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This paper intends to study the following degenerate fractional Schrödinger–Kirchhoff–Poisson equations with critical nonlinearity and electromagnetic fields in R 3 : { ε 2 s M ([ u ] s , A 2) (− Δ) A s u + V (x) u + ϕ u = k (x) | u | r − 2 u + (I μ ∗ | u | 2 s ♯ ) | u | 2 s ♯ − 2 u , x ∈ R 3 , (− Δ) t ϕ = u 2 , x ∈ R 3 , where ε > 0 is a positive parameter, 3 / 4 < s < 1 , 0

In this article, we study the existence of infinitely many nontrivial solutions for the following doubly nonlocal problem involving fractional (p (x) , p +) -Laplacian. (− Δ) p (x) s u + (− Δ) p + s u = | u | r (x) − 2 u + | u | p s ∗ (x) − 2 u + λ f (x , u) in Ω , u = 0 in R N ∖ Ω. Here, Ω ⊂ R N is a bounded Lipschitz domain, λ > 0 , q s ∈ (0 , 1) , p (⋅ , ⋅) is a continuous, bounded, symmetric function in R N × R N such that p + = sup (x , y) ∈ R N × R N { p (x , y) } < N s , r ∈ C (Ω ¯) , p s ∗ (x) = N p (x , x) N − s p (x , x) for every x ∈ R N and the function f satisfies certain assumptions which will be made precise later. Further, the exponents p s ∗ (⋅) and r (⋅) are two critical exponents with the assumption that the critical sets { x ∈ Ω : p s ∗ (x) = (p +) s ∗ } and { x ∈ Ω : r (x) = (p +) s ∗ } are nonempty, where (p +) s ∗ = N p + N − s p + . We also develop a concentration compactness type principle in the process. [ABSTRACT FROM AUTHOR]

We analyze the existence and non-existence of cylindrical solutions as well as ground state solutions to a class of Kirchhoff-type problems with critical Sobolev exponent, subcritical Sobolev exponent, and critical Sobolev–Hardy exponent. To the best of our knowledge, this is the first time to study the existence of cylindrically symmetric solutions to the Kirchhoff equations in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of critical p(x)-Kirchhoff problems with a singular term. A nontrivial positive solution is obtained by combining variational methods with an appropriate truncation argument. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: ( − Δ ) α u + V ( x ) u = k ( x ) f ( u ) + λ | u | 2 α ∗ − 2 u in R N , where ( − Δ ) α is the fractional Laplacian operator with α ∈ ( 0 , 1 ) , N ≥ 2 , λ is a positive real parameter and 2 α ∗ = 2 N / ( N − 2 α ) is the critical Sobolev exponent, V ( x ) and k ( x ) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the asymptotic behavior of the ground state solutions of the Hénon type biharmonic equation in , in and on , where is the unit ball in , . We prove that the ground state solution concentrates on a boundary point and has a unique maximum point as , which deduce that the ground state solution is not radially symmetric. [Copyright &y& Elsevier]

Abstract: In this paper we consider the following Schrödinger equation: where is a periodic continuous real function with 0 in a gap of the spectrum and the classical Ambrosetti–Rabinowitz superlinear condition on is replaced by a general super-quadratic condition. [Copyright &y& Elsevier]