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.

scholarly journals Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Potential Term ◽  
Indefinite Potential

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When $$q=2$$ q = 2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.

Arbitrarily small nodal solutions for nonhomogeneous Robin problems

2021 ◽  
pp. 1-15
Author(s):  
Shengda Zeng ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Variational Tools

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).

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 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.

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.

scholarly journals Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Calogero Vetro
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Robin Problem ◽  
Admissible Range ◽  
Indefinite Potential ◽  
Multiplicity Of Positive Solutions

AbstractWe consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation $$f(z,\cdot )$$ f ( z , · ) is $$(p-1)$$ ( p - 1 ) -sublinear and then the case where it is $$(p-1)$$ ( p - 1 ) -superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter $$\lambda \in {\mathbb {R}}$$ λ ∈ R which we specify exactly in terms of principal eigenvalue of the differential operator.

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.

scholarly journals Asymmetric Robin Problems with Indefinite Potential and Concave Terms

2019 ◽  
Vol 19 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Variational Methods ◽  
Morse Theory ◽  
Critical Groups ◽  
Robin Problem ◽  
Perturbation Techniques ◽  
Smooth Solutions ◽  
Asymptotically Linear ◽  
Unbounded Potential ◽  
Indefinite Potential ◽  
Concave Term

Abstract We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing four and five, respectively, nontrivial smooth solutions when the parameter {\lambda>0} is small.

Sign in / Sign up

Export Citation Format

Share Document