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This study considers the quasilinear elliptic equation with a damping term, $$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) =… Click to show full abstract
This study considers the quasilinear elliptic equation with a damping term, $$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$div(D(u)∇u)+k(|x|)|x|x·(D(u)∇u)+ω2(|u|p-2u+|u|q-2u)=0,where $${\mathbf {x}}$$x is an N-dimensional vector in $$\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}$${x∈RN:|x|≥α} for some $$\alpha > 0$$α>0 and $$N \in {\mathbb {N}}\setminus \{1\}$$N∈N\{1}; $$D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}$$D(u)=|∇u|p-2+|∇u|q-2 with $$1 < q \le p$$1
Journal Title: Journal of Dynamics and Differential Equations
Year Published: 2018
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