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Solutions for fourth order elliptic equations on \(\mathbb{R}^N\) involving \(u \Delta(u^{2})\) and sign-changing potentials. (English) Zbl 1418.35128
Summary: We obtain existence and multiplicity results for fourth order elliptic equations on \(\mathbb{R}^N\) involving \(u \Delta(u^2)\) and sign-changing potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential \(V\) is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential \(V\) is bounded so that the working space is exactly \(H^2(\mathbb{R}^N)\), which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term \(u \Delta(u^2)\) in the equation.
MSC:
35J30 Higher-order elliptic equations
35J62 Quasilinear elliptic equations
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