共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper we establish existence of radial and nonradial solutions to the system where \(F_1,F_2\) are nonlinearities with critical behavior.
相似文献
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_1 = F_1(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ -\Delta u_2 = F_2(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0 &{}\quad \text {in }{\mathbb R}^N,\\ u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$
2.
We prove the existence of infinitely many solutions for where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).
相似文献
$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$
3.
Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator
Using variational methods, we establish existence of multi-bump solutions for the following class of problems where \(N \ge 1\), \(\Delta ^2\) is the biharmonic operator, f is a continuous function with subcritical growth, \(V : \mathbb {R}^N \rightarrow \mathbb {R}\) is a continuous function verifying some conditions and \(\lambda >0\) is a real constant large enough.
相似文献
$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N},\\ u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$
4.
The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion \(2N/(N+1)<p<2\). Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as \(|x|\rightarrow \infty \), it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form as \(t\rightarrow T_e\), where \(T_e\) denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile \(f_*\) is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional.
相似文献
$$\begin{aligned} \partial _tu-\Delta _p u+|\nabla u|^{p-1}=0 \quad \hbox {in}~ (0,\infty )\times \mathbb {R}^N, \end{aligned}$$
$$\begin{aligned} U(t,x)=(T_e-t)^{1/(2-p)}f_*(|x|), \quad (t,x)\in (0,T_e)\times \mathbb {R}^N, \end{aligned}$$
5.
Silvia Cingolani Louis Jeanjean Kazunaga Tanaka 《Journal of Fixed Point Theory and Applications》2017,19(1):37-66
We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).
相似文献
$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$
(0.1)
$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$
(0.2)
$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$
6.
7.
In this paper, the prescribed \(\sigma \)-curvature problem is considered. When \({\tilde{K}}(x)\) is some axis symmetric function on \({\mathbb {S}}^N\), by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for \(0<\sigma \le 1\) and \(N> 2\sigma +2\).
相似文献
$$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$
8.
In this paper we consider classical solutions u of the semilinear fractional problem \((-\Delta )^s u = f(u)\) in \({\mathbb {R}}^N_+\) with \(u=0\) in \({\mathbb {R}}^N {\setminus } {\mathbb {R}}^N_+\), where \((-\Delta )^s\), \(0<s<1\), stands for the fractional laplacian, \(N\ge 2\), \({\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N{:}\ x_N>0\}\) is the half-space and \(f\in C^1\) is a given function. With no additional restriction on the function f, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in \({\mathbb {R}}^N_+\) and verify This is in contrast with previously known results for the local case \(s=1\), where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when \(f(0)<0\).
相似文献
$$\begin{aligned} \frac{\partial u}{\partial x_N}>0 \quad \hbox {in } {\mathbb {R}}^N_+. \end{aligned}$$
9.
Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: For \(N\ge 4\) and \(\sigma \in (\frac{1}{2}, 1),\) we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$
(0.1)
10.
Andrea Davini Albert Fathi Renato Iturriaga Maxime Zavidovique 《Mathematische Zeitschrift》2016,284(3-4):1021-1034
We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, \(c : M\times M \rightarrow \mathbb {R}\) a continuous cost function and \(\lambda \in (0,1)\), the unique solution to the discrete \(\lambda \)-discounted equation is the only function \(u_\lambda : M\rightarrow \mathbb {R}\) such that We prove that there exists a unique constant \(\alpha \in \mathbb {R}\) such that the family of \(u_\lambda +\alpha /(1-\lambda )\) is bounded as \(\lambda \rightarrow 1\) and that for this \(\alpha \), the family uniformly converges to a function \(u_0 : M\rightarrow \mathbb {R}\) which then verifies The proofs make use of Discrete Weak KAM theory. We also characterize \(u_0\) in terms of Peierls barrier and projected Mather measures.
相似文献
$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$
$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$
11.
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)\({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$
$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$
12.
In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.
相似文献
$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$
(1)
13.
We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form where \(\Omega \) is a bounded Lipschitz domain, \(A\subset \mathbb {R}^N\) is a Borel set, \(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\), \(\mathscr {A}\) is an operator of gradient form, and \(\sigma _1, \sigma _2\) are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on f, we show for a solution (w, A), that the topological boundary of \(A \cap \Omega \) is locally a \(\mathrm {C}^1\)-hypersurface up to a closed set of zero \(\mathscr {H}^{N-1}\)-measure.
相似文献
$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x \; - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; + \; \text {Per }(A;\overline{\Omega }), \end{aligned}$$
14.
Existence and uniqueness of global strong solutions to fully nonlinear second order elliptic systems
We consider the problem of existence and uniqueness of strong a.e. solutions \({u: \mathbb{R}^n \longrightarrow \mathbb{R}^N}\) to the fully nonlinear PDE systemwhen \({ f\in L^2(\mathbb{R}^n)^N}\) and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1).
相似文献
$$\label{equ1}F(\cdot,D^2u ) = f, \quad \text{ a.e. on }\mathbb{R}^n,\quad \quad(1)$$
15.
In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem where \(\lambda >0\) and \(\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)\) uniformly on any compact subset of \([0,\infty )\). We prove that under suitable conditions on f and h, there exists \(0<\lambda ^*<+\infty \) such that the problem has at least two positive solutions if \(\lambda \in (0,\lambda ^*)\), a unique positive solution if \(\lambda =\lambda ^*\), and no solution if \(\lambda >\lambda ^*\). We also obtain the bifurcation of positive solutions for the problem at \((\lambda ^*,u^*)\) and further analyse the set of positive solutions.
相似文献
$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$
16.
In this paper we study perturbed Ornstein–Uhlenbeck operators for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by One key assumption is a new \(L^p\)-dissipativity condition for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.
相似文献
$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$
$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$
$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$
17.
Let \(\Omega := ( a,b ) \subset \mathbb {R}\), \(m\in L^{1} ( \Omega ) \) and \(\phi :\mathbb {R\rightarrow R}\) be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form where \(f: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is a continuous function which is, roughly speaking, superlinear with respect to \(\phi \). Our approach combines the Guo-Krasnoselski? fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case \(m\ge 0\).
相似文献
$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$
18.
In this paper, we are concerned with the following nonlocal problem where \(N\ge 3, \alpha \in ((N-4)_+,N), 2\le p<\frac{N+\alpha }{N-2}\) and q(x) is a given potential. Using comparison arguments and variational approach, we obtain the existence of positive ground-state solution for the Choquard-type equations with some restrictions on the potential q.
相似文献
$$\begin{aligned} -\Delta u+u=q(x)\left( \int _{\mathbb {R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha }}\mathrm{d}y\right) |u|^{p-2}u,\quad x\in \mathbb {R}^N, \end{aligned}$$
19.
In this paper, we study the existence of multiple and infinite homoclinic solutions for the following perturbed dynamical systems where \(t\in {\mathbb R}, x\in {\mathbb R}^N,\) A is an antisymmetric constant matrix, the matrix L(t) is not necessary positive definite for all \(t\in {\mathbb R}\) nor coercive, the nonlinearity \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing and \(f\in C({\mathbb R},{\mathbb R}^{N})\cap L^{2}({\mathbb R},{\mathbb R}^{N}).\) Recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.
相似文献
$$\begin{aligned} \ddot{x}+A\cdot \dot{x}-L(t)\cdot x+\nabla W(t,x)=f(t), \end{aligned}$$
20.
In this paper we discuss the existence and the global behavior of positive solutions of the following generalized Lane–Emden system of differential equations: where \(r,\,s\in {\mathbb {R}}\), \(\alpha ,\,\beta <1\) such that \(\gamma :=(1-\alpha )(1-\beta )-rs>0\) and the nonnegative functions \(a,\,b\) satisfy some conditions related to the Karamata regular variation theory.
相似文献
$$\begin{aligned} -u''= & {} a(x)u^{\alpha }\,v^{r}\quad \text{ in } (0,1), \\ -v''= & {} b(x)u^{s}\,v^{\beta }\quad \, \text{ in } (0,1), \\ u'(0)= & {} v'(0)=0; \quad \, u(1)=v(1)=0, \end{aligned}$$