Mathematica Bohemica, Vol. 147, No. 2, pp. 155-168, 2022
Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian
Makkia Dammak, Abir Amor Ben Ali, Said Taarabti
Received May 25, 2020. Published online May 17, 2021.
Abstract: We study the existence and nonexistence of positive solutions of the nonlinear equation -\Delta_{p(x)} u = \lambda k(x) u^q \pm h(x) u^r \text{in} \Omega,\quad u=0 \text{on} \partial\Omega, \tag{\rm Q} where \Omega\subset\mathbb{R}^N$, $N\geq2$, is a regular bounded open domain in $\mathbb{R}^N$ and the $p(x)$-Laplacian
\Delta_{p(x)} u := div( |\nabla u|^{p(x)-2} \nabla u) is introduced for a continuous function $p(x)>1$ defined on $\Omega$. The positive parameter $\lambda$ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive solutions to the problem (Q) in $W_0^{1,p(x)}(\Omega)$. When we prove the existence of minimal solution, we use the sub-super solutions method.
Affiliations: Makkia Dammak, Mathematics Department, College of Science, Taibah University, Medina, Saudi Arabia, and Mathematics Department, Faculty of Sciences, University of Sfax, Sfax 3029, Tunisia, e-mail: makkia.dammak@gmail.com; Abir Amor Ben Ali, Mathematics Department, Faculty of Mathematical Sciences, Physics and Natural Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia, e-mail: abir.amorbenali@gmail.com; Said Taarabti, Laboratory of Systems Engineering and Information Technologies (LISTI), National School of Applied Sciences of Agadir, Ibn Zohr University, Agadir 80000, Morocco, e-mail: taarabti@gmail.com
