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Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions

This work is supported National Science Foundation of China No. 11871250
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  • We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities $ f(x) |u|^{q-1} u $ and $ h(x) |u|^{p-1} u $ under certain conditions on $ f(x), \, h(x) $, $ p $ and $ q $. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of $ f(x) $ and $ h(x) $ on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When $ h(x)^+ \neq 0 $, we prove that the equation has at least one nontrivial solution if $ f(x)^+ = 0 $ and that the equation has at least two nontrivial solutions if $ \int_\Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $, where $ r $ and $ \varLambda $ are explicit numbers. These results are novel, which improve and extend the existing results in the literature.

    Mathematics Subject Classification: 35J35, 35J40, 35J65.

    Citation:

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