Search

In this paper, we study the following kind of Schrödinger–Poisson equations without any growth and Ambrosetti–Rabinowitz conditions: $$\begin{equation*} -\Delta u+V(x)u+(|x|^{-1}*u^2)u-\lambda u=f(u), x\in \mathbb{R}^N, \end{equation*} $$ − Δ u + V (x) u + (| x | − 1 ∗ u 2) u − λu = f (u) , x ∈ R N , where $ \lambda λ < 0 , $ V(x)\in C[0,+\infty) $ V (x) ∈ C [ 0 , + ∞) is a potential function and satisfies certain conditions. By using variational method, truncation function and Fountain Theorem, we get the existence of infinitely many solutions to revised equations. Then we use the Moser iteration to obtain the existence of infinitely many solutions to original Schrödinger–Poisson equations. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following Kirchhoff type equation 0.1 [ a + λ (∫ R 3 (| ∇ u | 2 + V (| x |) u 2) d x ) α ] (- Δ u + V (| x |) u) = | u | p - 2 u in R 3 , u ∈ H 1 (R 3) , where a , λ > 0 , α ∈ (0 , 2) and p ∈ (2 α + 2 , 6). The potential V(|x|) is radial and bounded below by a positive number. By introducing the Gersgorin Disc’s theorem, we prove that for each positive integer k, Eq. (0.1) has a radial nodal solution U k λ with exactly k nodes. Moreover, the energy of U k λ is strictly increasing in k and for any sequence { λ n } with λ n → 0 + , up to a subsequence, U k λ n converges to U k 0 in H 1 (R 3) , which is also a radial nodal solution with exactly k nodes to the classical Schrödinger equation - a Δ u + a V (| x |) u = | u | p - 2 u in R 3 , u ∈ H 1 (R 3). Our results can be viewed as an extension of Kirchhoff equation concerning the existence of nodal solutions with any prescribed numbers of nodes. [ABSTRACT FROM AUTHOR]

This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence of infinitely many solutions for a class of anisotropic p → -Laplacian problems with Dirichlet boundary conditions, depending on two real parameters. These problems have a variational structure and some critical point theorems are applied. We also give an example to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two solutions to some semilinear elliptic equation defined in the whole euclidean space R 2. The nonlinearity consists of two terms: one presents a singularity at the origin and the other has a doubly exponential behavior at infinity. This new exponential behavior is governed by a recently established weighted inequality of Trudinger–Moser type. A combination of perturbation arguments and variational tools is employed to obtain our multiplicity result. [ABSTRACT FROM AUTHOR]

In this paper, we study the following kind of Schrödinger–Poisson equations without any growth and Ambrosetti–Rabinowitz conditions: $$\begin{equation*} -\Delta u+V(x)u+(|x|^{-1}*u^2)u-\lambda u=f(u), x\in \mathbb{R}^N, \end{equation*} $$ − Δ u + V (x) u + (| x | − 1 ∗ u 2) u − λu = f (u) , x ∈ R N , where $ \lambda λ < 0 , $ V(x)\in C[0,+\infty) $ V (x) ∈ C [ 0 , + ∞) is a potential function and satisfies certain conditions. By using variational method, truncation function and Fountain Theorem, we get the existence of infinitely many solutions to revised equations. Then we use the Moser iteration to obtain the existence of infinitely many solutions to original Schrödinger–Poisson equations. [ABSTRACT FROM AUTHOR]

We study radial symmetric point defects with degree k 2 in the 2-D disk or ℝ2 in the Q-tensor framework with a singular bulk energy, which is defined by Bingham closure. First, we obtain the existence of solutions for the profiles of radial symmetric point defects with degree k 2 in the 2-D disk or ℝ2. Then, we prove that the solution is stable for ∣k∣ = 1 and unstable for ∣k∣ > 1. Some identities are derived and utilized throughout the proof of existence and stability/instability. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following HLS lower critical Choquard equation with a nonlocal perturbation - Δ u - (I α ∗ [ h | u | N + α N ]) h | u | N + α N - 2 u - μ (I α ∗ | u | q ) | u | q - 2 u = λ u in R N , ∫ R N u 2 d x = c , where α ∈ (0 , N) , N ≥ 3 , μ , c > 0 , N + α N < q < N + α + 2 N , λ ∈ R is an unknown Lagrange multiplier and h : R N → (0 , ∞) is a continuous function. The novelty of this paper is that we not only investigate autonomous case but also handle nonautonomous situation for the above problem. For both cases, we prove the existence and discuss asymptotic behavior of ground state normalized solutions. Compared with the existing references, we extend the recent results obtained by Ye et al. (J Geom Anal 32:242, 2022) to the HLS lower critical case. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the following N-Laplace Choquard equation - ε N Δ N v + V (x) | v | N - 2 v = ε μ - N (| x | - μ ∗ F (v)) f (v) , x ∈ R N , u ∈ W 1 , N (R N) , where N ≥ 2 , μ ∈ (0 , N) , Δ N v = div (| ∇ v | N - 2 ∇ v) is the N-Laplace operator, ε is a positive parameter, V is a differentiable potential, F is the primitive of f with critical exponential growth in the sense of Trudinger–Moser. To address the challenges stemming from the presence of an exponential growth given by f and the quasilinear nature of the equation, we conduct meticulous analyses. This enables us to establish an intricate threshold for the Mountain-Pass minimax level and demonstrate the presence and concentration of semiclassical ground state solutions for the equation in question. [ABSTRACT FROM AUTHOR]

We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the Γ convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the congested PKS model (which includes a density constraint ρ ≤ 1 and a pressure term): It shows that the congestion constraint is not necessary to observe phase separation and surface tension phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following class of (p 1 , p 2) -Laplacian problems: - Δ p 1 u - Δ p 2 u = g (u) in R N , u ∈ W 1 , p 1 (R N) ∩ W 1 , p 2 (R N) , where N ≥ 2 , 1 < p 1 < p 2 ≤ N , Δ p i is the p i -Laplacian operator, for i = 1 , 2 , and g : R → R is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems. Furthermore, the arguments performed throughout the paper are rather flexible and can be also applied to study other p-Laplacian and (p 1 , p 2) -Laplacian equations with general nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of a normalized ground-state solution for the Choquard–Kirchhoff equation: { − (a + b ∫ R 3 | ∇ u | 2 d x) Δ u = λ u + μ (I α ∗ | u | p) | u | p − 2 u + ω | u | 4 u , in R 3 , u > 0 , ∫ R 3 | u | 2 = m 2 , in R 3 , where a, b, m, μ, ω > 0 , p ∈ (2 , 7 + α 3) , λ ∈ R , α ∈ (0 , 3) , and I α is a Riesz potential. Utilizing approximation methods and Schwartz symmetrization rearrangements, we establish the existence of normalized ground states for these kinds of mass-constrained Choquard–Kirchhoff problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified Kirchhoff type equation, that is, the Kirchhoff type equation with a quasilinear term (1) $$\begin{equation} \begin{cases} -\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\right)\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u\\\noalign{\vskip2.6pt} \quad=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\ u\to 0 \qquad as\ |x|\to\infty, \end{cases} \end{equation}$$ { − (a + b ∫ R 3 | ∇u | 2 d x) Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞ , where a, b>0, $ p\in (4,22^*) $ p ∈ (4 , 22 ∗) and $ V(|x|) $ V (| x |) is a radial potential function and bounded below by a positive number. The appearance of nonlocal term $ b\int _{\mathbb {R}^3}|\nabla u|^2{\rm d}x\Delta u $ b ∫ R 3 | ∇u | 2 d x Δu and quasilinear term $ \frac {1}{2}\Delta (|u|^2)u $ 1 2 Δ (| u | 2) u makes the variational functional of (1) totally different from the classical Schrödinger equation. By introducing the Miranda theorem, via a construction and gluing method, for any given integer $ k\geq ~1 $ k ≥ 1 , we prove that Equation (1) admits a radial nodal solution $ U_{k}^b $ U k b having exactly k nodes. Moreover, the energy of $ U_{k}^b $ U k b is monotonically increasing in k and for any sequence $ \{b_n\} $ { b n } , and up to a subsequence, $ U_{k}^{b_n} $ U k b n converges strongly to some $ U_{k}^0\neq ~0 $ U k 0 ≠ 0 as $ b_n\to 0_+ $ b n → 0 + , which is a nodal solution with exactly k nodes to the local quasilinear Schrödinger equation (2) $$\begin{equation} \begin{cases} -a\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\\noalign{\vskip2.6pt} u\to 0 \qquad as\ |x|\to\infty. \end{cases} \end{equation}$$ { − a Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞. These results improve and generalize the previous results in the literature from the local quasilinear Schrödinger equation to the nonlocal quasilinear Kirchhoff equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff-type problem with critical nonlinearity - a + b ∫ Ω | ∇ u | 2 d x Δ u = λ f (x) | u | p - 2 u + | u | 4 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R 3 , a > 0 is a constant, b , λ are positive parameters and 2 < p < 4 . Under different assumptions on the nonlinearity, the equation has been extensively considered in the case 4 < p < 6 . By contrast, there is no existence result of solutions for the case 2 < p < 4 since the appearance of the nonlocal term. By using some innovative analytical skills, we obtain the existence results about the sign-changing solutions of this problem. Furthermore, we also present asymptotic behaviors of the sign-changing solutions as b ↘ 0 or λ ↘ 0 . [ABSTRACT FROM AUTHOR]

A novel generalization of Γ -convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ -convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s (− Δ) s u + V (x) u + ϕ u = H (u − β) f (u) , in R 3 , ε 2 s (− Δ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1) , β > 0, H is the Heaviside function, (−Δ)su is the fractional Laplacian operator, V : R 3 → R is a continuous potential and f : R → R is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b (a , b are positive integers) and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf ηϱ(x) > 0 is not required. [ABSTRACT FROM AUTHOR]

We investigate the following fractional p-Laplacian convex-concave problem: ( P λ ) (− Δ) p s u = λ | u | q − 2 u + | u | p s * − 2 u in Ω , u = 0 in R n \ Ω , where Ω is a bounded C1,1 domain in R n , s ∈ (0, 1), p > q > 1, n > sp, λ > 0, and p s * = n p n − s p is the critical Sobolev exponent. Our analysis extends classical works (A. Ambrosetti, H. Brezis, and G. Cerami, "Combined effects of concave and convex nonlinearities in some elliptic problems," J. Funct. Anal., vol. 122, no. 2, pp. 519–543, 1994, B. Barrios, E. Colorado, R. Servadei, and F. Soria, "A critical fractional equation with concave-convex power nonlinearities," Ann. Inst. Henri Poincare Anal. Non Lineaire, vol. 32, no. 4, pp. 875–900, 2015, J. García Azorero, J. Manfredi, and I. Peral Alonso, "Sobolev versus Hölder local minimizer and global multiplicity for some quasilinear elliptic equations," Commun. Contemp. Math., vol. 2, no. 3, pp. 385–404, 2000) to fractional p-Laplacian. Owing to the nonlinear and nonlocal properties of (− Δ) p s , we need to overcome many difficulties and apply notably different approaches, due to the lack of Picone identity, the stability theory, and the strong comparison principle. We show first a dichotomy result: a positive W 0 s , p (Ω) solution of (Pλ) exists if and only if λ ∈ (0, Λ] with an extremal value Λ ∈ (0, ∞). The W 0 s , p (Ω) regularity for the extremal solution seems to be unknown regardless of whether s = 1 or s ∈ (0, 1). When p ≥ 2, p − 1 < q < p and n > s p (q + 1) q + 1 − p , we get two positive solutions for (Pλ) with small λ > 0. Here the mountain pass structure is more involved than the classical situations due to the lack of explicit minimizers for the Sobolev embedding, we should proceed carefully and simultaneously the construction of mountain pass geometry and the estimate for mountain pass level. Finally, we show another new result for (Pλ) and all p > q > 1: without sign constraint, (Pλ) possesses infinitely many solutions when λ > 0 is small enough. Here we use the Z 2 -genus theory, based on a space decomposition for reflexible and separable Banach spaces, which has its own interest. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems - ϵ 2 Δ u + V (x) u = λ u + f (u) , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, V : R N → [ 0 , ∞) is a continuous function, and f is a continuous function with L 2 -subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential V attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations 4, 121–137 1996). [ABSTRACT FROM AUTHOR]

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive 'Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. [ABSTRACT FROM AUTHOR]

The present paper is inspired by a previous work of the author, where the large data scattering problem for the focusing cubic nonlinear Schrödinger equation (NLS) on R 2 × T was studied. Nevertheless, the results from the companion paper are by no means sharp, as we could not even prove the existence of ground state solutions on the formulated threshold. By making use of the semivirial-vanishing geometry, we establish in this paper the sharpened scattering results. Yet due to the mass-critical nature of the model, we encounter the major challenge that the standard scaling arguments fail to perturb the energy functionals. We overcome this difficulty by proving a crucial Legendre–Fenchel identity for the variational problems with prescribed mass and frequency. More precisely, we build up a general framework based on the Legendre–Fenchel identity and show that the much harder or even unsolvable variational problem with prescribed mass, can in fact be equivalently solved by considering the much easier variational problem with prescribed frequency. As an application showing how the geometry of the domain affects the existence of the ground state solutions, we also prove that while all mass-critical ground states on R d must possess the fixed mass M ^ (Q) , the existence of mass-critical ground states on R d × T is ensured for a sequence of mass numbers approaching zero. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems. [ABSTRACT FROM AUTHOR]

In this work, we will study the existence of solutions for a class of problems involving the p-Laplacian operator with weight and a nonlinearity that changes sign in a domain $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N with $ \partial \Omega $ ∂Ω smooth and $ N\geq 3 $ N ≥ 3. For this class of problems with subcritical growth, we can show the existence of two solutions, one nodal and one that does not change sign in Ω. To class with critical case, we prove the existence of nontrivial solution which does not change sign in Ω. [ABSTRACT FROM AUTHOR]

In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional p-Laplace problem: { (- Δ) p s ⁢ v + 풱 ⁢ (ξ ⁢ x) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ (v) in ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 푑 x = a p , where a , ξ > 0 , p ≥ 2 , λ ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier, 풱 : ℝ N → [ 0 , ∞) is a continuous function, and f is a continuous function with L p -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 풱 , as ξ is small enough via Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

In this paper, with the help of potential function, we extend the classical Brezis–Lieb lemma on Euclidean space to graphs, which can be applied to the following Kirchhoff equation - 1 + b ∫ V | ∇ u | 2 d μ Δ u + λ V (x) + 1 u = | u | p - 2 u in V , u ∈ W 1 , 2 (V) , on a connected locally finite graph G = (V , E) , where b , λ > 0 , p > 2 and V(x) is a potential function defined on V . The purpose of this paper is four-fold. First of all, using the idea of the filtration Nehari manifold technique and a compactness result based on generalized Brezis–Lieb lemma on graphs, we prove that there admits a positive solution u λ , b ∈ E λ with positive energy for b ∈ (0 , b ∗) when 2 < p < 4 . In the sequel, when p ⩾ 4 , a positive ground state solution w λ , b ∈ E λ is also obtained by using standard variational methods. What’s more, we explore various asymptotic behaviors of u λ , b , w λ , b ∈ E λ by separately controlling the parameters λ → ∞ and b → 0 + , as well as jointly controlling both parameters. Finally, we utilize iteration to obtain the L ∞ -norm estimates of the solution. [ABSTRACT FROM AUTHOR]

We discuss an optimal control problem for the Kirchhoff–Poisson type controlled system. By using variational methods and embedding theorem, we obtain the existence uniqueness of solutions to the state equation and existence of an optimal control. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following quasilinear Schrödinger equation involving Stein–Weiss type nonlinearity: - Δ u + V (x) u - Δ (u 2) u = 1 | x | α ∫ R N G (u (y)) | y - x | μ | y | α d y g (u (x)) , in R N , where N ≥ 3 , 0 < μ < N , α ≥ 0 and 2 α + μ < min { N + 2 2 , 4 } and G is the primitive of function g. The potential V : R N → R may decay to zero at infinity. By using variational methods, penalization technique and L ∞ -estimates, we obtain the existence of a positive solution for the above quasilinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities - Δ u + ϕ u = | u | p - 2 u log | u | 2 + λ f (u) + h (x) , in Ω , - Δ ϕ - ε 4 Δ 4 ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , where 4 < p < + ∞ , ε , λ > 0 are parameters, Δ 4 ϕ = div (| ∇ ϕ | 2 ∇ ϕ) , Ω ⊂ R 2 is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for λ large enough and ε fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters ε . [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation −ε2Δu+V(x)u=λu+ε−α(Iα*F(u))f(u),x∈RN,∫RN∣u∣2dx=a2εN,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a>0a\gt 0 is a constant, ε>0\varepsilon \gt 0 is a parameter, N≥3N\ge 3, α∈(0,N)\alpha \in \left(0,N), λ∈R\lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, ff is a continuous function with L2{L}^{2}-subcritical growth, and V:RN→[0,∞)V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential VV attains its minimum value if the parameter ε>0\varepsilon \gt 0 is small.

This paper focuses on the following class of fractional magnetic Schrödinger equations: \[ (-\Delta)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+\lambda\vert u\vert^{q-2}u, \quad \hbox{in } \mathbb{R}^{N}, \] (− Δ) A s u + V (x) u = g (| u | 2) u + λ | u | q − 2 u , in R N , where $ (-\Delta)_{A}^{s} $ (− Δ) A s is the fractional magnetic Laplacian, $ A :\mathbb {R}^N \rightarrow \mathbb {R}^N $ A : R N → R N is the magnetic potential, $ s\in (0,1) $ s ∈ (0 , 1) , N>2s, $ \lambda \geq 0 $ λ ≥ 0 is a parameter, $ V:\mathbb {R}^N \rightarrow \mathbb {R} $ V : R N → R is a potential function that may decay to zero at infinity and $ g: \mathbb {R}_{+} \rightarrow \mathbb {R} $ g : R + → R is a continuous function with subcritical growth. We deal with supercritical case $ q\geq ~2^*_s:=2N/(N-2s) $ q ≥ 2 s ∗ := 2 N / (N − 2 s). Our approach is based on variational methods combined with penalization technique and $ L^{\infty } $ L ∞ -estimates. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many radially symmetric solutions to the problem \[ (-\Delta_p)^s u=g(u) \quad {\rm in} \quad \mathbb{R}^N, \quad u\in W^{s,p}(\mathbb{R}^N), \] (− Δ p) s u = g (u) in R N , u ∈ W s , p (R N) , where $ s\in (0,1) $ s ∈ (0 , 1) , $ 2 \leq p 2 ≤ p < ∞ , $ sp \leq N $ sp ≤ N , $ 2 \leq N \in \mathbb {N} $ 2 ≤ N ∈ N and $ (-\Delta _p)^s $ (− Δ p) s is the fractional p-Laplacian operator. We treat both the cases sp = N and sp

This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation - Δ u - μ (I α ∗ [ h | u | p ]) h | u | p - 2 u - (I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ - 2 u = λ u in R N , ∫ R N u 2 d x = c , where μ , c > 0 , N ≥ 3 , 0 < α < N , 2 α : = N + α N < p < 2 α ∗ : = N + α N - 2 , λ ∈ R is a Lagrange multiplier, I α is the Riesz potential and h : R N → (0 , ∞) is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case N + α + 2 N ≤ p < N + α N - 2 and discuss its asymptotical behaviors as μ → 0 + and c → 0 + respectively. When N + α N < p < N + α + 2 N , we obtain the existence of one local minimizer after considering a suitable minimization problem. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence of normalized ground state solution for Schrödinger–Poisson system with doubly critical growth - Δ u - ϕ | u | 3 u = λ u + f u + | u | 4 u , x ∈ R 3 , - Δ ϕ = | u | 5 , x ∈ R 3 , having prescribed mass ∫ R 3 | u | 2 d x = m 2 , where m > 0 is a constant, λ ∈ R is unknown and appears as a Lagrange multiplier, f is a nonlinear term of Sobolev subcritical and is mass supercritical. We show that there exists a normalized ground state solution for the above system when the mass m is large enough. In addition, we also discuss the continuity and monotonicity of the ground state energy E m , and explain the asymptotic property of E m when m is large enough. Our studies complement and extend the researches of Meng and He (Methods Nonlinear Anal 62:509–534, 2023), in the sense we deal with the effect of mass-changing on the existence of normalized solutions. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of normalized solutions for the following N-Kirchhoff equation - a ϵ N + b ϵ 2 N ∫ R N ∇ u N d x div ∇ u N - 2 ∇ u = λ u N - 2 u + ϵ - (N - 1) f (ϵ u) | x | β , x ∈ R N , u ∈ W 1 , N (R N) , ∫ R N | u | N d x = ρ , where N ≥ 2 , a , b > 0 , 0 < β < N , ρ > 0 , λ ∈ R is a Lagrange multiplier, ϵ > 0 and f ∈ C (R) behaves like exp (α t N N - 1 ) with some α > 0 as t → ∞ . Under some suitable assumptions, the existence of normalized ground state solutions is studied by restricting the discussion on Nehari–Pohozaev manifold and using the singular Trudinger–Moser inequality. As by-products, the regularity and exponential decay estimates are obtained. Moreover, another solution is discussed by the mountain pass theorem. In addition, the asymptotic behavior of solutions is also investigated as ϵ → 0 . The main novelty of this paper is that we consider the Kirchhoff problem in the borderline case and the nonlinearity is singular. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equations with critical growth and potential - Δ u + V (ϵ x) u = λ u + μ | u | q - 2 u + f (u) in R N , ∫ R N | u | 2 d x = a , where a > 0 , μ > 0 , q ∈ (2 , 2 + 4 / N) , ϵ > 0 is a small parameter, λ ∈ R will arise as Lagrange multiplier, the potential V : R N → 0 , + ∞ is a continuous function that satisfies some suitable conditions and f ∈ C (R , R) enjoys critical exponential growth of Trudinger–Moser type when N = 2 , and f (u) = | u | 2 ∗ - 2 u when N ≥ 3 and 2 ∗ = 2 N N - 2 . When ϵ > 0 is sufficiently small, we prove that the existence of normalized solutions by applying truncation techniques, and the relationship between the number of positive solutions and the topology of the set where V attains its minimum is obtained by using Ljusternik–Schnirelmann theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of normalized solutions to the following Gross-Pitaevskii equation arising in trapped dipolar quantum gases: - 1 2 Δ u + λ 1 | u | 2 u + λ 2 (K ∗ | u | 2 ) u - h (ε x) | u | p - 2 u + μ u = 0 in R 3 under the mass constraint ∫ R 3 | u | 2 d x = c 2 , where (λ 1 , λ 2) ∈ R 2 , c , ε > 0 , 2 < p < 10 3 , h : R 3 → [ 0 , ∞) is a continuous function and K is the classical dipole-dipole interaction kernel. In the unstable regime, that is, (λ 1 , λ 2) ∈ D = { (λ 1 , λ 2) ∈ R 2 : λ 1 < 4 π 3 λ 2 ≤ 0 or λ 1 < - 8 π 3 λ 2 ≤ 0 } , we establish the existence of a normalized solution that corresponds to an interior local minimizer of the constraint functional provided that inf x ∈ R 3 h (x) = lim | x | → ∞ h (x) = h ∞ > 0. Moreover, if the origin is a maximum point of function h, we obtain that the number of normalized solutions is not less than the number of global maximum points of h when ε > 0 is sufficiently small. [ABSTRACT FROM AUTHOR]

This article deals with the fractional Choquard equation involving exponential growth as follows ε N (- Δ) p s u + Z (x) | u | p - 2 u = ε μ - N 1 | x | μ ∗ H (u) h (u) in R N , where ε a positive parameter, s ∈ (0 , 1) , μ ∈ (0 , N) and (- Δ) p s is a fractional p-Laplace operator with p = N s ≥ 2. The function h is only continuous and has exponential growth. In addition, the potential function Z satisfies some appropriate conditions. We use the Trudinger–Moser inequality to deal with the function h involving exponential growth. Together with the Ljusternik–Schnirelmann category theory and variational method, the multiplicity and concentration behavior of positive solutions are obtained for the above problem. As far as we know, our results seem to be new for the fractional N s -Laplacian Choquard equation. [ABSTRACT FROM AUTHOR]

In this article, we consider the double-phase problem with nonlocal reaction. For the autonomous case, we introduce the methods of the Pohozaev manifold, Hardy-Littlewood Sobolev subcritical approximation, adding the mass term to prove the existence and nonexistence of normalized solutions to this problem. For the nonautonomous case, we show the existence of normalized solutions to the double-phase problem by using the Pohozaev restrict method and describing the relationship between the energy of this problem and its limit problem. Moreover, we study the existence of normalized solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical exponents.

We investigate the existence and stability of standing waves with prescribed mass c>0c\gt 0 for Choquard equation with noncritical rotation in Bose-Einstein condensation. Then, we consider the mass collapse behavior of standing waves, the ratio of energy to mass and the Lagrange multiplier, as c→0+c\to {0}^{+}. Our results extend the existing results.

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain Ω such that V has no critical point in ∂ Ω . To overcome the "lack of compactness" of the problem, we combine Del Pino-Felmer's penalization technique with Moser's iteration method and some ideas from Alves [1]. [ABSTRACT FROM AUTHOR]

In this paper, by using the variational methods, we consider the existence of nontrivial solitary wave for a generalized Kadomtsev–Petviashvili equation with a constant potential and a periodic potential in $ \mathbb {R}^{2} $ R 2 , respectively. In our problem, the nonlinear term f is only continuous, which allows to consider larger classes of nonlinearities than usually assumed. [ABSTRACT FROM AUTHOR]

The paper concerns with the existence of positive solutions for the following Kirchhoff problem \[ \begin{cases} -\Bigl(a+b\displaystyle \int_{\Omega}|\nabla u|^2\,{\rm d}x\Bigr)\Delta u+u=|u|^{p-2}u+\varepsilon |u|^4 u & \mbox{in}\ \Omega,\\ u\in H_0^1(\Omega), \end{cases} \] { − (a + b ∫ Ω | ∇u | 2 d x) Δu + u = | u | p − 2 u + ε | u | 4 u in Ω , u ∈ H 0 1 (Ω) , where a, b>0, 40 $ ε > 0 is a parameter and $ \Omega \subset \mathbb {R}^3 $ Ω ⊂ R 3 is an exterior domain, that is, Ω is an unbounded domain with $ \mathbb {R}^3\backslash \Omega $ R 3 ∖Ω nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when $ \mathbb {R}^3\backslash \Omega $ R 3 ∖Ω is contained in a small ball and $ \varepsilon >0 $ ε > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the following double-phase equation - div (| ∇ u | p - 2 ∇ u + μ ε (x) | ∇ u | q - 2 ∇ u) + V ε (x) (| u | p - 2 u + μ ε (x) | u | q - 2 u) = f (u) in R N , where N ≥ 2 , 1 < p < q < N , q < p ∗ with p ∗ = Np N - p , μ : R N → R is a continuous non-negative function, μ ε (x) = μ (ε x) , V : R N → R is a positive potential satisfying a local minimum condition, V ε (x) = V (ε x) , and the nonlinearity f : R → R is a continuous function with subcritical growth. Under natural assumptions on μ , V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: - M ∫ R 2 | ∇ u | 2 d x Δ u + μ V (x) + h (x) u = λ f (u) in R 2 , where M is continuous with inf R + M > 0 , V ≥ 0 and its zero set has several disjoint bounded components, μ , λ are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions. [ABSTRACT FROM AUTHOR]

AbstractThis article concerns the following Bibarmonic elliptic equation with critical growthwhere Ω ⊂ ℝN is a smooth bounded domain , g(x) is a positive continuous function on is a real parameter, is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. By combining the Nehari manifold and the variational methods, we prove the existence of k nontrivial solutions for this equation. Our results extend the results of Rani and Goyal in Topological Methods in Nonlinear Analysis 59 (2022), 221–260. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of strongly degenerate problems with critical exponential growth in R N , N ≥ 2 . We do not assume ellipticity condition on the operator and thus the maximum principle given by Lieberman (Commun Partial Differ Equ 16:311–361, 1991) can not be accessed. Therefore, a careful and delicate analysis is necessary and some ideas can not be applied in our scenario. The arguments developed in this paper are variational and our main result completes the study made in the current literature about the subject. Moreover, when N = 2 or N = 3 the solutions model the slow steady-state flow of a fluid of Prandtl-Eyring type. [ABSTRACT FROM AUTHOR]

For the f-weighted area-functional E f (M) = ∫ M f (x) d H n (x) we prove non-existence of compact stationary surfaces when f = | x | α and α > - n , while for α = - n all spheres S R (0) are shown to be stable and even minimizers for E f . Moreover any compact and f-stable surface is a sphere S R (0) in case α = - n . Furthermore we prove stability of the minimal cones over products of spheres under suitable conditions on α and show non-existence of nontrivial cones in case these conditions do not hold. Finally we show that the cones over products of spheres S k - 1 × S k - 1 ⊂ R k × R k , k ≥ 2 , are in fact minimizers for E f , if 1 ≤ α ≤ 2 (k - 1) . In particular the cone over the Clifford torus minimizes E f , f = | x | α and 1 ≤ α ≤ 2 . [ABSTRACT FROM AUTHOR]

Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation. [ABSTRACT FROM AUTHOR]

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, \[ \begin{cases} -(1+b\|u\|^{2})\Delta u=\left(\int_{\Omega}\frac{u^{2}(y)}{|x-y|^{4}}\,{\rm d}y\right)u +\lambda u & {\rm in}\ \Omega,\\ u=0 & {\rm on}\ \partial\Omega, \end{cases} \] { − (1 + b ‖ u ‖ 2) Δu = (∫ Ω u 2 (y) | x − y | 4 d y) u + λu in Ω , u = 0 on ∂Ω , where $ \Omega \subset \mathbb {R}^{N}(N\geq 5) $ Ω ⊂ R N (N ≥ 5) is a smooth bounded domain and $ \|\cdot \| $ ‖ ⋅ ‖ is the standard norm of $ H_{0}^{1}(\Omega) $ H 0 1 (Ω). Under the suitable assumptions on the constant $ b\geq 0 $ b ≥ 0 , we prove the existence of solutions for $ 0 \lt \lambda \leq \lambda _{1} $ 0 < λ ≤ λ 1 , where $ \lambda _{1} \gt 0 $ λ 1 > 0 is the first eigenvalue of $ -\Delta $ − Δ on Ω. Moreover, we prove the multiplicity of solutions for $ \lambda \gt \lambda _{1} $ λ > λ 1 and b>0 in suitable intervals. [ABSTRACT FROM AUTHOR]