TY  - JOUR
ID  - Candela2019
LB  - Candela2019
AU  - Candela, Anna Maria
AU  - Salvatore, Addolorata
TI  - Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
ST  - Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2019
VL  - 39
IS  - 2
SP  - 175
EP  - 194
DO  - https://doi.org/10.7494/OpMath.2019.39.2.175
UR  - https://doi.org/10.7494/OpMath.2019.39.2.175
AB  - 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.
KW  - quasilinear elliptic equation
KW  - weak Cerami-Palais-Smale condition
KW  - Ambrosetti-Rabinowitz condition
KW  - break of symmetry
KW  - perturbation method
KW  - supercritical growth
SN  - 1232-9274
ER  -