scholarly journals Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2465-2481 ◽  
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Bitao Cheng ◽  
Huxiao Luo
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Compact Embedding ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Unbounded Potential ◽  
Superquadratic Growth

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations (-?u + V(x)u = K(x)f(u), x ? RN, u ? D1,2(RN), where N ? 3, V(x) and K(x) are both unbounded potential functions and f is a function with a superquadratic growth. Firstly, we prove the existence of infinitely many solutions with compact embedding and by means of symmetric mountain pass theorem. Moreover, we prove the existence of nontrivial solutions without compact embedding in weighted Sobolev spaces and by means of mountain pass theorem. Our results extend and generalize some existing results.

Equations of p-Laplacian Type in Unbounded Domains

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
Pablo L. De Nápoli ◽  
M. Cristina Mariani
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Solutions To Equations

AbstractThis work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

scholarly journals Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 83 ◽  
Author(s):  
Mohammed Massar ◽  
EL Miloud Hssini ◽  
Najib Tsouli
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Saddle Point Theorem ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

2015 ◽  
Vol 4 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Fractional Hamiltonian Systems ◽  
Symmetric Mountain Pass Theorem

AbstractIn this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form ${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$, where ${\alpha \in (\frac{1}{2},1)}$, ${t\in \mathbb {R}}$, ${u\in \mathbb {R}^n}$, ${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in \mathbb {R}}$, ${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ${\nabla W(t,u)}$ is the gradient of ${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants ${0<\tau _1<\tau _2< \infty }$ such that ${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all ${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and ${W(t,u)}$ is of the form ${({a(t)}/({p+1}))|u|^{p+1}}$ such that ${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and ${0<p<1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

scholarly journals GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

2020 ◽  
Vol 6 (1) ◽  
pp. 30
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Mountain Pass Theorem ◽  
Type Equation ◽  
Robin Boundary Condition ◽  
Mountain Pass ◽  
Technical Approach ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Quasilinear Problems ◽  
Robin Boundary ◽  
Laplace Type

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$\left\{\begin{array}{lll}-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.\end{array}\right.$$ Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

scholarly journals Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1792
Author(s):  
Yun-Ho Kim
Keyword(s):  
Potential Function ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Laplacian Operator ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Continuous Potential ◽  
The Given ◽  
Laplacian Equations

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qiongfen Zhang
Keyword(s):  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Impulsive Effects ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Vibration Problems ◽  
Damped Vibration Problems

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

scholarly journals Multiple solutions for Grushin operator without odd nonlinearity

2019 ◽  
Vol 13 (07) ◽  
pp. 2050131 ◽  
Author(s):  
Mohamed Karim Hamdani
Keyword(s):  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Grushin Operator ◽  
Clark’S Theorem ◽  
Superlinear Growth ◽  
Degenerate Operator ◽  
Strongly Degenerate ◽  
Odd Nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

scholarly journals Nontrivial Solutions of the Kirchhoff-Type Fractional p-Laplacian Dirichlet Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Taiyong Chen ◽  
Wenbin Liu ◽  
Hua Jin
Keyword(s):  
Dirichlet Problem ◽  
Fractional Derivative ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Fractional Derivative Operators

In this article, we consider the new results for the Kirchhoff-type p-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.

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