Cyril Joel Batkam

The aim of this note is to investigate the existence of signed and sign-changing solutions to the Kirchhoff type problem {-(a+b∫Ω|∇u|²Δu=∫Ω1/p|u|p)2/p|u|p-2uinΩu=0on∂Ω, where Ω is a bounded smooth domain in RN(N = 1,2,3), a,b > 0 and 2 < p < 2∗, with 2∗=+∞ if N = 1,2 and 2∗=6 if N = 3. Using variational methods, we show that (0.1) possesses three s...

In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} -\Delta \phi= u^{2} &\text{in }\Omega \hbox{} u=\phi=0&\text{on }\partial\Omega, \hbox{} \end{array} \right. \end{equation*} where $\Omega$ is a...

This paper is concerned with the existence of sign-changing solutions to nonlocal Kirchhoff type problems of the form \begin{equation}\label{s}\tag{S} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\, \text{ in }\Omega,\quad\quad u=0 \text{ on }\partial\Omega, \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$) wi...

We prove the existence of infinitely many high energy sign-changing solutions for some classes of Schr\"{o}dinger-Poisson systems in bounded domains, with nonlinearities having subcritical or critical growth. Our approach is variational and relies on an application of a new sign-changing version of the symmetric mountain pass theorem.

We prove some multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. The nonlinear terms are not assumed to satisfy the Ambrosetti-Rabinowitz superquadratic condition. To establish the existence of soluti...

In this paper, we obtain a nonsmooth version of the infinite-dimensional Fountain Theorem established by Batkam and Colin (2013). No symmetry condition on the energy functional is needed in our formulation. As an application, we prove the existence of multiple solutions for the following class of elliptic system (S)Δu−u∈[f̲(x,u,v),f¯(x,u,v)]a.e inR...

We consider a class of elliptic systems leading to strongly indefinite functionals, with nonlinearities which involve a combination of concave and convex terms. Using variational methods, we prove the existence of infinitely many large and small energy solutions. Our approach relies on new critical point theorems which guarantee the existence of in...

We study the existence of multiple solutions of a strongly indefinite elliptic system involving the critical Sobolev exponent and concave-convex nonlinearities. By using a suitable version of the dual fountain theorem established in this paper, we prove the existence of infinitely many small energy solutions. MSC: 35A02, 35J50, 35J60.

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a general $4-$superlinear condition on the nonlinearity $f$, we prove the existence of a ground state solution; that...

By using the degree theory and the ττ-topology of Kryszewski and Szulkin, we establish a version of the Fountain Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear Schrödinger equation.

This paper is concerned with the semilinear Schrödinger equation (S)−Δu+V(x)u=f(x,u),u∈H1(RN), where VV and ff are periodic in the xx-variables, ff is a superlinear and subcritical nonlinearity, and 0 lies in a spectral gap of −Δu+V−Δu+V. It is shown that, if ff is odd in uu then (S) has infinitely many large energy solutions. The proof relies on a...

In this paper, we consider the elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u=g(x,v)\,\, \textnormal{in}\Omega, & \hbox{} -\Delta v=f(x,u)\,\,\textnormal{in}\Omega, & \hbox{} u=v=0\textnormal{on}\partial\Omega, & \hbox{} \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, and $f$ an...

In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system {equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. {equation*} Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely m...

This paper is concerned with the following system of elliptic equations {equation*} \{{array}{ll} -\Delta u+u= F_u(|x|,u,v), & \hbox{} -\Delta v+v=- F_v(|x|,u,v), & \hbox{} \,\,\,\,\,u,v\in H^1(\mathbb{R}^N). & \hbox{} {array}. {equation*} It is shown that if $F$ is odd in $(u,v)$ and satisfy some growth conditions, then $(\mathcal{S})$ has infinit...

In this paper, we study the existence of a ground state solution, that is, a non trivial solution with least energy, of a noncooperative semilinear elliptic system on a bounded domain. By using the method of the generalized Nehari manifold developed recently by Szulkin and Weth, we prove the existence of a ground state solution when the nonlinearit...

We prove a global in time existence theorem, for the initial valued problem for the Einstein equations, in the case of strictly positive cosmological constant.

Global existence is proved in the case of a positive cosmological constant in the Einstein equations and asymptotic behavior is investigated.