} \right. $$ for $μ\in \R $ and a nonlinearity $f$ with exponential critical growth. Here $ λ\in \R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in \cite{Ji} and \cite{[cjj]}.
We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times L^{\infty}$, we derive the blow-up rate near a non-characteristic point in the smaller space, and give some bounds near other points. Our result generalize those proved by Godin under high regularity assumptions on initial data.
In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left\{\begin{split} M\left(\|u\|^{\frac{n}{s}}\right)(-Δ)^s_{n/s}u & = μu^{-q}+ u^{r-1}\exp( u^β)\;\text{in}\;\Om, u&>0,\;\text{in}\; \Om, u &= 0,\;\text{in}\; \mb R^n \setminus{\Om}, \end{split} \right. \end{equation*} where $\Om$ is smooth bounded domain in $\mb R^n$, {$n\geq 1$}, $s\in (0,1)$, $μ>0$ is a real parameter, $β<\frac{n}{n-s}$ and $q\in (0,1)$. We have considered the degenerate Kirchhoff case here…
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {Δ^{2}u-ρ^{4}k(x)e^{u}=0 & \hbox{in}Ω, u=Δu=0 & \hbox{on}\partialΩ, has a solution blowing-up, as $ρ\to0$, at $m$ points of $Ω$, for any given number $m$.
In this paper we consider the following quasilinear Schrödinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - Δu +ϕu = f(u) &\ \mbox{in } Ω, -Δϕ- \varepsilon^{4}Δ_4 ϕ= u^{2} & \ \mbox{in } Ω, u=ϕ=0 & \ \mbox{on } \partialΩ\end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, ϕ_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associa…
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{δE}{δh}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution prese…
We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. We prove that any energy solution has a global bound of the $L^6_{t,x}$ space-time norm, and hence scatters in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$ as $t\rightarrow\pm \infty$. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364 (2012), no. 3, 1571--1631). However, since well-posedness in $H^{1/2}(\…
We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation \[iu_t+Δu=u(e^{4π|u|^2}-1)\] in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao--Visan--Zhang. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson and the work of Ibrahim--Majdoub--Masmoudi--Nakanishi on a related problem to prove corresponding spacetime estimates which imply sc…
Let $Ω$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - Δu \pm g(u) &= μ\quad &\text{in } Q_T:=Ω\times (0,T) \\ \phantom{------,} u&=0 &\text{on } \partial Ω\times (0,T)\\ \phantom{----,} u(.,0) &=ω&\text{in } Ω. \end{array} \right. \end{equation} where $μ$ and $ω$ are bounded Radon measures in $Q_T$ and $Ω$ respectively and $g(u) \sim e^{a |u|^q} $ with $a>0$ and $q \geq 1$. We provide a sufficient condition in terms of fractional maximal potentials of $μ$ and $ω$ for solving (P).
This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential …
We study the existence of nontrivial solutions for a nonlinear fractional elliptic equation in presence of logarithmic and critical exponential nonlinearities. This problem extends [5] to fractional $N/s$-Laplacian equations with logarithmic nonlinearity. We overcome the lack of compactness due to the critical exponential nonlinearity by using the fractional Trudinger-Moser inequality. The existence result is established via critical point theory.
We investigate the defocusing inhomogeneous nonlinear Schrödinger equation $$ i \partial_tu + Δu = |x|^{-b} \left({\rm e}^{α|u|^2} - 1- α|u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and $α=2π(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $Σ(\mathbb{R}^2)=\{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$. Then we combine the local theory with the decay estimate to obtain scattering in $Σ$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.
We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-Δ)^{\frac{1}{2}} u &=\frac{p}{p+q}λf(x)|u|^{p-2}u|v|^q + h_1(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1),\\ (-Δ)^{\frac{1}{2}} v &=\frac{q}{p+q}λf(x)|u|^p|v|^{q-2}v + h_2(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1), u,v&>0 \;\textrm{in } \; (-1,1), u&=v=0 \; \text{in} \; \mathbb R\setminus (-1,1). \end{array} \end{equation*} where {$1<p+q<2$}, $h_1(u,v)=(α{+}2u^2)|u|^{α-2}u|v|^β, h_2(u,v)=(β{+}2v^2) |u|^α|v|^{β-2}v$ and ${α+β>2}$. Here $(-Δ)^{\frac{1}{2}}$ i…
The present work is concerned with the following version of Choquard Logarithmic equations $ -Δ_p u -Δ_N u + a|u|^{p-2}u + b|u|^{N-2}u + λ(\ln|\cdot|\ast G(u))g(u) = f(u) \textrm{ in } \mathbb{R}^N $ , where $ a, b, λ>0 $, $ \max\{\frac{N}{2}, 2 \} < p< N $, $f, g: \mathbb{R} \rightarrow \mathbb{R} $ are continuous functions that behave like $ \exp(α|u|^{\frac{N}{N-1}}) $ at infinity, for $ α>0 $, and that has polynomial growth, respectively, and $ G(s)=\int\limits_{0}^{s}g(τ)dτ$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution.…
We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-Δu + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is loca…
We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical $α$-stable noises, $α\in (0,2)$, introduced in Priola-Zabczyk "Structural properties of semilinear SPDEs driven by cylindrical stable processes" (PTRF to appear). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are suffici…
In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-Δ)^{s}_{n/s} u=\left(\displaystyle\int_Ω\frac{G(y,u)}{|x-y|^μ}~dy\right)g(x,u) \; \text{in}\; Ω,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus Ω, \end{array} \right. $$ where $(-Δ)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $Ω\subset \mathbb R^n$ is a bounded …
We consider the Cauchy problem for semi-linear heat equations with exponential nonlinearity. The main purpose of this paper is to prove the existence of solutions lying on the borderline between global existence and blow-up infinite time. The existence has been shown for semi-linear heat equations with power type nonlinearity. We explain the main strategy to prove the existence. By using the definition of exponential function, we approximate the solution to exponential type equation by that of power type equation. Then we can use directly the knowledge for power type equation.
By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends results in the literature for the N-Laplacian operator.
In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDE's which use Arhenious rates, (and also have a fourth order exponential nonlinearity) the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even…
We study a class of nonlinear non-autonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.
In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger-Moser type inequality. Under some assumptions on the potential and the nonlinearity, it is proved that there is a nontrivial positive weak solution to this equation. Also it is shown that there are two distinct positive weak solutions to a perturbation of the equation. The method of proving these results is combining Trudinger-Moser type inequality, Mountain-pass theorem and Ekeland's varia…
Let $n \geq 2$ and $Ω\subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(Ω)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $γ_0 > 0$, and a corresponding $R_0(γ_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$…
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.
In this paper we study existence of ground state solution to the following problem $$ (- Δ)^αu = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^α(\mathbb R^N) $$ where $(-Δ)^α$ is the fractional Laplacian, $α\in (0,1)$. We treat both cases $N\geq2$ and $N=1$ with $α=1/2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\geq2$, exponential if $N=1$.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
In this paper we consider the initial value {problem $\partial_{t} u- Δu=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|\sim \mbox{e}^{|u|^q}$ as $|u|\to \infty$,} $|f(u)|\sim |u|^{m}$ as $u\to 0,$ $0<q\leq p\leq\,m,\;{N(m-1)\over 2}\geq p>1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$
