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Given a compact Riemannian manifold (M, g) of dimension n ≥ 3 without boundary, using the variational methods, we study the existence of solutions for the elliptic equation P g k u = f | u | N − 2 u + λ h | u | q − 2 u , (1) where P g k is the GJMS operator of order 2k < n, h, f ∈ C ∞(M), 1 < q < 2, λ > 0 and N is the critical Sobolev exponent for the space H k 2 ( M ) . We apply Ljusternik-Schnirelmann theory on C¹-manifolds to prove that under some conditions, the equation (1) admits infinitely many solutions. At the end, we give two applications, one for Paneitz-Branson operator and the second is for the GJMS operator when k = 3.

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: where and are positive bounded functions defined in , is the unit ball in , and is a pair of positive numbers lying on the critical hyperbola Under some suitable further assumptions on the functions and , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

We consider the following fractional Schrödinger equation involving critical exponent: ( - Δ ) s u + V ( y ) u = Q ( y ) u 2 s ∗ - 1 , u > 0 , in R N , u ∈ D s ( R N ) , where 2 s ∗ = 2 N N - 2 s , ( y ′ , y ′ ′ ) ∈ R 2 × R N - 2 and V ( y ) = V ( | y ′ | , y ′ ′ ) and Q ( y ) = Q ( | y ′ | , y ′ ′ ) are bounded nonnegative functions in R + × R N - 2 . By using finite-dimensional reduction method and local Pohozaev-type identities, we show that if 2 + N - N 2 + 4 4 < s < min { N 4 , 1 } and Q ( r , y ′ ′ ) has a stable critical point ( r 0 , y 0 ′ ′ ) with r 0 > 0 , Q ( r 0 , y 0 ′ ′ ) > 0 and V ( r 0 , y 0 ′ ′ ) > 0 , then the above problem has infinitely many solutions, whose energy can be arbitrarily large.

In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with multiple Hardy–Sobolev critical exponents -Δu=μ|u|2∗-2u+∑i=1l|u|2∗(si)-2u|x|si+a(x)|u|q-2uinΩ,u=0on∂Ω,where Ω is a smooth bounded domain in RN with 0∈∂Ω and all the principle curvatures of ∂Ω at 0 are negative, a∈C1(Ω¯,R∗+),μ>0,02q+1q-1. By 2∗:=2NN-2 and 2∗(si):=2(N-si)N-2 we denote the critical Sobolev exponent and Hardy–Sobolev exponents, respectively.

In this paper, we consider the following elliptic systems with critical Sobolev growth -Δu=2αα+β|u|α-2u|v|β+λ|u|q-2uinΩ,-Δv=2βα+β|u|α|v|β-2v+μ|v|q-2vinΩ,u=v=0on∂Ω,where Ω is a smooth bounded domain in RN, 10, α, β>1 satisfying α+β=2∗, 2∗=2NN-2. We prove that if N>2q+1q-1, then the above problem has two disjoint and infinite sets of solutions.

In this manuscript, we deal with a class of fractional non-local problems involving a singular term and vanishing potential of the form: { L p ( x, . ) , q ( x, . ) s 1 , s 2 w ( x ) = g ( x,w ( x ) ) w ( x ) ξ ( x ) + V ( x ) | w ( x ) | σ ( x ) − 2 w ( x ) in   U , w   >   0 in   U , w   = 0 in   ℝ N \\ U , , where L p ( x, . ) , q ( x, . ) s 1 , s 2 is a (p(x, .), q(x, .)) − fractional double-phase operator with s₁, s₂ ∈ (0, 1), g, and 𝓥 are functions that satisfy some conditions. The strategy of the proof for these results is to approach the problem proximatively and calculate the critical groups. Moreover, using Morse’s theory to prove our problem has infinitely many solutions.