Infinitely many homoclinic solutions for perturbed second-order Hamiltonian systems with subquadratic potentials

In this paper, we consider the following perturbed second-order Hamiltonian system $$-\ddot{u}(t)+L(t)u=\nabla W(t,u(t))+\nabla G(t,u(t)), \qquad \forall \ t\in \mathbb{R},$$ where $W(t,u)$ is subquadratic near origin with respect to $u$; the perturbation term $G(t,u)$ is only locally defined near the origin and may not be even in $u$. By using the variant Rabinowitz's perturbation method, we establish a new criterion for guaranteeing that this perturbed second-order Hamiltonian system has infinitely many homoclinic solutions under broken symmetry situations. Our result improves some related results in the literature

Infinitely many solutions for some nonlinear supercritical problems with break of symmetry

In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $$\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}$$ where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $N\geq 3$, and $A(x,t,\xi)$, $g(x,t)$, $h(x)$ are given functions, with $A_t = \frac{\partial A}{\partial t}$, $a = \nabla_{\xi} A$, such that $A(x,\cdot,\cdot)$ is even and $g(x,\cdot)$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if $A(x,t,\xi)$ grows fast enough with respect to $t$, then the nonlinear term related to $g(x,t)$ may have also a supercritical growth

Infinitely many solutions for some nonlinear supercritical problems with break of symmetry

In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $$\left\{\begin{array}{ll}- {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = g(x,u) + h(x) &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,}\end{array}\right.$$ where $\Omega \subset \R^N$ is an open bounded domain, $N\ge 3$, and $A(x,t,\xi)$, $g(x,t)$, $h(x)$ are given functions, with $A_t = \frac{\partial A}{\partial t}$, $a = \nabla_\xi A$, such that $A(x,\cdot,\cdot)$ is even and $g(x,\cdot)$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami--Palais--Smale condition. Furthermore, if $A(x,t,\xi)$ grows fast enough with respect to $t$, then the nonlinear term related to $g(x,t)$ may have also a supercritical growth