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Mohsen Timoumi 대한수학회 2020 대한수학회지 Vol.57 No.4
We obtain infinitely many solutions for a class of fractional Schr\"odinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.
Timoumi, Mohsen Korean Mathematical Society 2020 대한수학회지 Vol.57 No.4
We obtain infinitely many solutions for a class of fractional Schrödinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.
Infinitely many homoclinic solutions for different classes of fourth-order differential equations
Mohsen Timoumi 대한수학회 2022 대한수학회논문집 Vol.37 No.1
In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation $$u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in\mathbb{R} \leqno(1)$$ where $a(x)$ is not required to be either positive or coercive, and $F(x,u)=\int^{u}_{0}f(x,v)dv$ is of subquadratic or superquadratic growth as $\left|u\right|\rightarrow\infty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|u|\rightarrow\infty$). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.
Infinitely many homoclinic solutions for damped vibration systems with locally defined potentials
Wafa Selmi,Mohsen Timoumi 대한수학회 2022 대한수학회논문집 Vol.37 No.3
In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R}, \leqno(1)$$ where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric and positive definite matix-valued function and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that $(1)$ possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.