Symmetric And Nonsymmetric Solutions For a Class of Quasilinear Schrödinger Equations

2008 ◽  
Vol 8 (2) ◽  
Author(s):  
Uberlandio B. Severo
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Quasilinear Elliptic Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Multiplicity Of Solutions ◽  
Principle Of Symmetric Criticality ◽  
Quasilinear Elliptic

AbstractWe use the mountain-pass theorem combined with the principle of symmetric criticality to establish multiplicity of solutions for the class of quasilinear elliptic equations-Δu + V(z)u - Δ(uwhere N ≥ 4, the potential V : ℝ

scholarly journals Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Qilin Xie ◽  
Huafeng Xiao
Keyword(s):  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
High Energy ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Nonlocal Term ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Symmetric Mountain Pass Theorem

AbstractIn the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

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 lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

2017 ◽  
Vol 15 (1) ◽  
pp. 578-586
Author(s):  
Peiluan Li ◽  
Youlin Shang
Keyword(s):  
Variational Methods ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Symmetric Mountain Pass Theorem ◽  
Fractional Schrödinger Equations ◽  
Existence Criteria

Abstract Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

Mountain pass type solutions for quasilinear elliptic equations

2000 ◽  
Vol 11 (1) ◽  
pp. 33-62 ◽  
Author(s):  
Ph. Cl�ment ◽  
M. Garc�a-Huidobro ◽  
R. Man�sevich ◽  
K. Schmitt
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass ◽  
Quasilinear Elliptic Equations ◽  
Quasilinear Elliptic
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