Multiple bound states of higher topological type for semi-classical Choquard equations

2020 ◽  
pp. 1-27
Author(s):  
Xiaonan Liu ◽  
Shiwang Ma ◽  
Jiankang Xia
Keyword(s):  
Bound States ◽  
Local Minimum ◽  
Mountain Pass Theorem ◽  
Riesz Potential ◽  
Topological Type ◽  
Mountain Pass ◽  
Choquard Equation ◽  
Symmetric Mountain Pass Theorem ◽  
Image Position ◽  
Minimum Points

Abstract We are concerned with the semi-classical states for the Choquard equation $$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$ where N ⩾ 2, I α is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < (N + α)/(N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.

Multiplicity of solutions to the weighted critical quasilinear problems

2012 ◽  
Vol 55 (1) ◽  
pp. 181-195 ◽  
Author(s):  
Sihua Liang ◽  
Jihui Zhang
Keyword(s):  
Bounded Domain ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Positive Parameter ◽  
Symmetric Mountain Pass Theorem ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Quasilinear Problems ◽  
Image Position

AbstractWe consider a class of critical quasilinear problemswhere 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain and 1 < p < N, a < N/p, a ≤ b < a + 1, λ is a positive parameter, 0 ≤ μ < $\bar{\mu}$ ≡ ((N − p)/p − a)p, q = q*(a, b) ≡ Np/[N − pd] and d ≡ a+1 − b. Infinitely many small solutions are obtained by using a version of the symmetric Mountain Pass Theorem and a variant of the concentration-compactness principle. We deal with a problem that extends some results involving singularities not only in the nonlinearities but also in the operator.

The existence of a positive solution to asymptotically linear scalar field equations

2000 ◽  
Vol 130 (1) ◽  
pp. 81-105 ◽  
Author(s):  
G. Li ◽  
H.-S. Zhou
Keyword(s):  
Positive Solution ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Field Equations ◽  
Usual Condition ◽  
Image Position ◽  
Scalar Field Equations ◽  
Constraint Method ◽  
Linear Scalar

We consider the following elliptic equation: where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

2018 ◽  
Vol 61 (4) ◽  
pp. 943-959 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Reaction Function ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Potential Term ◽  
Symmetric Mountain Pass Theorem ◽  
Indefinite Potential ◽  
Concave Term

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

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 ${0&lt;\tau _1&lt;\tau _2&lt; \infty }$ 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 ${0&lt;p&lt;1}$ 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 of solutions for critical Choquard equations via the concentration-compactness method

2019 ◽  
Vol 150 (2) ◽  
pp. 921-954 ◽  
Author(s):  
Fashun Gao ◽  
Edcarlos D. da Silva ◽  
Minbo Yang ◽  
Jiazheng Zhou
Keyword(s):  
Sobolev Inequality ◽  
Existence Of Solutions ◽  
High Energy ◽  
Mountain Pass ◽  
Concentration Compactness ◽  
Choquard Equation ◽  
Concentration Compactness Principle ◽  
Nonlinear Choquard Equation ◽  
Compactness Principle ◽  
Image Position

AbstractIn this paper, we consider the nonlinear Choquard equation $$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$ where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.

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.

Infinitely Many Solutions for Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations

2017 ◽  
Vol 60 (4) ◽  
pp. 1003-1020 ◽  
Author(s):  
Hongxue Song ◽  
Caisheng Chen
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Biharmonic Operator ◽  
Kirchhoff Type ◽  
Symmetric Mountain Pass Theorem ◽  
Image Position ◽  
Fourth Order Elliptic Equations ◽  
Biharmonic Problems

AbstractThis paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problemswhere Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

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