Ground State Solutions Research Articles

Ground State Solution for the Logarithmic Schrödinger–Poisson System with Critical Growth

Ground State Solution for the Logarithmic Schrödinger–Poisson System with Critical Growth

Existence and asymptotic behavior of normalized solutions for fourth‐order equations of Kirchhoff type

In this paper, we consider the following fourth‐order equation of Kirchhoff type under the normalized constraint , where are constants, and is a Lagrange multiplier. Firstly, we obtain the existence of ground state solutions for or with . Moreover, we show the nonexistence of solutions for . Finally, we investigate the asymptotic behavior of normalized solutions obtained above with respect to the parameter .

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s. Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities

Abstract We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math. 79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

Existence of ground state sign-changing solutions for a class of quasilinear scalar field equations originating from nonlinear optics

This paper is mainly concerned with the existence of ground state sign-changing solutions for a class of second order quasilinear elliptic equations in bounded domains, which derived from nonlinear optics models. Combining a non-Nehari manifold method due to Tang and Cheng [J. Differ. Equations 261, 2384–2402 (2016)] and a quantitative deformation lemma with Miranda theorem, we obtain that the problem has at least one ground state solution and one ground state sign-changing solution with two precise nodal domains and we present the concrete lower bounds of ground state solution and ground state energy. We also obtain that any weak solution of the problem has C1,σ-regularity for some σ ∈ (0, 1). With the help of Maximum Principle, we reach the conclusion that the energy of the ground state sign-changing solutions is strictly larger than twice that of the ground state solutions.

Concentration of ground state solutions for supercritical zero-mass (N, q)-equations of Choquard reaction

We study the following singularly perturbed (N, q)-equation of Choquard type -εNΔNu-εqΔqu=εμ-N(∫RNK(y)F(u(y))|x-y|μdy)K(x)f(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^N\\Delta _Nu-\\varepsilon ^q\\Delta _qu=\\varepsilon ^{\\mu -N} \\bigg (\\int _{\\mathbb {R}^N}\\frac{K(y)F(u(y))}{|x-y|^{\\mu }}dy\\bigg )K(x)f(u),~x\\in \\mathbb {R}^N, \\end{aligned}$$\\end{document}where Δru=div(|∇u|r-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _ru = \ ext {div}(|\ abla u|^{r-2}\ abla u)$$\\end{document} denotes the usual r-Laplacian operator with r∈{q,N}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\in \\{q,N\\}$$\\end{document} and 1<q<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<q<N$$\\end{document}, ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} is a sufficiently small parameter, K∈C0(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\in C^0(\\mathbb {R}^N)$$\\end{document} satisfies some technical assumptions, 0<μ<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\mu <N$$\\end{document} and F is the primitive of f that fulfills a supercritical exponential growth in the Trudinger–Moser sense. Due to the new version of Trudinger–Moser type inequality introduced in Shen and Rădulescu (Zero-mass (N, q)-Laplacian equation with Stein-Weiss convolution part in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document}: supercritical exponential case. submitted), we aim to derive the existence and concentration of ground state solutions for the given equation using variational method, where the concentrating phenomenon appears at the maximum point set of K as ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+$$\\end{document}.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

Normalized ground state solutions of Schrödinger-KdV system in $$\mathbb {R}^3$$

Normalized ground state solutions of Schrödinger-KdV system in $$\mathbb {R}^3$$

Ground state solutions for elliptic Kirchhoff–Boussinesq type problems with supercritical exponential growth

In this work, we are interested in studying the existence of a ground state solution for the problem Δ(wβ(x)Δu)±div(wβ(x)|∇u|p−2∇u)=f(x,u) in B,andu=∂u∂ν=0on∂B,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta (w_{\\beta}(x)\\Delta u ) \\pm \ ext{div}(w_{\\beta} (x) | \ abla u |^{p-2} \ abla u)= f(x,u) \ ext{ in }B,\\quad \ ext{and} \\ u= \\dfrac{\\partial u}{\\partial \ u}=0 \\ \ ext{on}\\ \\partial B, $$\\end{document} where B is the unit ball in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{4}$\\end{document}, wβ(x)=(loge|x|)β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$w_{\\beta}(x)=\\big(\\log \\frac{e}{|x|}\\big)^{\\beta}$\\end{document} or wβ(x)=(log(1|x|)β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$w_{\\beta}(x)=\\big(\\log (\\frac{1}{|x|}\\big)^{\\beta}$\\end{document} for β∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta \\in (0,1)$\\end{document}, 2<p<4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2< p < 4$\\end{document} and f:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:\\mathbb{R}\\rightarrow \\mathbb{R}$\\end{document} is a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result.

Scattering and blow up for nonlinear Schrödinger equation with the averaged nonlinearity

We consider the 3-dimensional nonlinear Schrödinger equation (NLS) with average nonlinearity. This is a limiting model of NLS with strong magnetic confinement and a generalized model of the resonant system of NLS with a partial harmonic oscillator in terms of nonlinear power. We provide a new proof for the conservation law of kinetic energy and remove the restriction on nonlinearity. Moreover, in the case of focusing, super-quintic, and sub-nonic, we construct a new ground-state solution and classify the behavior of the solutions below the ground state. We demonstrate a sharp threshold for scattering and blow-up.

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Concentrating Solutions for Fractional Schrödinger–Poisson Systems with Critical Growth

In this paper, we consider a class of fractional Schrödinger–Poisson systems (−Δ)su+λV(x)u+ϕu=f(u)+|u|2s*−2u and (−Δ)tϕ=u2 in R3, where s,t∈(0,1) with 2s+2t&gt;3, λ&gt;0 denotes a parameter, V:R3→R admits a potential well Ω≜intV−1(0) and 2s*≜63−2s is the fractional Sobolev critical exponent. Given some reasonable assumptions as to the potential V and the nonlinearity f, with the help of a constrained manifold argument, we conclude the existence of positive ground state solutions for some sufficiently large λ. Upon relaxing the restrictions on V and f, we utilize the minimax technique to show that the system has a positive mountain-pass type by introducing some analytic tricks. Moreover, we investigate the asymptotical behavior of the obtained solutions when λ→+∞.

Existence of Semiclassical Ground State Solutions for a Class of N-Laplace Choquard Equation with Critical Exponential Growth

Existence of Semiclassical Ground State Solutions for a Class of N-Laplace Choquard Equation with Critical Exponential Growth

Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$ - \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where $ N \geq 2 $, $ V, K $ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.

Discrete Schrödinger–Newton equations with power nonlinearity

We study Schrödinger–Newton equations with general power nonlinearity on the lattice Z d . After the proof of well-posedness of the Cauchy problem, our interest is focused on the existence of stable standing (stationary) solutions. Employing variational methods we prove the existence of a ground state solution solving the associated stationary system. Orbital stability as well as exponential localization of the ground state is verified. Finally, we discuss the existence of excitation thresholds for ground state solutions in dependence of the conserved mass ( l 2 -norm).

Solutions for a Logarithmic Fractional Schrödinger-Poisson System with Asymptotic Potential

In this paper, we consider a logarithmic fractional Schrödinger-Poisson system where the potential is a sign-changing function. When the potential is coercive, we get the existence of infinitely many solutions for the system. When the potential is bounded, we get the existence of a ground state solution for the system.

Nonlinear dynamics as a ground-state solution on quantum computers

For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results. Published by the American Physical Society 2024

Nonlinear scalar field (p1,p2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p_{1}, p_{2})$$\\end{document}-Laplacian equations in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{N}$$\\end{document}: existence and multiplicity

In this paper, we deal with the following class of (p1,p2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p_{1}, p_{2})$$\\end{document}-Laplacian problems: -Δp1u-Δp2u=g(u)inRN,u∈W1,p1(RN)∩W1,p2(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta _{p_{1}}u-\\Delta _{p_{2}}u= g(u) \ ext{ in } \\mathbb {R}^{N},\\\\ u\\in W^{1, p_{1}}(\\mathbb {R}^{N})\\cap W^{1, p_{2}}(\\mathbb {R}^{N}), \\end{array} \\right. \\end{aligned}$$\\end{document}where N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, 1<p1<p2≤N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p_{1}<p_{2}\\le N$$\\end{document}, Δpi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{p_{i}}$$\\end{document} is the pi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{i}$$\\end{document}-Laplacian operator, for i=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1, 2$$\\end{document}, and g:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:\\mathbb {R}\\rightarrow \\mathbb {R}$$\\end{document} 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 (p1,p2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p_1, p_2)$$\\end{document}-Laplacian equations with general nonlinearities.

Asymptotic profiles for Choquard equations with combined attractive nonlinearities

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation(Pε)−Δu+εu=(Iα⁎|u|p)|u|p−2u+|u|q−2u,inRN,where N≥3 is an integer, p∈[N+αN,N+αN−2], q∈(2,2NN−2), Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. We show that as ε→0 (resp. ε→∞), the ground state solutions of (Pε), after appropriate rescalings dependent on parameter regimes, converge in H1(RN) to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of uε(0), ‖∇uε‖22, ‖uε‖22, ∫RN(Iα⁎|uε|p)|uε|p and ‖uε‖qq, which depend in a non-trivial way on the exponents p, q and the space dimension N. Further, we discuss a connection of our results with a mass constrained problem, associated to (Pε) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, multiplicity and precise asymptotic behaviour of positive normalized solutions of such a problem as c→0 and c→∞.