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Periodic nonlinear Schrödinger equation with application to photonic crystals
 Milan J. Math
, 2005
"... We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems. 0 ..."
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We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schrödinger equations. We also sketch an application to nonlinear optics and discuss some open problems. 0
Multiple Bound States of Nonlinear Schrödinger Systems
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2008
"... This paper is concerned with existence of bound states for Schrödinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of ..."
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This paper is concerned with existence of bound states for Schrödinger systems which have appeared as several models from mathematical physics. We establish multiplicity results of bound states for both small and large interactions. This is done by different approaches depending upon the sizes of the interaction parameters in the systems. For small interactions we give a new approach to deal with multiple bound states. The novelty of our approach lies in establishing a certain type of invariant sets of the associated gradient flows. For large interactions we use a minimax procedure to distinguish solutions by analyzing their Morse indices.
On Decay of Solutions to Nonlinear Schrödinger Equations
, 2006
"... We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. AMS Subject Classification (2000): 35J60, 35B40 In this note we consider the equation −∆u + V (x)u = f(x, u), x ∈ R n and, under rather general assumptions, derive exponential de ..."
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We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. AMS Subject Classification (2000): 35J60, 35B40 In this note we consider the equation −∆u + V (x)u = f(x, u), x ∈ R n and, under rather general assumptions, derive exponential decay estimates for its solutions. We suppose that (i) The potential V belongs to L ∞ loc (Rn) and is bounded below, i.e. V (x) ≥ −c0 for some c0 ∈ R. Under assumption (i) the left hand side of equation (1) defines a selfadjoint operator in L 2 (R n) denoted by H. The operator H is bounded below. We suppose that (ii) The essential spectrum σess(H) of the operator H does not contain the point 0. Note, however, that 0 can be an eigenvalue of finite multiplicity. The nonlinearity f is supposed to satisfy the following assumption. (iii)The function f(x, u) is a Carathéodory function, i.e. it is Lebesgue measurable with respect to x ∈ R n for all u ∈ R and continuous with respect to u ∈ R for almost all x ∈ R n. Furthermore, f(x, u)  ≤ c(1 + u  p−1), x ∈ R n u ∈ R, (2) with c> 0 and 2 ≤ p < 2 ∗ , where
Nodal type bound states of Schrödinger equations via invariant set and minimax methods
, 2005
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Existence of nontrivial solutions for pLaplacian equations in
 R N , J. Math. Anal. Appl
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ftp ejde.math.txstate.edu (login: ftp) A DOUBLE EIGENVALUE PROBLEM FOR SCHRÖDINGER EQUATIONS INVOLVING SUBLINEAR NONLINEARITIES AT INFINITY
"... Abstract. We present some multiplicity results concerning parameterized Schrödinger type equations which involve nonlinearities with sublinear growth at infinity. Some stability properties of solutions with respect to the parameters are also established in an appropriate Sobolev space. 1. ..."
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Abstract. We present some multiplicity results concerning parameterized Schrödinger type equations which involve nonlinearities with sublinear growth at infinity. Some stability properties of solutions with respect to the parameters are also established in an appropriate Sobolev space. 1.
ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE AND SIGNCHANGING SOLUTIONS FOR pLAPLACE EQUATIONS WITH POTENTIALS IN R N
"... Abstract. We study the perturbed equation −ε p div(∇u  p−2 ∇u) + V (x)u  p−2 u = h(x, u) + K(x)u  p∗−2 N u, x ∈ R u(x) → 0 as x  → ∞. where 2 ≤ p < N, p ∗ = pN N−p, p < q < p ∗. Under proper conditions on V (x) and h(x, u), we obtain the existence and multiplicity of solutions. W ..."
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Abstract. We study the perturbed equation −ε p div(∇u  p−2 ∇u) + V (x)u  p−2 u = h(x, u) + K(x)u  p∗−2 N u, x ∈ R u(x) → 0 as x  → ∞. where 2 ≤ p < N, p ∗ = pN N−p, p < q < p ∗. Under proper conditions on V (x) and h(x, u), we obtain the existence and multiplicity of solutions. We also study the existence of solutions which change sign. In this article, we study the equation 1.
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"... We consider the Newtonian system −¨q + B(t)q = Wq(q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q0 having a nontrivial critical group (in the sense of Morse theory) multibump solutions (with 2 ≤ k ≤ ∞ bumps) can be constructed by gluing tran ..."
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We consider the Newtonian system −¨q + B(t)q = Wq(q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q0 having a nontrivial critical group (in the sense of Morse theory) multibump solutions (with 2 ≤ k ≤ ∞ bumps) can be constructed by gluing translates of q0. Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equation −∆u + V (x)u = g(x, u) in R N, where V, g are periodic in x1,..., xN, σ(− ∆ + V) ⊂ (0, ∞), and we show that similar results hold in this case as well. In particular, if g(x, u) = u  2 ∗ −2 u, N ≥ 4 and V changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated. 1 Introduction and statement of main result In this paper we will be concerned with the existence of multibump solutions for Newtonian systems of ordinary differential equations and for semilinear partial differential equations of Schrödinger type. Consider first the Newtonian system