Search

This paper is concerned with the following logarithmic Schrödinger system { − Δ u 1 + ω 1 u 1 = μ 1 u 1 log u 1 2 + 2 p p + q ∣ u 2 ∣ q ∣ u 1 ∣ p − 2 u 1 , − Δ u 2 + ω 2 u 2 = μ 2 u 2 log u 2 2 + 2 q p + q ∣ u 1 ∣ p ∣ u 2 ∣ q − 2 u 2 , ∫ Ω ∣ u i ∣ 2 d x = ρ i , i = 1 , 2 , (u 1 , u 2) ∈ H 0 1 (Ω ; ℝ 2) , where Ω = ℝN or Ω ⊂ ℝN (N ⩾ 3) is a bounded smooth domain, and ωi ∈ ℝ, μi, ρi > 0 for i = 1, 2. Moreover, p,q ⩾ 1, and 2 ⩽ p + q ⩽ 2*, where 2 ∗ : = 2 N N − 2 . By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u2, firstly, we provide a unified proof of the existence of the normalized ground state solution for all 2 ⩽ p + q ⩽ 2*. Secondly, we consider the stability of normalized ground state solutions. Finally, we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p + q approaches 2*. Notably, the uncertainty of the sign of u log u2 in (0, +∞) is one of the difficulties of this paper, and also one of the motivations we are interested in. In particular, we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations (i.e., p + q = 2*). In addition, our study includes proving the existence of solutions to the logarithmic type Brézis-Nirenberg problem with and without the L2-mass constraint ∫Ω∣ui∣2dx = ρi (i = 1, 2) by two different methods, respectively. Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrödinger system with logarithmic perturbations. [ABSTRACT FROM AUTHOR]

This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of normalized solutions to the Kirchhoff equation with L 2 -subcritical or critical nonlinearities - a + b ∫ R 2 | ∇ u | 2 d x Δ u = λ u + | u | p - 2 u + μ | u | q - 2 u in R 2 , where a , b , μ > 0 , 2 < q < p ≤ 6 . By minimizing methods and the concentration compactness principle, we prove the existence and nonexistence of normalized solutions when (p, q) belongs to a certain domain in R 2 , and discuss how μ affects the existence of normalized solutions. Our main results may be illustrated by the red areas and green areas shown in Figure 1. Our results partially extend the results of Soave (J. Differ. Equ. 2020). [ABSTRACT FROM AUTHOR]

In this paper, we deal with the boundary value problem - Δ u = | u | 4 / (n - 2) u / [ ln (e + | u |) ] ε in a bounded smooth domain Ω in R n , n ≥ 3 with homogenous Dirichlet boundary condition. Here ε > 0 . Clapp et al. (J Differ Equ 275:418–446, 2021) built a family of solution blowing up if n ≥ 4 and ε small enough. They conjectured in their paper the existence of sign changing solutions which blow up and blow down at the same point. Here we give a confirmative answer by proving that our slightly subcritical problem has a solution with the shape of sign changing bubbles concentrating on a stable critical point of the Robin function for ε sufficiently small. [ABSTRACT FROM AUTHOR]

The article is intend to study the asymptotic behavior of the large solutions near the boundary to the following infinity Laplace equation ▵ ∞ h u = b (x) f (u) , x ∈ Ω , u | ∂ Ω = + ∞ , where ▵ ∞ h u : = | D u | h - 3 ⟨ D 2 u D u , D u ⟩ for all h > 1 , Ω is a bounded domain with smooth boundary in R N , b ∈ C (Ω ¯) which is positive in Ω , and f ∈ C 1 (0 , ∞) is positive increasing. The main feature of this paper is that the nonlinearity f is regularly varying at infinity with the critical index h. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of fractional Laplacian problems involving fractional q i (x) -laplacian operators (i = 1 : 2) , and a singular nonlinearity. By using variational methods and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces, we prove the existence of solutions for such problems. An illustrative example is presented to validate the main results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [ABSTRACT FROM AUTHOR]

In this present paper, we concern investigating nonlinear Kirchhoff-type problems subject to Dirichlet boundary conditions, incorporating nonlocal terms and logarithmic nonlinearity in the ϕ -Hilfer fractional spaces with the η (·) -Laplacian operator by means of the do Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and uniqueness results for a class of non-coercive Dirichlet elliptic problems whose model example is - div (1 (m - u) β (1 + | u |) q | ∇ u | p - 2 ∇ u + c (x) | u | p - 2 u sin (u - m)) + g (u) = f in Ω , u = 0 on ∂ Ω , where Ω is a bounded open subset of R N (N ≥ 2) , 1 < p < N , m > 0 , 0 < β < 1 , q > 0 , |c| belongs to L N p - 1 (Ω) and g is a continuous function in R which satisfies the sign condition and the data f belongs to L 1 (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and a superlinear reaction which need not satisfy the Ambrosetti–Robinowitz condition. By using variational tools together with truncation and comparison techniques and critical groups, we show the existence of at least five nontrivial smooth solutions, all with sign information: two positive, two negative and a nodal (sign-changing). [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]

This paper is concerned to studying the quasilinear Schrödinger equation: - Δ u + V (x) u - γ 2 Δ (u 2) u = | u | p - 2 u , x ∈ R N , where V(x) is a given potential, γ > 0 and either p ∈ (2 , 2 ∗) , 2 ∗ = 2 N N - 2 for N ⩾ 4 or p ∈ (2 , 4) for N = 3 . We establish the existence of arbitrary multiple nodal solutions for the above equations. [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]

In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities $ a_{i}(u) $ a i (u) that satisfy $ \lim _{s\rightarrow +\infty }a_{i}(s)=0 $ lim s → + ∞ a i (s) = 0 for all $ i=1,\ldots,N $ i = 1 , ... , N. We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution. [ABSTRACT FROM AUTHOR]

In this paper we study existence and uniqueness of solutions to Dirichlet problems as g (u) - div Du 1 + | D u | 2 = f in Ω , u = 0 on ∂ Ω , where Ω is an open bounded subset of R N ( N ≥ 2 ) with Lipschitz boundary, g : R → R is a continuous function and f belongs to some Lebesgue space. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get solutions for data f merely belonging to L 1 (Ω) and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in L N (Ω) as well as uniqueness if g is increasing. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasi-linear elliptic equation { − d i v (ϕ ( ∣ ∇ u ∣ ) ∇ u) = λ ψ ( ∣ u ∣ ) u + φ ( ∣ u ∣ ) u , i n Ω , u = 0 , o n ∂ Ω , where Ω ⊂ ℝN is a bounded domain, λ > 0 is a parameter. The function ψ(∣t∣)t is the subcritical term, and ϕ(∣t∣)t is the critical Orlicz-Sobolev growth term with respect to φ. Under appropriate conditions on φ, ψ and ϕ, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for λ ∈ (0, λ0), where λ0 > 0 is a fixed constant. [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]

This paper is concerned with the following fractional electromagnetic Kirchhoff equation with competing potentials and critical nonlinearity a ε 2 s + b ε 4 s - 3 [ u ] A / ε 2 (- Δ) A / ε s u + V (x) u = f (| u | 2 ) u + K (x) | u | 2 s ∗ - 2 u in R 3 , where ε > 0 is a small parameter, A ∈ C 0 , α (R 3 , R 3) with exponent α ∈ (0 , 1 ] , (- Δ) A / ε s is the fractional magnetic operator with s ∈ (3 4 , 1) , 2 s ∗ = 6 3 - 2 s is the fractional critical exponent, and a , b > 0 are fixed constants. Assuming that V, K and f satisfy some suitable conditions, we establish the multiplicity and concentration of solutions by variational methods and Ljusternik–Schnirelmann theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and multiplicity of normalized solutions for the following Kirchhoff equation, P - a + b ∫ R 3 | ∇ u | 2 d x Δ u - λ u = f (u) , in R 3 , ∫ R 3 | u | 2 d x = c , where a, b, c are positive constants and λ ∈ R is an unknown parameter that appears as a Lagrange multiplier. Two normal solutions, manifesting as a local minimizer or mountain pass solution, have been obtained under the mass subcritical conditions on the nonlinearity f and some suitable mass c. Additionally, we employ the Symmetric Mountain Pass Theorem to establish the multiplicity of normalized solutions for problem (P). To the best of our knowledge, we extend and complement the research success in the Kirchhoff equation for a general nonlinearity with weaker subcritical mass growth. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of uniform W 1 , np n - p - and W 2 , p -estimates for periodic homogenization problems of fully nonlinear elliptic equations. We establish sharp, global, large-scale estimates under the Dirichlet boundary conditions. The main novelty of this paper can be found in the characterization of the size of the "effective" Hessian and gradient of viscosity solutions to homogenization problems. Moreover, the large-scale estimates work in a large class of non-convex problems. It should be stressed that our global estimates are new even for the standard problems without homogenization. [ABSTRACT FROM AUTHOR]

In this paper, we provide an inhomogeneous version of the Carleson estimate for quasilinear elliptic equations with g-Laplace type growth and unbounded right-hand side. We use this result to extend exponential growth theorems in cylindrical unbounded domains proven by Berestycki et al. (Duke Math J 81:467–494, 1996). We finish this paper showing a boundary Harnack-type inequality. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the the Schrödinger-Newton system with L2-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical points of V(x) under certain assumptions on asymptotic behavior of V(x) and its first derivatives near these points. Especially, the critical points of V(x) in this paper must be degenerate. The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term. [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]

In this paper, we consider a class of asymptotically linear damped vibration problems with resonance at infinity. Compared with the existing results, under this resonance condition the functional corresponding to the problem may not satisfy the compactness condition. By combining the penalized functional technique, Morse theory and two critical point theorems, we obtain the existence and multiplicity of nontrivial periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity for the following Schrödinger–Poisson equation - Δ u + λ u - κ (| x | - 1 ∗ | u | 2) u = f (u) , in R 3 , u > 0 , ∫ R 3 u 2 d x = a 2 , where a > 0 is a prescribed mass, κ ∈ R \ { 0 } and λ ∈ R is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case κ < 0 , we obtain the normalized ground state solution for a > 0 small by working on the Pohozaev manifold, where f satisfies the L 2 -supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy c a is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for κ > 0 and f (u) = | u | 4 u , we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the L 2 -constraint setting. [ABSTRACT FROM AUTHOR]

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L ∞ and in the Musielak–Orlicz Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

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 analyze the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

This paper is devoted to deriving the multiplicity result of solutions to the nonlinear elliptic equations of Kirchhoff–Schrödinger type on a class of a nonlocal Kirchhoff coefficient which slightly differs from the previous related works. More precisely, the main purpose of this paper, under the various conditions for a nonlinear term, is to show that our problem has a sequence of infinitely many small energy solutions. In order to obtain such a multiplicity result, the dual fountain theorem is used as the primary tool. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two-dimensional capillary-gravity water waves of small amplitude, which propagate over a flat bed. We prove the existence of a local curve of solutions by using the Crandall–Rabinowitz local bifurcation theory, and show the uniqueness for the capillary-gravity water waves. Furthermore, we recover the dispersion relation for the constant vorticity setting. Moreover, we present a formal stability result for the bifurcation of the laminar solution. In addition, we prove the analyticity of the free surface. [ABSTRACT FROM AUTHOR]

In this paper, we study the discrete nonlinear Schrödinger equation - Δ u + V (x) - ρ (| x | 2 + 1) u = f (x , u) , u ∈ ℓ 2 (Z N) , where N ≥ 3 , V is a bounded periodic potential and 0 lies in a spectral gap of the Schrödinger operator - Δ + V . The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is the lack of compactness of the Cerami sequence. We overcome these two major difficulties by the generalized linking theorem and Lions lemma. This enables us to establish the existence and asymptotic behavior of ground state solutions for small ρ ≥ 0 . [ABSTRACT FROM AUTHOR]

In a bounded open domain $ \Omega \subset \mathbb {R}^{N} $ Ω ⊂ R N , where $ N\geq 2 $ N ≥ 2 , with Lipschitz boundary $ \partial \Omega $ ∂Ω , we consider the Dirichlet problem for the elliptic system given by \[ \left \{ \begin{array}{@{}ll} \displaystyle-\sum^{N}_{i=1}D_{i}\left(a_{i}(x,u(x),D_{i}u(x))\right)+F(x,u)=f(x), & x\quad \in \Omega,\\ u(x)=0, & x\quad \in \partial \Omega, \end{array} \right. \] { − ∑ i = 1 N D i (a i (x , u (x) , D i u (x))) + F (x , u) = f (x) , x ∈ Ω , u (x) = 0 , x ∈ ∂Ω , here, $ u:\Omega \rightarrow \mathbb {R}^{n} $ u : Ω → R n , $ n\geq 2 $ n ≥ 2 , represents a vector-valued function, $ D_{i}u=\frac {\partial u}{\partial x_{i}} $ D i u = ∂ u ∂ x i denotes the partial derivative of u with respect to $ x_i $ x i , and the vector fields $ a_i:\Omega \times \mathbb {R}^{n}\times \mathbb {R}^{n} \rightarrow \mathbb {R}^{n} $ a i : Ω × R n × R n → R n and $ F:\Omega \times \mathbb {R}^n\rightarrow \mathbb {R}^n $ F : Ω × R n → R n are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and $ L^{m} $ L m data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to $ L^{m}(\Omega ;\mathbb {R}^{n}) $ L m (Ω ; R n) with 1