The Nehari manifold approach for singular equations involving the $p(x)$-Laplace operatorRepovš, Dušan (Avtor) Saoudi, Kamel (Avtor) Nehari manifoldgeneralized Lebesgue-Sobolev spacetopological methodsingular equationp(x)-Laplace operatormultiplicityWe study the following singular problem involving the $p(x)$-Laplace operator $\Delta p(x)u = \mathrm{div}(|\nabla u|^{p(x)-2} \nabla u)$, where $p(x)$ is a nonconstant continuous function $$ (P_\lambda) \quad \begin{cases} -\Delta_{p(x)}u = a(x)|u|^{q(x)-2}u(x) + \frac{\lambda b(x)}{u^{\delta(x)}} & \text{in} \; \Omega, \\ u>0 & \text{in} \; \Omega, \\ u=0 &\text{on} \; \partial\Omega. \end{cases} $$ Here, $\Omega$ is a bounded domain in $\mathbb{R}^{N \ge 2}$ with $C^2$-boundary, $\lambda$ is a positive parameter, $a(x), b(x) \in C(\overline{\Omega})$ are positive weight functions with compact support in $\Omega$, and $\delta(x), p(x), q(x) \in C(\overline{\Omega})$ satisfy certain hypotheses $(A_0)$ and $(A_1)$. We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $(P_\lambda)$.20232026-05-12 12:02:57Članek v reviji182455sl