共查询到20条相似文献,搜索用时 46 毫秒
1.
Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity 下载免费PDF全文
Francisco Odair de Paiva Sandra Machado de Souza Lima & Olimpio Hiroshi Miyagaki 《分析论及其应用》2022,38(2):148-177
In this work we consider the following class of elliptic problems $\begin{cases} −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N), \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions. 相似文献
2.
对变分不等式的奇异摄动问题进行了探索,证明了解的重合集Iε={x∈Ωuε(x)=φ}在Hausdorff距离意义下收敛到ε=0时解的重合集。 相似文献
3.
This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially. 相似文献
4.
In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential. 相似文献
5.
On ground state of fractional $p$-Kirchhoff equation involving subcritical and critical exponential growth 下载免费PDF全文
In this paper, we concern the existence of nontrivial ground state solutions of
fractional $p$-Kirchhoff equation
$$\left\{\begin{array}{ll}
m\left(\|u\|^p\right) [(-\Delta)_p^su+V(x)|u|^{p-2}u]
=f(x,u) \quad\text{in}\, \mathbb{R}^N, \vspace{0.2
cm}\\ \|u\|=\left(\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy +\int_{\mathbb{R}^N}V(x)|u|^pdx\right)^{\frac{1}{p}},
\end{array}\right.$$
where $m:[0,+\infty)\rightarrow [0,+\infty)$ is a continuous function, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator with $0相似文献
6.
Zhaoli Liu Zhi-Qiang Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(4):609-629
We obtain the existence of infinitely many nodal solutions for the Schrödinger type equation on
with
Here,
The nonlinearity f is symmetric in the sense of being odd in u, and may involve a combination of concave and convex terms.Received: November 11, 2003; revised: December 12, 2004Supported by NSFC:10441003 相似文献
7.
We study the existence of weak solutions for a nonlinear elliptic system of Lane-Emden type $$\left\{\begin{array}{ll} -\Delta u \; = \; {\rm sgn}(v)|v|^{p-1} & {\rm in}\;\mathbb{R}^N, \\ -\Delta v \; = \; -\rho(x){\rm sgn}(u)|u|^{\frac{1}{p-1}} + f(x, u) & {\rm in}\;\mathbb{R}^N, \\ u, v \to 0 \quad {\rm as} \quad |x| \to +\infty, \end{array}\right.$$ by means of the Mountain Pass Theorem and some compact imbeddings in weighted Sobolev spaces. 相似文献
8.
Yongtao Jing & Zhaoli Liu 《数学研究》2015,48(3):290-305
Let $1
0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results. 相似文献
9.
We consider the following prescribed curvature problem for polyharmonic operator: $$\left\{\begin{array}{llll} D_{m} u = \tilde{K}(y)|u|^{m^*-2}u\; {\rm in}\; \mathbb{S}^N\\ u \quad\; >0\qquad\quad\quad\quad\quad{\rm on}\; \mathbb{S}^N\\ u \quad\; \in H^{m}(\mathbb{S}^N), \end{array} \right.$$ where ${m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}$ is positive and rationally symmetric, ${\mathbb{S}^N}$ is the unit sphere with the induced Riemannian metric ${g=g_{\mathbb{S}^N},}$ and D m is the elliptic differential operator of 2m order given by $$\begin{array}{lll}D_m={\prod\limits_{k=1}^m}{\left(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2)\right)}\end{array}$$ where Δ g is the Laplace-Beltrami operator on ${\mathbb{S}^N}$ . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large. 相似文献
10.
Let Ω IR^N, (N ≥ 2) be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- := inf pΩ(x) ≤ p+ = supp(x) Ω〈∞,
and f:Ω^-× IR be a C^1 function with f(x,s) ≥ 0, V (x,s) ∈Ω × R^+ and sup ∈Ωf(x,s) ≤ C(1+s)^q(x), Vs∈IR^+,Vx∈Ω for some 0〈q(x) ∈C(Ω^-) satisfying 1 〈p(x) 〈q(x) ≤p^* (x) -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^+ ≤ q- ≤ q+. As usual, p* (x) = Np(x)/N-p(x) if p(x) 〈 N and p^* (x) = ∞- if p(x) if p(x) 〉 N. Consider the functional I: W0^1,p(x) (Ω) →IR defined as
I(u) def= ∫Ω1/p(x)|△|^p(x)dx-∫ΩF(x,u^+)dx,Vu∈W0^1,p(x)(Ω),
where F (x, u) = ∫0^s f (x,s) ds. Theorem 1.1 proves that if u0 ∈ C^1 (Ω^-) is a local minimum of I in the C1 (Ω^-) ∩C0 (Ω^-)) topology, then it is also a local minimum in W0^1,p(x) (Ω)) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below. 相似文献
1 〈 p^- := inf pΩ(x) ≤ p+ = supp(x) Ω〈∞,
and f:Ω^-× IR be a C^1 function with f(x,s) ≥ 0, V (x,s) ∈Ω × R^+ and sup ∈Ωf(x,s) ≤ C(1+s)^q(x), Vs∈IR^+,Vx∈Ω for some 0〈q(x) ∈C(Ω^-) satisfying 1 〈p(x) 〈q(x) ≤p^* (x) -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^+ ≤ q- ≤ q+. As usual, p* (x) = Np(x)/N-p(x) if p(x) 〈 N and p^* (x) = ∞- if p(x) if p(x) 〉 N. Consider the functional I: W0^1,p(x) (Ω) →IR defined as
I(u) def= ∫Ω1/p(x)|△|^p(x)dx-∫ΩF(x,u^+)dx,Vu∈W0^1,p(x)(Ω),
where F (x, u) = ∫0^s f (x,s) ds. Theorem 1.1 proves that if u0 ∈ C^1 (Ω^-) is a local minimum of I in the C1 (Ω^-) ∩C0 (Ω^-)) topology, then it is also a local minimum in W0^1,p(x) (Ω)) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below. 相似文献
11.
On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$. 相似文献
12.
Haiyang He 《偏微分方程(英文版)》2015,28(2):120-127
In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$. 相似文献
13.
Pedro M. Santos 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(3):327-353
An integral representation for the functional
is obtained. This problem is motivated by equilibria issues in micromagnetics.
相似文献
14.
Regularity Results for Nonlinear Systems of Partial Differential Equations Under Weak Ellipticity Conditions 下载免费PDF全文
Yuesheng Zeng 《偏微分方程(英文版)》2000,13(3):217-225
We prove C^{1,α} almost everywhere regularity for weak solutions in the space W^{1,k} (Ω, R^N) of the systems - D_αA^i_α(x,u,Du) = B^i(z,u,Du) under the weak ellipticity condition ∫A(x_0 ,u, p + DΦ) ⋅ DΦdy ≥ λ ∫ (|DΦ|² + |DΦ|^k)dy. 相似文献
15.
16.
Thomas Bartsch Shuangjie Peng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,16(6):778-804
We study the radially symmetric Schr?dinger equation
$ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), $ - \varepsilon ^{2} \Delta u + V{\left( {|x|} \right)}u = W{\left( {|x|} \right)}u^{p} ,\quad u > 0,\;\;u \in H^{1} ({\mathbb{R}}^{N} ), 相似文献
17.
Fujita exponents for evolution problems with nonlocal diffusion 总被引:1,自引:0,他引:1
We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem
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