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- W3211190345 abstract "We study the behavior of solutions for the 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 a $C^2$-boundary $partial Omega$, $a_1,a_2 in L^infty(Omega)$ with $a_1(z),a_2(z)>0$ for a.a. $z in Omega$, $p,q in (1,infty)$ and $Delta_{p}^{a_1},Delta_{q}^{a_2}$ are weighted versions of $p$-Laplacian and $q$-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $f(z,x)$ asymptotically as $x to pm infty$ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $lambda$ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators." @default.
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- W3211190345 date "2021-01-01" @default.
- W3211190345 modified "2023-10-18" @default.
- W3211190345 title "The behavior of solutions of a parametric weighted $ (p, q) $-Laplacian equation" @default.
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- W3211190345 doi "https://doi.org/10.3934/math.2022032" @default.
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