Abstract
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="p"> <mml:mi>p</mml:mi> </mml:math>-biharmonic equation with Hardy–Sobolev exponent and without the Ambrosetti–Rabinowitz condition
Highlights
We consider the following p-biharmomic equations with clamped Dirichlet boundary conditions 2 p u = μ|u|r−2 u |x|s + f (x, u) u∂u ∂n in Ω, on ∂Ω (PD)and p-biharmomic equations with hinged Navier boundary conditions u= u=0 in Ω, on ∂Ω (PNa) where Ω ⊂ RN(N ≥ 3) is a smooth bounded domain, 0 ∈ Ω, 2 < 2p < N, p ≤ r < p∗(s) = (N−s)p N−2p ≤ p∗(0)
If we assume that f (x, t) is an odd function in t, we can prove the existence of infinitely many weak solutions to Problem (PD) and (PNa)
In order to use Theorem A.2 to study Eq (PD) and (PNa), we need to verify that the functionals Iμ satisfies the mountain pass geometry structure and compactness conditions
Summary
Zhao [25] studied the existence and multiplicity of solutions of p-biharmonic type equations with critical growth. To obtain such deformation results, some compactness condition of the functional is necessary. If we assume that f (x, t) is an odd function in t, we can prove the existence of infinitely many weak solutions to Problem (PD) and (PNa).
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