共查询到20条相似文献,搜索用时 31 毫秒
1.
D. D. Hai 《Ricerche di matematica》2018,67(2):525-532
We prove the \(C^{1,\beta }\)-boundary regularity and a comparison principle for weak solutions of the problem where \(\Omega \) is a bounded domain in \(\mathbb {R}^{N},N>1\ \)with smooth boundary \(\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ \)and f is allowed to be unbounded.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$
2.
César E Torres Ledesma 《Proceedings Mathematical Sciences》2018,128(4):50
In this article, we consider the following fractional Hamiltonian systems: where \(\alpha \in (1/2, 1)\), \(\lambda >0\) is a parameter, \(L\in C(\mathbb {R}, \mathbb {R}^{n\times n})\) and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})\). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all \(t\in \mathbb {R}\), that is, \(L(t) \equiv 0\) is allowed to occur in some finite interval \(\mathbb {I}\) of \(\mathbb {R}\). Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on \(\mathbb {R} \setminus \mathbb {I}\) as \(\lambda \rightarrow \infty ,\) and converge to \(\tilde{u}\) in \(H^{\alpha }(\mathbb {R})\); here \(\tilde{u} \in E_{0}^{\alpha }\) is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval \(\mathbb {I}\). Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.
相似文献
$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$
3.
We are going to study the standing waves for the generalized Choquard equation with potential: where \(V(x)\) is a real function, \(p>2\) is close to 2 and ? standards for the convolution in \(\mathbb{R}^{3}\). The stability of the standing waves \(u(x)=e^{i\omega t}\varphi (x)\) is investigated under suitable assumptions on the potential and the frequency \(\omega \).
相似文献
$$ -i\partial_{t} u-\Delta u+V(x)u=\bigl(|x|^{-1}\ast |u|^{p}\bigr)|u|^{p-2}u,\quad \hbox{in } \mathbb{R}\times \mathbb{R}^{3}, $$
4.
Vincent Bruneau Konstantin Pankrashkin Nicolas Popoff 《Journal of Geometric Analysis》2018,28(1):123-151
We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.
相似文献
$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$
5.
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
$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$
6.
Alexei Yu. Karlovich Yuri I. Karlovich Amarino B. Lebre 《Complex Analysis and Operator Theory》2016,10(6):1101-1131
Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.
相似文献
$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$
$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$
7.
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
$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$
8.
In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$
9.
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
$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$
10.
The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems where \(\epsilon\) is a positive parameter, \(N\geq2\), \(V\), \(f\) are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\)-function.
相似文献
$$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon|\nabla u|\bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$
11.
We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem where \(\lambda \) is a large number, \(p\in (2,\frac{2N}{N-2})\) for \(N\ge 3\), f(x, u) is subcritical about u and Q is positive and has some non-degenerate critical points in \(\mathbb {R}^{N}_{+}\). For \(\lambda \) large, we can get solutions which have peaks near the non-degenerate critical points of Q.
相似文献
$$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\lambda ^{2} u&=Q(x)|u|^{p-2}u \qquad&\text {in} ~~~~\mathbb {R}^{N}_{+}, \\ \frac{\partial u }{\partial n}&=f(x,u) \qquad&\text {on}~~\partial \mathbb {R}^{N}_{+}, \end{aligned}\right. \end{aligned}$$
12.
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
$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$
13.
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
$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$
$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$
14.
We study both the existence and uniqueness of nonnegative solution to a singular elliptic problem of Kirchhoff type, whose model is: where \(\Omega \subset \mathbb {R}^n(n\ge 1)\) is a smooth bounded domain, \(\gamma >1\), \(h\in L^1(\Omega )\) is positive (i.e., \(h(x)>0\) a.e. in \(\Omega \)), \(B : \mathbb {R}^+\rightarrow \mathbb {R}^+\) is a \(C^1\)-continuous function with positive lower bound. A necessary and sufficient condition will be given for the existence of weak solution of the general nonlocal singular elliptic with strong singularity. In addition, we prove that the solution is unique under some suitable conditions.
相似文献
$$\begin{aligned} {\left\{ \begin{array}{ll} -B\left( \dfrac{1}{2}\displaystyle \int _\Omega |\nabla u|^2\mathrm {d}x\right) \Delta u=\dfrac{h(x)}{u^\gamma }, &{}\quad x\in \Omega ,\\ u>0, &{}\quad x\in \Omega ,\\ u=0, &{}\quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$
15.
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
$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x \; - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; + \; \text {Per }(A;\overline{\Omega }), \end{aligned}$$
16.
Monica Musso 《Journal of Fixed Point Theory and Applications》2017,19(1):345-361
We consider the equation \(-\Delta u = |u| ^{\frac{4}{n-2}}u + \varepsilon f(x) \) under zero Dirichlet boundary conditions in a bounded domain \(\Omega \) in \(\mathbb {R}^{n}\), \(n \ge 3\), with \(f\ge 0\), \(f\ne 0\). We find sign-changing solutions with large energy. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem recently constructed in del Pino et al. (J Differ Equ 251(9):2568–2597, 2011).
相似文献
$$\begin{aligned} -\Delta w = |w|^{\frac{4}{n-2}} w, \quad w \in {\mathcal D}^{1,2} (\mathbb {R}^n) \end{aligned}$$
17.
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
$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$
18.
We consider the following fractional \( p \& q\) Laplacian problem with critical Sobolev–Hardy exponents where \(0<s<1\), \(1\le q<p<\frac{N}{s}\), \((-\Delta )^{s}_{r}\), with \(r\in \{p,q\}\), is the fractional r-Laplacian operator, \(\lambda \) is a positive parameter, \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with smooth boundary, \(0\le \alpha <sp\), and \(p^{*}_{s}(\alpha )=\frac{p(N-\alpha )}{N-sp}\) is the so-called Hardy–Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya [23], we show the existence of infinitely many solutions which tend to be zero provided that \(\lambda \) belongs to a suitable range.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p} u + (-\Delta )^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha )-2}u}{|x|^{\alpha }}+ \lambda f(x, u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^{N}{\setminus } \Omega , \end{array} \right. \end{aligned}$$
19.
20.
In this paper, we study the following fractional Schrödinger–Poisson system where \(\varepsilon >0\) is a small parameter, \(\frac{3}{4}<s<1\), \(4<p<2_s^*:=\frac{6}{3-2s}\), \(V(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)\) has positive global minimum, and \(K(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)\) is positive and has global maximum. We prove the existence of a positive ground state solution by using variational methods for each \(\varepsilon >0\) sufficiently small, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as \(\varepsilon \rightarrow 0\). Moreover, we considered some properties of these ground state solutions, such as convergence and decay estimate.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon ^{2s}(-\Delta )^s u +V(x)u+\phi u=K(x)|u|^{p-2}u,\,\,\text {in}~\mathbb {R}^3,\\ \\ \varepsilon ^{2s}(-\Delta )^s \phi =u^2,\,\,\text {in}~\mathbb {R}^3, \end{array} \right. \end{aligned}$$
(0.1)