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In this paper, we study a type of p-Laplacian equation - Δ p u = λ u p - 2 u + u q - 2 u , x ∈ R N , with prescribed mass ∫ R N | u | p 1 p = c > 0 , where 1 < p < q < p ∗ : = pN N - p , p < N , λ ∈ R is a Lagrange multiplier. Firstly, we prove the existence of normalized solutions to p-Laplacian equations and provide accurate descriptions; secondly, we discuss the existence of ground states; finally, we study the radial symmetry of normalized solutions in the mass supercritical case. Besides, we also study normalized solutions to p-Laplacian equation with a potential function V(x) - Δ p u + V (x) u p - 2 u = λ u p - 2 u + u q - 2 u , x ∈ R N , under different assumptions on q and the constraint norm c, we prove the existence, nonexistence, concentration phenomenon and exponential decay of normalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kirchhoff equation with Sobolev critical exponent$ -\left(a+b\int_{ {\mathbb{R}}^3}|\nabla u|^2\right)\Delta u = \lambda u+\mu|u|^{q-2}u+|u|^{4}u\ \ {\rm in}\ {\mathbb{R}}^3 $under the normalized constraint$ \int_{ {\mathbb{R}}^3}u^2 = c^2, $where $ a, \, b, \, c>0 $ are constants, $ \lambda, \, \mu\in{\mathbb{R}} $ and $ 20 $, we distinguish the problem into four cases: $ 2

In this paper, we study the p-Laplacian equation with a L p -norm constraint: âˆ' Î" p u = λ | u | p âˆ' 2 u + ÎĽ | u | q âˆ' 2 u + g (u) in R N , ∫ R N | u | p d x = a p , where N â©ľ 2, a > 0, 1 < p < q â©˝ p ÂŻ â‰" p + p 2 N , ÎĽ ∈ R , g ∈ C (R , R) and λ ∈ R is a Lagrange multiplier, which appears due to the mass constraint ‖ u ‖ p = a. We assume that g is odd and L p -supercritical. When q < p ÂŻ and ÎĽ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable ÎĽ. When q = p ÂŻ and ÎĽ > 0 or q â©˝ p ÂŻ and ÎĽ â©˝ 0, with an additional condition of g, we obtain a positive radial ground state if ÎĽ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable ÎĽ ∈ R , we obtain infinitely many radial solutions for any N â©ľ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N â©ľ 6. In any case mentioned above, the range of ÎĽ depends on the value of a : | ÎĽ | can be large if a > 0 is small. [ABSTRACT FROM AUTHOR]

We study quasilinear elliptic equations of the form âˆ' div A (u) ∇ u + 1 2 A ′ (u) | ∇ u | 2 + V (x) u = h (u) , u ∈ H 1 ( R N ) , where N â©ľ 3 , A ∈ C 1 (R , R + ) is a bounded function, V (x) is allowed to be singular at the origin, and h ∈ C (R , R) is a general nonlinearity. Such type of equations has been derived as models of several physical phenomena corresponding to various types of A. Despite the lack of a priori compactness condition for the energy functional, we develop a new variational approach to deal with the issues of multiple radial solutions, nonradial solutions and sign-changing solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence, uniqueness and asymptotic stability of time periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to investigate the asymptotic behavior of time periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method. [ABSTRACT FROM AUTHOR]

In this paper we show that the local minimizers of a class of functionals in the C1-topology are still their local minimizers in W1,p0(Ω). Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory. [Copyright &y& Elsevier]

Abstract: We study the differential systems arising from Bose–Einstein condensates under the assumption that all the diffusion constants are not equal. We give interesting addenda to our previous papers: [J. Funct. Anal. 262 (3) (2012) 1087–1131] and [J. Differential Equations 251 (2011) 2737–2769]. [Copyright &y& Elsevier]

Abstract: We establish the limit system for the Gross–Pitaevskii equations when the segregation phenomenon appears, and shows this limit is the one arising from the competing systems in population dynamics. This covers and verifies a conjecture of S. Terracini et al., both in the parabolic case and the elliptic case. [Copyright &y& Elsevier]