Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

2010 ◽  
Vol 72 (6) ◽  
pp. 2935-2949 ◽  
Author(s):  
Elves A.B. Silva ◽  
Gilberto F. Vieira
Keyword(s):  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Subcritical Growth ◽  
Asymptotically Periodic

scholarly journals Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations

2021 ◽  
Vol 7 (1) ◽  
pp. 1015-1034
Author(s):  
Shulin Zhang ◽  
◽  
Keyword(s):  
Variational Method ◽  
Ground State ◽  
Periodic Potential ◽  
Ground State Solution ◽  
State Solution ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Ground State Solutions ◽  
Asymptotically Periodic

<abstract><p>In this paper, we study the existence of a positive ground state solution for a class of generalized quasilinear Schrödinger equations with asymptotically periodic potential. By the variational method, a positive ground state solution is obtained. Compared with the existing results, our results improve and generalize some existing related results.</p></abstract>

scholarly journals Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

2019 ◽  
Vol 9 (1) ◽  
pp. 1066-1091 ◽  
Author(s):  
Zhi Chen ◽  
Xianhua Tang ◽  
Jian Zhang
Keyword(s):  
Continuous Function ◽  
External Potential ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Potential Well ◽  
Subcritical Growth ◽  
Nonnegative Continuous Function ◽  
Concentration Behavior ◽  
Chern Simons ◽  
General Nonlinearity

Abstract In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2, \end{array}$$ where λ > 0, V is an external potential and $$\begin{array}{} \displaystyle h(s)=\frac{1}{2}\int\limits^s_0ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s}u^2(x)dx \end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

scholarly journals Schrödinger equations with asymptotically periodic terms

2015 ◽  
Vol 145 (4) ◽  
pp. 745-757 ◽  
Author(s):  
Reinaldo de Marchi
Keyword(s):  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Schrodinger Equations ◽  
Periodic Case ◽  
Local Version ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Concentration Compactness Principle ◽  
Asymptotically Periodic ◽  
Compactness Principle

We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.

A class of asymptotically periodic fractional Schrödinger equations with critical growth

2018 ◽  
Vol 20 (03) ◽  
pp. 1750011
Author(s):  
Manassés de Souza ◽  
Yane Lísley Araújo
Keyword(s):  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Positive Potential ◽  
Fractional Schrödinger Equations ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we study a class of fractional Schrödinger equations in [Formula: see text] of the form [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is the critical Sobolev exponent, [Formula: see text] is a positive potential bounded away from zero, and the nonlinearity [Formula: see text] behaves like [Formula: see text] at infinity for some [Formula: see text], and does not satisfy the usual Ambrosetti–Rabinowitz condition. We also assume that the potential [Formula: see text] and the nonlinearity [Formula: see text] are asymptotically periodic at infinity. We prove the existence of at least one solution [Formula: see text] by combining a version of the mountain-pass theorem and a result due to Lions for critical growth.

scholarly journals New Super-quadratic Conditions for Asymptotically Periodic Schrödinger Equations

2017 ◽  
Vol 60 (2) ◽  
pp. 422-435 ◽  
Author(s):  
Xianhua Tang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Asymptotically Periodic ◽  
Subcritical Nonlinearity ◽  
Semilinear Schrödinger Equation

AbstractWe study the semilinear Schrödinger equationwherefis a superlinear, subcritical nonlinearity. It focuses on the casewhereV(x) =V0(x)+V1(x),V0∊C(RN),V0(x) is 1-periodic in each of x1 , x2 , . . . , xN, supinf, and. A new super-quadratic condition is obtained that is weaker than some well-known results.

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