Browsing by Subject "weak Cerami-Palais-Smale condition"

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Access status: Open Access ,
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
(Wydawnictwa AGH, 2019) Candela, Anna Maria; Salvatore, Addolorata
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 $AA_t = \frac{\partial A}{\partial t}$, $a = \nabla_{\xi} A$, 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.