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 continuous function, $a(x)\geq 1$ in $\mathbb{R}^{N}$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{\dot{H}^s(\mathbb{R}^{N})'}$ is small enough (but $f\not\equiv 0$), we establish existence of at least two positive solutions to ($\mathcal E$).
We consider the existence and multiplicity of solutions for a class of quasi-linear Schrödinger equations which include the modified nonlinear Schrödinger equations. A new perturbation approach is used to treat the sub-cubic nonlinearity.
We consider the initial value problem for the semilinear plate equation with nonlocal nonlinearity. We prove the existence of global attractor and then establish the regularity and finite dimensionality of this attractor.
We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.
We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional $p$-Laplacian operator and singular nonlinearities.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable pseudo-gradient vector field of the corresponding energy functional.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
The aim of this paper is to provide the first application of Theorem 3 of [2] in a case where the dependence of the underlying equation from the real parameter is not of affine type.
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.
We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and a stationary ergodic nonlinearity. By employing and adapting the theory of stochastic homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
We study a class of elliptic problems with homogeneous Dirichlet boundary condition and a nonlinear reaction term $f$ which is nonlocal depending on the $L^{p}$-norm of the unknown function. The nonlinearity $f$ can make the problem degenerate since it may even have multiple singularities in the nonlocal variable. We use fixed point arguments for an appropriately defined solution map to produce multiplicity of classical positive solutions with ordered norms.
Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$Δu +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degre…
In this paper we give a multiplicity result for the following Chern-Simons-Schrödinger equation \[ -Δu+2q u \int_{|x|}^{\infty}\frac{u^{2}(s)}{s}h_u(s)ds +q u\frac{h^{2}_u(|x|)}{|x|^{2}} = g(u), \quad\hbox{in }\mathbb{R}^2, \] where $\displaystyle h_u(s)=\int_0^s τu^2(τ) \ d τ$, under very general assumptions on the nonlinearity $g$. In particular, for every $n\in \mathbb N$, we prove the existence of (at least) $n$ distinct solutions, for every $q\in (0,q_{n})$, for a suitable $q_n$.
In this article, we consider nonlinear Schrödinger equation with nonlocal nonlinearity which is a generalized model of the Schrödinger-Poisson system (Schrödinger-Newton equations) in low dimensions. We first prove the global well-posedness in wider space than in previous result and show the stability of standing waves including excites states. It turns out that an example of stable excite states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.
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 study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.
The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -Δu+V(x)u- ϕ|u|^3u= f(u), & x\in\mathbb{R}^3, -Δϕ= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Pohožaev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary.
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.
In this paper, we study the long-time dynamics for the wave equation with nonlocal weak damping and sup-cubic nonlinearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah-Struwe solutions. Then we establish the existence of the global attractor for the Shatah-Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.
In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-Δ)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small.
In this research, we would like to study the global (in time) existence of small data solutions to the following damped $σ$-evolution equations with nonlocal (in space) nonlinearity: \begin{equation*} \partial_{t}^{2}u+(-Δ)^σu+\partial_{t}u+(-Δ)^σ\partial_{t}u=I_α(|u|^{p}), \ \ t>0, \ \ x\in \mathbb{R}^{n}, \end{equation*} where $σ\geq1$, $p>1$ and $I_α$ is the Riesz potential of power nonlinearity $|u|^{p}$ for any $α\in (0,n)$. More precisely, by using the $(L^{m}\cap L^{2})-L^{2}$ and $L^{2}-L^{2}$ linear estimates, where $m\in[1,2]$, we show the new influence of the parameter $α$ on the ad…
We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.
In this paper we are interested in propagation phenomena for nonlocal reaction-diffusion equations of the type: $δ_tu = J \times u - u + f (x, u) t \in R^+, x \in R^N$, where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.
We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Δ_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-Δ_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
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…
This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi-Dirac statistics as a singular perturbation of Maxwell-Boltzmann one, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allows us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by a some numerical computations. …
We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) Δu + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem.
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem \begin{equation*} \left\{ \begin{array}[c]{ll} -Δu - Δ(u^2)u = |u|^{p-2}u & \mbox{ in } Ωu= 0 &\mbox{ on } \partialΩ, \end{array} \right. \end{equation*} where $Ω$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More specifically we prove that, for $p$ near the critical exponent $22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $Ω$: the Ljusternick-Schnirelmann category and the Poincaré polynomial.
We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u, \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation \begin{equation*} (-Δ)^{\fracα{2}} u=λf(x)|u|^{q-2}u+|u|^{2^{*}_α-2}u, \quad\text{in}\,\,Ω, u=0,\text{on}\,\,\partialΩ, \end{equation*} where $Ω\subset \mathbb{R}^{N}(N\geq 2)$ is a bounded domain with smooth boundary, $0<α<2$, $(-Δ)^{\fracα{2}}$ stands for the fractional Laplacian operator, $f\in C(Ω\times\mathbb{R},\mathbb{R})$ may be sign changing and $λ$ is a positive parameter. We will prove that there exists $λ_{*}>0$ such that the problem has at least two positive solutions for each $λ…
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 $$ \mathcal{N}_{1/2}u(x) = \frac{1}π \int_{\R\setminus (a…
In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz conditio…
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）
We prove convergence of positive solutions to \[ u_t = uΔu + u\int_Ω |\nabla u|^2, \qquad u\rvert_{\partialΩ} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $Ω\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_Ωu_0=1$ and identify the $W_0^{1,2}(Ω)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_Ωu_0<1$ and $\int_Ωu_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_Ω |\nabla u…
For an open, bounded domain $\Om$ in $\mathbb{R}^N$ which is strictly convex with $C^2$ boundary, we show that there exists a $\land>0$ such that the singular quasilinear problem \begin{eqnarray*} &-\delp u =\cfracλ{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atleast two solution $ u$ and $v$ in $W^{1,p}_{loc}(\Om)\cap L^{\infty}(\Om)$ for any $\del>0$ and $0<\lam<\land$ provided $1<p<N$ and $p-1<q<\frac{p(N-1)}{N-p}-1$.\\ Moreover the solutions $u$ and $v$ are such that $u^{\alp}$ and $v^{\alp}$ are in $W^{1,p}_0(\…
In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence. We are also interested in the differences arising when the nonlinearity in the flux changes. Thus, we perform some numerics corresponding to different convexities for the nonlinearity in the equation.
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.
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 $\la \in (0,\La)$ …
