In this paper, we classify the solutions of the critical semilinear problem involving the logarithmic Laplacian $$(E)\qquad \qquad\qquad\qquad\qquad \mathcal{L}_Δu= k u\log u,\qquad u\geq0 \quad \ {\rm in}\ \ \mathbb{R}^n, \qquad\qquad\qquad\qquad\qquad\qquad$$ where $k\in(0,+\infty)$, $\mathcal{L}_Δ$ is the logarithmic Laplacian in $\mathbb{R}^n$ with $n\in\mathbb{N}$, and $s\log s=0$ if $s=0$. When $k=\frac4n$, problem $(E)$ only has the solutions with the form $$u_{\tilde x,t}(x)=β_n \Big(\frac{t}{t^2+|x-\tilde x|^2)}\Big)^{\frac{n}{2}}\quad \text{ for any $t>0$, $\tilde x\in\mathbb{R}^n$},$$ where $n\in\mathbb{N}$, $β_n=2^{\frac n2} e^{\frac n2ψ(\frac n2) }>0$. When $k\in(0,+\infty)\setminus\{\frac 4n\}$, problem $(E)$ has no any positive solution.
We study bounded solutions to the fractional equation $(-Δ)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).
We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open sets.
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the non-local operator and boundary behaviour of Green and Poisson potentials.
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractiona…
The purpose of this paper is to study the existence of solutions for semilinear elliptic system driven by fractional Laplacian and establish some new existence results which are obtained by virtue of the local linking theorem and the saddle point theorem. To make the nonlinear scheme feasible, rigorous analysis of the function space involved and corresponding energy functional is necessary.
The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniquen…
Interior regularity of mixed local-nonlocal parabolic semilinear equations
Universidad Complutense de Madrid（马德里康普顿斯大学）；Facultad de Ciencias Económicas y de Administración, Universidad de la República（共和国大学经济与行政学院）；PEDECIBA Matemática（PEDECIBA数学研究所）
We study stable solutions to the equation $(-Δ)^{1/2} u = f(u)$, posed in a bounded domain of $\mathbb{R}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\leq 4$. This result, which was known only for $n=1$, follows from a new interior Hölder estimate that is completely independent of the nonlinearity $f$. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobol…
We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_Δu(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_Δv(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and $\mathcal{L}_Δ$ denotes the Logarithmic Laplacian arising as a formal derivative $\partial_s|_{s=0}(-Δ)^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the m…
We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text). Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester [Commun. Pure Appl. Math. 47 (1994), 1403-1410].
We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.
We provide bounds for the sequence of eigenvalues $\{λ_i(Ω)\}_i$ of the Dirichlet problem $$ L_Δu=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus Ω,$$ where $L_Δ$ is the logarithmic Laplacian operator with Fourier transform symbol $2\ln |ζ|$. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first $k$ eigenvalues by extending the Li-Yau method and Kröger's method respectively. Moreover, we show the limit of the sum of the first $k$ eigenva…
In this article, we consider a semilinear pseudo parabolic heat equation with the nonlinearity which is the product of logarithmic and polynomial functions. Here we prove the global existence of solution to the problem for arbitrary dimension $n \geq 1$ and power index $p>1$. Asymptotic behaviour of the solution has been addressed at different energy levels. Moreover, we prove that the global solution indeed decays with an exponential rate. Finally, sufficient conditions are provided under which blow up of solutions take place.
On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Brandenburg University of Technology Cottbus–Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
