共查询到20条相似文献,搜索用时 15 毫秒
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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
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Rong Cheng 《Applicable analysis》2017,96(5):778-798
In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity. 相似文献
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We study the following nonlinear Schrödinger equations $$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$ for ${w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}$ , where g(|w|)w is super linear and subcritical, 2* = 2N/(N ? 2) if N > 2 and = ∞ if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V < τ 0 for some τ0 > 0, or ${\max W > \kappa_{0}}$ for some ${\kappa_0 > 0}$ . 相似文献
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We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L 2-unit sphere, and we show the existence of infinitely many solutions. 相似文献
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Nakao Hayashi 《manuscripta mathematica》1986,55(2):171-190
We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iut–u+q(|u|2)u=0 in iut - u + (|u|2)u = in (t, x)xn for 6n11. 相似文献
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We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions. 相似文献
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We consider the quasilinear Schrödinger equations of the form ?ε2Δu + V(x)u ? ε2Δ(u2)u = g(u), x∈ RN, where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C1(R) is an odd function with subcritical growth and V(x) is a positive Hölder continuous function which is bounded from below, away from zero, and infΛV(x) < inf?ΛV(x) for some open bounded subset Λ of RN. We prove that there is an ε0 > 0 such that for all ε ∈ (0, ε0], the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε → 0+. 相似文献
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《Nonlinear Analysis: Theory, Methods & Applications》2005,61(5):839-855
In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equationon . We present the -concentration property for general initial data and investigate the -minimality. 相似文献
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Peter Y. H. Pang Hongyan Tang Youde Wang 《Calculus of Variations and Partial Differential Equations》2006,26(2):137-169
In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2
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In this paper, we study the following quasilinear Schrödinger equation where , , is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition , we only need to assume that . 相似文献
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We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work. 相似文献
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In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved. 相似文献
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Yanheng Ding Cheng Lee Fukun Zhao 《Calculus of Variations and Partial Differential Equations》2014,51(3-4):725-760
This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) . 相似文献
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