The behavior of solutions of a parametric weighted $ (p, q) $-Laplacian equation
نویسندگان
چکیده
We study the behavior of solutions for parametric equation $$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet condition, where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain with $C^2$-boundary $\partial \Omega$, $a_1,a_2 \in L^\infty(\Omega)$ $a_1(z),a_2(z)>0$ a.a. $z $p,q (1,\infty)$ and $\Delta_{p}^{a_1},\Delta_{q}^{a_2}$ are weighted versions $p$-Laplacian $q$-Laplacian. prove existence nonexistence nontrivial solutions, when $f(z,x)$ asymptotically as $x \to \pm \infty$ can be resonant. In studied cases, we adopt variational approach use truncation comparison techniques. When $\lambda$ large, establish at least three smooth sign information ordered. Moreover, critical parameter value determined in terms spectrum one differential operators.
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ژورنال
عنوان ژورنال: AIMS mathematics
سال: 2021
ISSN: ['2473-6988']
DOI: https://doi.org/10.3934/math.2022032