Solitary Waves for a Class of Quasilinear Schrödinger Equations Involving Vanishing Potentials

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
João Marcos do Ó ◽  
Elisandra Gloss ◽  
Cláudia Santana
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Solitary Waves ◽  
Iteration Scheme ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonnegative Continuous Function ◽  
Vanishing Potentials

AbstractIn this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations−Δu + V(x)u − [Δ(uwhere h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.

Hamiltonian systems of Schrödinger equations with vanishing potentials

2020 ◽  
pp. 2050074
Author(s):  
E. Toon ◽  
P. Ubilla
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Positive Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Cerami Sequence ◽  
Cerami Sequences ◽  
Vanishing Potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

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 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.

scholarly journals Multiplicity and concentration results for magnetic relativistic Schrödinger equations

2019 ◽  
Vol 9 (1) ◽  
pp. 1161-1186 ◽  
Author(s):  
Aliang Xia
Keyword(s):  
Continuous Function ◽  
Schrödinger Equation ◽  
Variational Methods ◽  
Schrodinger Equation ◽  
Scalar Potential ◽  
Local Condition ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Penalization Techniques

Abstract In this paper, we consider the following magnetic pseudo-relativistic Schrödinger equation $$\begin{array}{} \displaystyle \sqrt{\left(\frac{\varepsilon}{i}\nabla-A(x)\right)^2+m^2}u+V(x)u= f(|u|)u \quad {\rm in}\,\,\mathbb{R}^N, \end{array}$$ where ε > 0 is a parameter, m > 0, N ≥ 1, V : ℝN → ℝ is a continuous scalar potential satisfies V(x) ≥ − V0 > − m for any x ∈ ℝN and f : ℝN → ℝ is a continuous function. Under a local condition imposed on the potential V, we discuss the number of nontrivial solutions with the topology of the set where the potential attains its minimum. We proof our results via variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

scholarly journals Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .

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