共查询到20条相似文献,搜索用时 31 毫秒
1.
Raúl Ferreira Mayte Pérez-Llanos 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(2):14
The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases: We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:where
相似文献
$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$
- (i)\({f=f(x).}\)
- (ii)\({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$
$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$
2.
In this article we study the problem where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.
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$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$
3.
We study the solution \({u(r,\rho)}\) of the quasilinear elliptic problemThe usual Laplace, \({m}\)-Laplace, and \({k}\)-Hessian operators are included in the differential operator \({r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'}\). Under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and \({p}\), the equation has a singular positive solution \({u^*(r)}\) and the solution \({u(r,\rho)}\) is positive for \({r\ge 0}\). We study the intersection numbers between \({u(r,\rho)}\) and \({u^*(r)}\) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\). A generalized Joseph–Lundgren exponent \({p^*_{JL}}\) plays a crucial role. The main technique is a phase plane analysis. In particular, we use two changes of variables which transform the equation into two autonomous systems.
相似文献
$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'+|u|^{p-1}u=0, & 0 < r < \infty, \\u(0)=\rho > 0,\ u'(0)=0.\end{cases}$$
4.
G. Cupini Bernard Dacorogna O. Kneuss 《Calculus of Variations and Partial Differential Equations》2009,36(2):251-283
We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfyingwhere k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfieswith no sign hypothesis on f.
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$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$
$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$
5.
Jun-ichi Segata Keishu Watanabe 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(5):51
We consider the generalized Korteweg-de Vries (gKdV) equation with the time oscillating nonlinearity:Under the suitable assumption on g, we show that if the nonlinear term is mass critical or supercritical i.e., \({p \geq 5}\) and \({u(0) \in \dot{H}^{s_{p}}}\), where \({s_{p} = 1/2 - 2/(p-1)}\) is a scale critical exponent, then there exists a unique global solution to (gKdV) provided that \({|\omega|}\) is sufficiently large. We also obtain the behavior of the solution to (gKdV) as \({|\omega| \to \infty}\).
相似文献
$${\partial}_t u+{\partial}_x^3 u+ g(\omega t) {\partial}_x (|u|^{p-1}u)= 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}.$$
6.
7.
Daniel M. Elton 《Annales Henri Poincare》2016,17(10):2951-2973
We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on \({\mathbb{R}^3}\). In particular, we are interested in those operators \({\mathcal{D}_B}\) for which the associated magnetic field \({B}\) is given by pulling back a two-form \({\beta}\) from the sphere \({\mathbb{S}^2}\) to \({\mathbb{R}^3}\) using a combination of the Hopf fibration and inverse stereographic projection. If \({\int_{\mathbb{s}^2} \beta \neq 0}\), we show that as \({T\to+\infty}\). The result relies on Erd?s and Solovej’s characterisation of the spectrum of \({\mathcal{D}_{tB}}\) in terms of a family of Dirac operators on \({\mathbb{S}^2}\), together with information about the strong field localisation of the Aharonov–Casher zero modes of the latter.
相似文献
$$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big| \int_{\mathbb{S}^2}\beta\Big|\,\int_{\mathbb{S}^2}|{\beta}| +o(T^2)$$
8.
Francesco Della Pietra Gianpaolo Piscitelli 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(6):62
Given \(1\le q \le 2\) and \(\alpha \in \mathbb {R}\), we study the properties of the solutions of the minimum problem In particular, depending on \(\alpha \) and q, we show that the minimizers have constant sign up to a critical value of \(\alpha =\alpha _{q}\), and when \(\alpha >\alpha _{q}\) the minimizers are odd.
相似文献
$$\begin{aligned} \lambda (\alpha ,q)=\min \left\{ \dfrac{\displaystyle \int _{-1}^{1}|u'|^{2}dx+\alpha \left| \int _{-1}^{1}|u|^{q-1}u\, dx\right| ^{\frac{2}{q}}}{\displaystyle \int _{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not \equiv 0\right\} . \end{aligned}$$
9.
Giovany M. Figueiredo Gaetano Siciliano 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(2):12
In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\) where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).
相似文献
$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$
10.
We study the positive solution \({u(r,\rho)}\) of the quasilinear elliptic equationThis class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u *(r) under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and p. A generalized Joseph–Lundgren exponent, which we denote by \({p^*_{JL}}\), is obtained. We study the intersection numbers between \({u(r,\rho)}\) and u *(r) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\), and see that \({p^*_{JL}}\) plays an important role. We also determine the bifurcation diagram of the problemThe main technique used in the proofs is a phase plane analysis.
相似文献
$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime}+|u|^{p-1}u=0, & 0 < r < \infty,\\ u(0) = \rho > 0,\ u^{\prime}(0)=0.\end{cases}$$
$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime} + \lambda(u+1)^p=0, & 0 < r < 1,\\ u(r) > 0, & 0 \le r < 1,\\ u^{\prime}(0)=0,\ u(1)=0.\end{cases}$$
11.
We show that a realization of the operator \({L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}}\) generates a semigroup in \({L^p(\mathbb{R}^N)}\) if and only if \({D_c=b+(N-2+c)^2/4 > 0}\) and \({s_1+\min\{0,2-\alpha\} < N/p < s_2+\max\{0,2-\alpha\}}\), where \({s_i}\) are the roots of the equation \({b+s(N-2+c-s)=0}\), or \({D_c=0}\) and \({s_0+\min\{0,2-\alpha\} < N/p < s_0+\max\{0,2-\alpha\}}\), where \({s_0}\) is the unique root of the above equation. The domain of the generator is also characterized. 相似文献
12.
MOUSOMI BHAKTA 《Proceedings Mathematical Sciences》2017,127(2):337-347
We study the existence and multiplicity of sign-changing solutions of the following equation where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).
相似文献
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$
13.
Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence
Marino Badiale Michela Guida Sergio Rolando 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(6):67
We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equationwhere \({\Omega}\) is a radial domain (bounded or unbounded) and u satisfies u = 0 on \({\partial\Omega}\) if \({\Omega \neq\mathbb{R}^{N}}\) and \({u \rightarrow 0}\) as \({\left|x\right| \rightarrow \infty}\) if \({\Omega}\) is unbounded. The potential V may be vanishing or unbounded at zero or at infinity and the nonlinearity g may be superlinear or sublinear. If g is sublinear, the case with a forcing term \({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with V and g exhibiting behaviours at zero or at infinity which are new in the literature and, when \({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.
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$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$
14.
The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution towhere \({0\in \Omega \subseteq \mathbb{H}^4}\) is a bounded domain, \(0 \leq a \leq Q,\,(Q=10).\) The special feature of this problem is that it contains an exponential nonlinearity and singular potential.
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$$\Delta_{\mathbb{H}^n}^2 u=\frac{f(\xi,u)}{\rho(\xi)^a}\,\,\text{ in}\Omega,\,\, u|_{\partial\Omega}=0=\left.\frac{\partial u}{\partial\nu}\right|_{\partial\Omega},$$
15.
In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).
相似文献
$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$
16.
In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.
相似文献
$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$
17.
We consider the problem where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\ge 3,\) \(V_{\infty }>0,\) \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),\) \(\inf _{\mathbb {R}^{N}}V>-V_{\infty }\) and \(2<p<\frac{2N}{N-2}\). We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega \) is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that \(\Omega \) and V are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)
相似文献
$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$
18.
For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\). 相似文献
19.
Yanheng Ding Fanghua Lin 《Calculus of Variations and Partial Differential Equations》2007,30(2):231-249
We consider the perturbed Schrödinger equationwhere \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .
相似文献
$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$
20.
Luiz G. Farah 《Journal of Evolution Equations》2016,16(1):193-208
We consider the supercritical inhomogeneous nonlinear Schrödinger equationwhere \({(2 - b)/N < \sigma < (2 - b)/(N-2)}\) and \({0 < b < \rm min\{2,N\}}\). We prove a Gagliardo–Nirenberg-type estimate and use it to establish sufficient conditions for global existence and blow-up in \({H^1(\mathbb{R}^N)}\).
相似文献
$$i\partial_t u+\Delta u+|x|^{-b}|u|^{2\sigma}u=0,$$