In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R \setminus (a,b)\\ \mathcal{N}_{1/2}u(x) &= 0\;\;\mbox{in}\;\;(a,b), \end{aligned} \right. \end{equation} where $a,b\in \R$ with $a<b$, $(-Δ)^{\frac{1}{2}}$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_{1/2}u(x) = \frac{1}π \int_{\R\setminus (a,b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy,\;\;x\in [a,b]. $$
We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
The classical local Neumann problem is well studied and solutions of this problem lie, in general, in a Sobolev space. In this work, we focus on nonlocal Neumann problems with measurable, nonnegative kernels, whose solutions require less regularity assumptions. For kernels of this kind we formulate and study the weak formulation of the nonlocal Neumann problem and we investigate a nonlocal counterpart of the Sobolev space $H^{1}$ as well as a resulting nonlocal trace space. We further establish, mainly for symmetric kernels, various existence results for the weak solution of the Neumann proble…
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $β$ is sufficiently large.
In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.
We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincaré-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
We prove trace and extension results for fractional Sobolev spaces of order $s\in(0,1)$. These spaces are used in the study of nonlocal Dirichlet and Neumann problems on bounded domains. The results are robust in the sense that the continuity of the trace and extension operators is uniform as $s$ approaches $1$ and our trace spaces converge to $H^{1/2}(\partial Ω)$. We apply these results in order to study the convergence of solutions of nonlocal Neumann problems as the integro-differential operators localize to a symmetric, second order operator in divergence form.
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside $Ω$, decreasing ener…
Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(Ω)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll} -a(\int_Ωu^{p}dx)Δu = f(u) & \mbox{in $Ω$,}\\ u=0 & \mbox{on $\partialΩ$,} \end{array}\right. \end{equation*} where $Ω$ is a bounded domain and $a$, $f$ are continuous real functions with $a$ vanishing in many positive points.
We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. Th…
We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function $u$ on $\mathbb R^n$ and of its perturbation $uφ_μ$, where $φ_μ$ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere $\mathbb S^{n-1}$, thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum p…
The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractional $g-$Laplacian $(-Δ_g)^s$ with different boundary conditions, namely, Dirichlet, Neumann and Robin.
Let $1<p<+\infty$ and let $Ω\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -Δ_p u = f(u), \quad u>0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular…
Mixed local-nonlocal eigenvalue problems in the Heisenberg group: spectral theory and singular multiplicity
Indian Institute of Science Education and Research Berhampur（印度科学教育研究学院伯赫拉普尔分校）；AGH University of Kraków（克拉科夫AGH科技大学）；Brno University of Technology（布尔诺理工大学）
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is a bounded set with smooth boundary, $(-Δ)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus Ω…
We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, tha…
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-Δ)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a contin…
In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):\;\quad (-\De)^s u = u^{-q} + \la u^{{2^*_s}-1}, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2s,\; s \in (0,1),\; \la >0,\; q>0$ satisfies $q(2s-1)<(2s+1)$ and $2^*_s=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(P_\la)$ in $X_0$ when $\la \in (0,\La)$ …
In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $λ>0$, $0<γ<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that f…
In this paper we study the existence and multiplicity of two distinct nontrivial weak solutions of the following equation in Nehari manifold. We have also proved that these solutions are in $L^{\infty}(Ω)$. \begin{align*} \begin{split} -Δ_{p(x,y)}^{s(x,y)}u &= β|u|^{α(x)-2}u+λf(x,u)\,\,\mbox{in}\,\,Ω,\\ u &= 0\,\, \mbox{in}\,\, \mathbb{R}^{N}\setminusΩ\end{split} \end{align*} Here, $λ, β> 0$ are parameters and $f(x,u)$ is a general nonlinear term satisfying certain conditions. The domain $Ω\subset\mathbb{R}^N (N\geq 2)$ is smooth and bounded. The relation between the exponents are assumed in t…
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_λ) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscr{P}_λ) and we show that the problem tr…
In this communication, we prove some important limits of the principal eigenvalue for nonlocal operator of Neumann type with respect to the parameters, which are significant in the understanding of dynamics of biological populations. We obtained a complete picture about limits of the principal eigenvalue in term of the large and small dispersal rate and dispersal range classified by "Ecological Stable Strategy" of persistence. This solves some open problems remainning in the series of work [3, 28, 29], in which we have to overcome the new difficulties comparing to [3, 28, 29] since principal e…
We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.
We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric d…
We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left\{\begin{aligned} (-Δ)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, u &\in H^{s}{(\mathbb{R}^{N})} \end{aligned} \right. \end{equation} where $s \in (0,1)$, $N>2s$, $1<p<2_s^*-1:=\frac{N+2s}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^N)$ and $f\in H^{-s}(\mathbb{R}^N)$ is a nonnegative functional i.e., ${\langle}f,u{\rangle} \geq 0$ whenever $u$ is a nonnegative function in $H^s(\mathbb{R}^N)$. We prove existence of a positive…
We propose here a motivation for a mixed local/nonlocal problem with a new type of Neumann condition. Our description is based on formal expansions and approximations. In a nutshell, a biological species is supposed to diffuse either by a random walk or by a jump process, according to prescribed probabilities. If the process makes an individual exit the niche, it must come to the niche right away, by selecting the return point according to the underlying stochastic process. More precisely, if the random particle exits the domain, it is forced to immediately reenter the domain, and the new poin…
We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton-Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron's method we get the existence and uniquen…
