This paper is devoted to the study of a class of nonlocal singular elliptic problems involving the fractional ψ-Hilfer p ( x ) -Laplace operator H L p ( x ) α , β , ψ u , with positive weight functions. The presence of both the singular nonlinearity and the generalized nonlocal operator introduces significant analytical challenges. To address these, we employ the Nehari manifold technique, which enables us to investigate the variational structure of the problem and establish the existence of at least two distinct solutions.

We provide existence results for the subelliptic critical problem involving a Hardy-type perturbation − Δ γ u − μ ψ 2 d 2 u = | u | 2 γ ∗ − 2 u + λu in Ω , u = 0 on ∂Ω , where Δ γ = Δ x + | x | 2 γ Δ y , 0 $ ]]> γ > 0 is the so-called Grushin operator, Ω is a smooth bounded domain of R N = R x m × R y n , 0 ∈ Ω , d is the gauge norm naturally associated with Δ γ , ψ = | ∇ γ d | , where ∇ γ is the Grushin gradient, 2 γ ∗ is the critical Sobolev exponent in this context, and 0 < μ < μ ¯ , where μ ¯ is the best Hardy constant for Δ γ . In particular, we extend to the Grushin context well-known results obtained by Jannelli in [J Differ Equ. 1999;156] for the classical Laplacian case, and generalize to the case with Hardy perturbation some of the existence results proved for the Grushin unperturbed operator by Alves-Gandal-Loiudice-Tyagi in [J Geom Anal. 2024;34(2):52].

This article is concerned with the existence of multiple solutions of the following biharmonic system involving critical nonlinearities with sign-changing weight functions { Δ 2 u = λf ( x ) | u | r − 2 u + β β + γ h ( x ) | u | β − 2 u | v | γ in Ω , Δ 2 v = μg ( x ) | v | r − 2 v + γ β + γ h ( x ) | u | β | v | γ − 2 v in Ω , u = v = ∇u = ∇v = 0 on ∂Ω , where Δ 2 denotes the biharmonic operator, Ω is a bounded domain in R N with smooth boundary ∂Ω , 1<r<2 and 1 $ ]]> β > 1 , 1 $ ]]> γ > 1 satisfying 2 < β + γ ≤ 2 2 ∗ with 2 2 ∗ = 2 N N − 4 as a critical Sobolev exponent and λ, 0 $ ]]> μ > 0 . The functions f, g and h : Ω ¯ → R are sign-changing weight functions satisfying f, g ∈ L α ( Ω ) and h ∈ L ∞ ( Ω ) respectively. Using the variational methods and the Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter ( λ , μ ) ∈ R + 2 ∖ { ( 0 , 0 ) } .

We prove several existence and multiplicity results for some nonlinear elliptic Kirchhoff equations { − ( 1 + γ G ′ ( ∫ Ω | ∇u | 2 d x ) ) Δu = f ( x , u , ∇u ) , x ∈ Ω , u ( x ) = 0 , x ∈ ∂Ω , where 0 $ ]]> γ > 0 is a parameter, G : R + → R and f : Ω × R + × R N → R are smooth functions satisfying C 1 t q − 1 ≤ G ′ ( t ) ≤ C 2 t q − 1 ∀ t ∈ [ 0 , ∞ ) , C 3 | u | p − 1 ≤ f ( x , u , v ) ≤ C 4 | u | p − 1 ∀ ( x , u , v ) ∈ Ω × R + × R N , C i ( i = 1 , 2 , 3 , 4 ) are constants and q>1, 2 < p < 2 ∗ with p ≤ 2 q . The proofs of our main results are based upon bifurcation techniques.

This paper investigates the zero distribution of two classes of differential-difference polynomials in the complex plane. The first class has the form A ( z ) f ( z ) n + B ( z ) [ g ( k ) ( z + c ) ] m − a ( z ) , while the second is structured as A ( z ) f ( z ) n [ g ( k ) ( z + c ) ] m − a ( z ) . Here f ( z ) , g ( z ) are transcendental meromorphic functions, a ( z ) , A ( z ) , B ( z ) are non-zero small functions with respect to f and g. Based on the derived distribution properties of zeros, we further analyze the existence of solutions for associated differential-difference equations. Representative examples are provided to validate the theoretical results.

We study the elliptic equation { − Δ u = u p − 1 + λ u q − 1 ln ⁡ u , x ∈ Ω , u ≥ 0 , x ∈ Ω , u = 0 , x ∈ ∂Ω , where Ω ⊂ R N is a bounded smooth domain, N ≥ 3 , λ ≥ 0 is a parameter, 1 < q < 2 ∗ , 1 < p ≤ 2 ∗ . Under different conditions on the parameters λ , p and q, we investigate the existence and multiplicity of solutions to this problem utilizing the mountain pass Theorem and the Nehari technique. Compared to u q − 1 , some interesting phenomena occur since the sign of the logarithmic term u q − 1 ln ⁡ u is indefinite.

In this paper, we consider the H e ´ non equation with supercritical exponent 0, & {\\rm in} \\ \\Omega,\\\\ u=0, & {\\rm on}\\ \\partial\\Omega, \\end{cases} \\] ]]> { − Δ u = | x | α u 2 ∗ − 1 + ϵ , u > 0 , in Ω , u = 0 , on ∂Ω , where Ω is a bounded smooth domain in R N containing the origin, N ≥ 3 , 0 $ ]]> α > 0 , 2 ∗ = 2 N N − 2 and ϵ is a positive constant. M. Ben Ayed et al. [A nonexistence result of single peaked solutions to a supercritical nonlinear problem. Commun Contemp Math. 2003;5:179–195] proved the equation without | x | α does not have a single peaked solution. Moreover, there are few works on the H e ´ non equation with supercritical exponent in a general domain. Inspired by them, we proved the non-existence of a single peak solution to the H e ´ non problem by Pohozaev identities.

In this paper, we investigate the normalized solutions to the fractional p-Laplacian equations with critical Hardy–Sobolev exponent 0, & \\end{array}\\right. \\] ]]> { ( − Δ ) p s u + λ | u | p − 2 u = μ | u | r − 2 u + | u | p β ∗ − 2 u | x | β , in R N , ∫ R N | u | p d x = a > 0 , where N ⩾ 2 , 0 $ ]]> μ > 0 is a parameter, λ ∈ R is an uncharted Lagrange multiplier. 0 < s < 1 < p , 0 < β < p 2 s 2 N < ps < N , r ∈ ( p , p β ∗ ) , p β ∗ = ( N − β ) p N − ps denotes the fractional critical Hardy–Sobolev exponent. By constrained variational methods, with the help of Pohožaev identity and asymptotic estimates for minimizers, we establish the existence of normalized solutions and reveal the behavior of solutions as μ → 0 + .

Let ( M , g ) be a smooth n-dimensional compact Riemannian manifold with totally geodesic boundary ∂ M , n ≥ 2 . We prove a sharp Sobolev inequality of W 1 , p ( M ) for all 1<p<n. When p = 2 and n ≥ 3 , this inequality was proved by Li-Zhu in [Sharp Sobolev inequalities involving boundary terms. Geom Funct Anal. 1998;8:59–87].

We study the unique solvability of the Steklov problem for the biharmonic equation in a half–space under the assumption that the generalized solutions of this problem have a finite weighted Dirichlet integral. Depending on the value of the weight parameter, we prove uniqueness theorems or give exact formulas for calculating the dimension of the solution space of the Steklov biharmonic problem in a half-space.