Opuscula Math. 39, no. 2 (2019), 175-194

Opuscula Mathematica

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

Anna Maria Candela
Addolorata Salvatore

Abstract. 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.

Keywords: quasilinear elliptic equation, weak Cerami-Palais-Smale condition, Ambrosetti-Rabinowitz condition, break of symmetry, perturbation method, supercritical growth.

Mathematics Subject Classification: 35J20, 35J62, 35J66, 58E05.

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  • Communicated by Giovanni Molica Bisci.
  • Received: 2018-05-04.
  • Revised: 2018-11-11.
  • Accepted: 2018-11-13.
  • Published online: 2018-12-07.
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Cite this article as:
Anna Maria Candela, Addolorata Salvatore, Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math. 39, no. 2 (2019), 175-194, https://doi.org/10.7494/OpMath.2019.39.2.175

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