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We study bound states of the following nonlinear Schr?dinger equation in the presence of a magnetic field: $$ \left\{\begin{array}{l} \left(-i\hbar\nabla+A(x)\right)^2u+V(x)u=g(x,|u|)u \\ |u|\in H^1(\mathbb{R}^N) \end{array} \right. $$ where ${A: \mathbb{R}^N\to\mathbb{R}^N, V: \mathbb{R}^N\to\mathbb{R}}$ and ${g: \mathbb{R}^N\times\mathbb{R}\to [0,\infty)}$ . We prove that if V is bounded below with the set ${\{x\in\mathbb{R}^N: V(x) < b\}\not=\emptyset}$ having finite measure for some b?>?0, inf V???0, and g satisfies some growth conditions, then for any integer m when ${\hbar >0 }$ is sufficiently small the problem has m geometrically different solutions. 相似文献
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Hiroyuki Chihara 《Mathematische Annalen》1999,315(4):529-567
We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998 相似文献
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Thomas Bartsch Zhi-Qiang Wang Juncheng Wei 《Journal of Fixed Point Theory and Applications》2007,2(2):353-367
We consider the existence of bound states for the coupled elliptic system
where n ≤ 3. Using the fixed point index in cones we prove the existence of a five-dimensional continuum of solutions (λ1, λ2, μ
1, μ
2, β, u
1, u
2) bifurcating from the set of semipositive solutions (where u
1 = 0 or u
2 = 0) and investigate the parameter range covered by .
Dedicated to Albrecht Dold and Edward Fadell 相似文献
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Xiangqing Liu Jiaquan Liu Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2013,46(3-4):641-669
For a class of quasilinear Schrödinger equations with critical exponent we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems from our earlier work (Liu et al. Commun Partial Differ Equ 29:879–901, 2004) and to pass limit as the exponent approaches to the critical exponent. 相似文献
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In this paper, we consider the critical quasilinear Schr?dinger equations of the form
-e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u), x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N, 相似文献
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Alexander Pankov 《Applicable analysis》2013,92(2):308-317
We consider the discrete nonlinear Schrödinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave 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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We study a class of logarithmic Schrödinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions. 相似文献
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We study the bound states to nonlinear Schrödinger equations with electro-magnetic fields
. Let
and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x,t) = e?lEt/hUh(χ) with Uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity. 相似文献
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