In this paper, we discuss the following nonlinear p -Laplacian problems with a Hardy term and Sobolev critical growth: { − Δ p u − μ | x | p | u | p − 2 u + λ | u | p − 2 u = | u | p * − 2 u + κ | u | q − 2 u , x ∈ R N ∖ { 0 } , ∫ R N | u | p d x = c , where N ≥ 2 , 1 < p < N , c , κ > 0 , p * := p + p 2 / N < q < p * := N p / ( N − p ) , μ < μ ¯ := ( ( N − p ) / p ) p and λ ∈ R is an unknown Lagrange multiplier. The central focus of this paper is to demonstrate the existence of normalized ground state solutions for this problem, which possesses an exponential decay property at infinity. Moreover, for any sequence { μ n } ⊂ ( 0 , + ∞ ) satisfying μ n → 0 + , we prove that the corresponding sequence of these ground state solutions converges to a ground state solution of { − Δ p u + λ | u | p − 2 u = | u | p * − 2 u + κ | u | q − 2 u , x ∈ R N ∖ { 0 } , ∫ R N | u | p d x = c .
We consider interior transmission problems with one complex-valued refraction index independent of the spectral parameter, that is, with a strong damping term, and we suppose that it does not vanish on the boundary. Under some quite general conditions, we obtain transmission eigenvalue-free regions similar to those obtained by the author in the case of weak damping.
We study a quasilinear elliptic system λ u | u | p − 2 − div ( a ( x , ∇ u ) ) + b ( x , u ) = f ( x ) , in a bounded domain Ω ⊂ R l ( l > 2 , 2 ≤ p < l ), with homogeneous Dirichlet boundary conditions. The principal part a depends only on the gradient and satisfies strong monotonicity and growth conditions, while the lower-order term b depends only on u and is monotone; its growth is controlled by a coefficient in the classical Kato class. Using the Minty–Browder theorem for monotone operators, we prove existence and uniqueness of a weak solution in W 0 1 , p ( Ω ) N . The results cover the p -Laplacian system and allow singular potentials.
This paper investigates the backward asymptotic autonomy of invariant measures for the velocity–vorticity-Voigt model of the three-dimensional Navier–Stokes equations with delays. First, by overcoming the difficulties arising from delays in the higher-order regularity estimates of solutions and the asymptotic compactness of solution operators, we prove the existence of pullback attractors. Second, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. We then prove the asymptotically autonomous robustness for invariant measures, that is, any limit of the sequence of invariant measures for such a non-autonomous system must be an invariant measure of the corresponding autonomous system as time tends to negative infinity. Moreover, the asymptotically autonomous limiting set of a pullback attractor can also be characterized by the global attractor of the corresponding autonomous system with the autonomous external force terms that satisfy the convergence relationship.
In this article, we investigate the existence and multiplicity of normalized solutions for the p -Laplacian Schrödinger–Poisson system: { − Δ p u + λ u p − 1 − κ ( | x | − 1 * | u | p ) u p − 1 = f ( u ) in R 3 , ∫ R 3 | u | p d x = a p , u > 0 in R 3 , where a > 0 represents the prescribed L p -norm, κ ∈ R ∖ { 0 } is a parameter, and λ ∈ R appears as an undetermined Lagrange multiplier. Our principal findings are summarized as follows. (i) For κ < 0 , under the conditions that f is odd and satisfies L p -supercritical yet Sobolev subcritical growth, we establish the existence of a normalized ground state solution for sufficiently small a > 0 by employing the Pohožaev manifold method combined with genus theory. In this setting, we prove that the problem admits infinitely many normalized solutions whose energies tend to infinity. The asymptotic behavior of the normalized ground state energy is also analyzed. (ii) For κ > 0 and the Sobolev critical nonlinearity f ( u ) = | u | p * − 2 u , we use a truncation technique together with the concentration-compactness principle to address the lack of lower boundedness of the energy functional. Under appropriate constraints on κ and a , we demonstrate the existence of infinitely many normalized solutions possessing negative energy. These results extend earlier work by some of the authors on p -Laplacian Schrödinger–Poisson systems.
