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This paper is concerned with the Schrödinger–Poisson–Slater equation involving the Coulomb–Sobolev exponent. We apply the concentration compactness principle and the Pohožaev-type identity to overcome loss of compactness caused by the Coulomb exponent and obtain a ground state solution, which generalizes and improves some recent results in the literature. [ABSTRACT FROM AUTHOR]

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n ≥ 3 and 1 < p ≤ n + 2 n - 2 , there exists a best constant S n , p , λ (B n) > 0 such that S n , p , λ (B n) ∫ B n | u | p + 1 d v B n 2 p + 1 ≤ ∫ B n | ∇ B n u | 2 - λ u 2 d v B n , holds for all u ∈ C c ∞ (B n) , and λ ≤ (n - 1) 2 4 , where (n - 1) 2 4 is the bottom of the L 2 -spectrum of - Δ B n. It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635–671, 2008) that under appropriate assumptions on n, p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S n , p , λ (B n). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in R n by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18–24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780–6797, 2018) to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744–789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial-boundary value problem for finitely degenerate pseudo-parabolic equation associated with the Hörmander type operator. We first specify the conditions under which the weak solutions of the equation exist globally and finite time blow up when the initial energy J (u 0) > d . Especially, the life span of the blow-up solution and the blow-up result with arbitrarily high initial energy level are derived. Additionally, we show the solution of degenerate pseudo-parabolic problem will converge to the steady state solution. Finally, we obtain the existence of ground state solution to the stationary problem. We prove that there exists initial value such that the instability occurs as a blow-up in finite time and the corresponding solution exists globally, respectively. [ABSTRACT FROM AUTHOR]

We discuss the existence and the Morse index of solutions to the problem Δ U + λ e U = 0 in Ω R , U = 0 on ∂ Ω R , where Ω R is a tubular domain in the plane with fixed width. We obtain the existence of an increasing sequence R k → ∞ as k → ∞ and a corresponding sequence of solutions { U R k } k ≥ 1 to the above problem. We investigate the energy of such solutions and show that their Morse index m (U R k) satisfies m (U R k) / R k → 2 - η 1 as k → ∞ , where η 1 < 0 is the first eigenvalue of the linearized 1D Gel’fand problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the following asymptotically critical Choquard equation 0.1 - Δ u = ∫ Ω u 2 α ∗ - ε (y) | x - y | α d y u 2 α ∗ - 1 - ε , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where Ω is a smooth bounded domain in R N , N ≥ 3 , α ∈ (0 , N) , 2 α ∗ = 2 N - α N - 2 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and ε > 0 is a small parameter. We prove that any family of solutions to (0.1), under a suitable constrain on their gradients, blows up exactly at a critical point of the Robin function. Conversely, by a reduction argument, we construct a family of solutions to (0.1) concentrating around a critical point of the Robin function as ε → 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the normalized solutions for the following fractional Schrödinger equation with combined nonlinearities { (- Δ) s ⁢ u = λ ⁢ u + μ ⁢ | u | q - 2 ⁢ u + | u | p - 2 ⁢ u in ⁢ ℝ N , ∫ ℝ N u 2 ⁢ 푑 x = a 2 , where 0 < s < 1 , N > 2 ⁢ s , 2 < q < p = 2 s * = 2 ⁢ N N - 2 ⁢ s , a , μ > 0 and λ ∈ ℝ is a Lagrange multiplier. Since the existence results for p < 2 s * have been proved, using an approximation method, that is, let p → 2 s * , we obtain several existence results. Moreover, we analyze the asymptotic behavior of solutions as μ → 0 and μ goes to its upper bound. [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]

Let u ≥ - 1 be a solution to the semilinear elliptic equation - Δ u = f (u) in R N such that lim x N → - ∞ u (x ′ , x N) = - 1 uniformly in x ′ ∈ R N - 1 , lim t → + ∞ inf x N > t u (x) > - 1 , and u is bounded in each half-space { x N < λ } , λ ∈ R . Here f : [ - 1 , + ∞) → R is a locally Lipschitz continuous function which satisfies some mild assumptions. We show that u is strictly monotonically increasing in the x N -direction. Under some further assumptions on f, we deduce that u depends only on x N and it is unique up to a translation. In particular, such a solution u to the problem Δ u = u + 1 in R N must have the form u (x) ≡ e x N + α - 1 for some α ∈ R . [ABSTRACT FROM AUTHOR]

In this article, we study the following Choquard equation: −Δu+u=(Iα⋆u2)u,x∈R3,-\Delta u+u=\left({{\rm{I}}}_{\alpha }\star {u}^{2})u,\hspace{1.0em}x\in {{\mathbb{R}}}^{3}, where Iα{{\rm{I}}}_{\alpha } is the Riesz potential and α\alpha is sufficiently close to 2. By investigating the limit profile of ground states of the equation as α→2\alpha \to 2, we prove the uniqueness and nondegeneracy of ground states.

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of δ -type at the unique graph-vertex. Additionally, we extend the linear instability result in Angulo and Cavalcante (Nonlinearity 34:3373–3410, 2021) to one of nonlinear instability. For the proof of local well posedness theory, the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class C 2 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following free boundary problem: \[ (P)\begin{cases} \Delta u = \lambda \phi(x)\sum_{i=1}^n H(u-\mu_i) & \quad \mbox{in}\ \Omega=\Omega_2\setminus \bar{\Omega}_1, \\ u =0 & \quad \mbox{on}\ \partial \Omega_2, \\ u =M & \quad \mbox{on}\ \partial \Omega_1. \end{cases}\] (P) { Δu = λϕ (x) ∑ i = 1 n H (u − μ i) in Ω = Ω 2 ∖ Ω ¯ 1 , u = 0 on ∂ Ω 2 , u = M on ∂ Ω 1. The domain Ω is a convex ring where $ \Omega _1 $ Ω 1 and $ \Omega _2 $ Ω 2 are bounded convex domains in $ \mathbb {R}^N $ R N , $ N\geq 2 $ N ≥ 2 , H is the Heaviside step function, $ \lambda, M,\mu _i $ λ , M , μ i $ (i=1,..,n) $ (i = 1 ,.. , n) are given positive real parameters and ϕ is a given function. We show that under suitable conditions, there exists a solution to problem (P) with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function ϕ which shows that there are more solutions. [ABSTRACT FROM AUTHOR]

Our main goal is to investigate supercritical Hardy–Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term. [ABSTRACT FROM AUTHOR]

In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler–Lagrange equation whose graphs produce a foliation. Then the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory. [ABSTRACT FROM AUTHOR]

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown. [ABSTRACT FROM AUTHOR]

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray–Schauder topological degree involving perturbation argument. [ABSTRACT FROM AUTHOR]

On a smooth, closed Einstein manifold (M, g) of dimension n ≥ 3 with positive scalar curvature and not conformally diffeomorphic to the standard sphere, we prove that the only conformal metrics to g with constant Q-curvature of order 4 are the metrics λ g with λ > 0 constant. [ABSTRACT FROM AUTHOR]

We study boundary value problems with measure data in smooth bounded domains Ω , for semilinear equations. Specifically we consider problems of the form - L V u + f (u) = τ in Ω and tr V u = ν on ∂ Ω , where L V = Δ + V , f ∈ C (R) is monotone increasing with f (0) = 0 and tr V u denotes the measure boundary trace of u associated with L V . The potential V ∈ C 1 (Ω) typically blows up at a set F ⊂ ∂ Ω as dist (x , F) - 2 . In general the above boundary value problem may not have a solution. We are interested in questions related to the concept of 'reduced measures', introduced in Brezis et al. (Ann Math Stud 163:55–109, 20072007) for V = 0 . Our results extend results of [4] and Brezis and Ponce (J Funct Anal 229(1):95–120, 2005) and apply to a larger class of nonlinear terms f. In the case of signed measures, some of the present results are new even for V = 0 . [ABSTRACT FROM AUTHOR]

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system \[ \left\{ \begin{array}{ll} & -\Delta u+u=vg(u) \quad {\rm in}\ \mathbb R^N, \\ & -\Delta v+v=G(u) \quad {\rm in}\ \mathbb R^N, \end{array} \right. \] { − Δ u + u = vg (u) in R N , − Δv + v = G (u) in R N , where $ N\geq 3 $ N ≥ 3 , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential \[ -\Delta u+ u=\left(\mathfrak{g}_2\star G(u)\right) g(u) \quad {\rm in}\ \mathbb R^N, \] − Δ u + u = (g 2 ⋆ G (u)) g (u) in R N , where $ \mathfrak {g}_2 $ g 2 is the fundamental solution of the linear operator $ -\Delta + I $ − Δ + I on $ \mathbb R^N $ R N . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method. [ABSTRACT FROM AUTHOR]

In this paper we consider semilinear equations - Δ u = f (u) with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution u has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument. [ABSTRACT FROM AUTHOR]

In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class C 2 rather than C 3 , particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper studies the upscaling of an elliptic problem with a highly oscillating quasilinear matrix coefficient, a quasilinear term, and a semilinear term in domains periodically perforated with holes of critical size. A Signorini boundary condition is imposed on the boundary of the holes, while a Dirichlet boundary condition is prescribed on the exterior boundary. Using the periodic unfolding method, we obtain an obstacle problem with a nonnegativity spreading effect. [ABSTRACT FROM AUTHOR]

In this paper, we extend the results of Brézis and Willem (J Funct Anal 255:2286–2298, 2008) to the equation with single or double weighted critical exponents, including Hardy–Sobolev, Sobolev and Hénon–Sobolev exponents. More precisely, we establish the existence or nonexistence of equation with different coefficient which has an important impact. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of the following 2-D Schrödinger–Poisson equation with critical exponential growth - ε 2 Δ u + V (x) u + ε - α (I α ∗ | u | q ) | u | q - 2 u = f (u) , where ε > 0 is a parameter, I α is the Riesz potential, 0 < α < 2 , V ∈ C (R 2 , R) , and f ∈ C (R , R) satisfies the critical exponential growth. By variational methods, we first prove the existence of ground state solutions for the above system with the periodic potential. Then we obtain that there exists a positive ground state solution of the above system concentrating at a global minimum of V in the semi-classical limit under some suitable conditions. Meanwhile, the exponential decay of this ground state solution is detected. Finally, we establish the multiplicity of positive solutions by using the Ljusternik–Schnirelmann theory. [ABSTRACT FROM AUTHOR]

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : $$\begin{align*} & \inf \left\{\vphantom{\frac{|u|^{2^*_s}}{|x|^s}} \int_\Omega |\nabla u|^2 \,{\rm d}x - \lambda_1 \int_\Omega \frac{u^2}{|x-P_1|^2}\,{\rm d}x \right. \\ & \quad - \left.\lambda_2 \int_\Omega \frac{u^2}{|x-P_2|^2}\,{\rm d}x\ \middle| \ u \in H^1_0(\Omega), \ \int_\Omega \frac{|u|^{2^*_s}}{|x|^s}\,{\rm d}x=1 \right\} \end{align*}$$ inf { | u | 2 s ∗ | x | s ∫ Ω | ∇ u | 2 d x − λ 1 ∫ Ω u 2 | x − P 1 | 2 d x − λ 2 ∫ Ω u 2 | x − P 2 | 2 d x | u ∈ H 0 1 (Ω) , ∫ Ω | u | 2 s ∗ | x | s d x = 1 } where $ N \geq ~3 $ N ≥ 3 , Ω is a smooth domain, $ \lambda _1, \lambda _2 \in \mathbb {R} $ λ 1 , λ 2 ∈ R , $ 0, P_1, P_2 \in \Omega $ 0 , P 1 , P 2 ∈ Ω , $ s \in (0,2) $ s ∈ (0 , 2) and $ 2^*_s = \frac {2(N-s)}{N-2} $ 2 s ∗ = 2 (N − s) N − 2 . Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems. [ABSTRACT FROM AUTHOR]

We look for solutions to the Schrödinger equation - Δ u + λ u = g (u) in R N coupled with the mass constraint ∫ R N | u | 2 d x = ρ 2 , with N ≥ 2 . The behaviour of g at the origin is allowed to be strongly sublinear, i.e., lim s → 0 g (s) / s = - ∞ , which includes the case g (s) = α s ln s 2 + μ | s | p - 2 s with α > 0 and μ ∈ R , 2 < p ≤ 2 ∗ properly chosen. We consider a family of approximating problems that can be set in H 1 (R N) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of H 1 (R N) , we prove the existence of infinitely, many solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e. 0.2 { - (Δ x u + (α + 1) 2 | x | 2 α Δ y u) = u Q + 2 Q - 2 + λ u log u 2 , u = 0 on ∂ Ω , where (x , y) ∈ Ω ⊂ R N = R m × R n with m ≥ 1 , n ≥ 0 , Ω ∩ { x = 0 } ≠ ∅ is a bounded domain, the parameter α ≥ 0 and Q = m + n (α + 1) denotes the "homogeneous dimension" of R N . When λ = 0 , we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for λ ∈ R \ { 0 } , we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions. [ABSTRACT FROM AUTHOR]

This paper investigates the mathematical programming formulation of semilinear elliptic optimal control problems with finitely many state constraints, this allows the use of results in parametric mathematical programming. By applying recent stability results in parametric mathematical programming, we will obtain some new results on differential stability and tilt stability for parametric control problems. On the one hand, we derive an explicit upper estimate for regular subdifferential of marginal function of control problems under basic parameter perturbations. On the other hand, we establish a characterization of tilt stability of control problems under tilt parameter perturbations. [ABSTRACT FROM AUTHOR]

We investigate a class of the nonlinear Schrödinger equation in R N - Δ u + V (x) u = | u | 2 ∗ - 2 u + λ | u | p - 2 u , where N ≥ 3 , λ > 0 and p ∈ (2 , 2 ∗) with 2 ∗ = 2 N N - 2 . Here, V (x) = V 1 (x) for x 1 > 0 and V (x) = V 2 (x) for x 1 < 0 , where V 1 , V 2 are periodic in each coordinate direction. By providing a splitting Lemma corresponding to non-periodic external potential, we obtain the existence of ground state solution for the above problem. It is worth to mention that the arguments used in this paper are also valid for the Sobolev subcritical problem studied by Dohnal et al. (Commun Math Phys 308:511–542, 2011). [ABSTRACT FROM AUTHOR]

We consider periodic solutions of the following nonlinear equation associated with the fractional Laplacian \[ (-\Delta)^s {u}(x)+V(x){u}(x)= {f(x,u)}, \quad x\in \mathbb{R}, \] (− Δ) s u (x) + V (x) u (x) = f (x , u) , x ∈ R , where $ V(x)\in C(\mathbb R) $ V (x) ∈ C (R) is T-periodic and 0 belongs to a spectral gap of the operator $ (-\Delta)^s+V $ (− Δ) s + V. We obtain the existence of a periodic solution by using a generalized linking theorem. [ABSTRACT FROM AUTHOR]

In this paper, we establish the monotonicity of the period map in terms of the energy levels for certain periodic solutions of the equation - φ ′ ′ + φ - φ k = 0 , where k > 1 is a real number. We present a new approach to demonstrate this property, utilizing spectral information of the corresponding linearized operator around the periodic solution and tools related to Floquet theory. [ABSTRACT FROM AUTHOR]

This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: −Δu=α(x)up+κμ,inRN,u>0,inRN,(Pκ)\hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where N≥3N\ge 3, p>1p\gt 1, κ>0\kappa \gt 0, and α\alpha is a positive continuous function in RN\{0}{{\mathbb{R}}}^{N}\setminus \left\{0\right\}, and μ\mu is a nonnegative Radon measure in RN{{\mathbb{R}}}^{N}. Under suitable assumptions on the exponent pp, the coefficient α\alpha , and the forcing term μ\mu , we give a complete classification of the existence/nonexistence of solutions to problem (Pκ{{\rm{P}}}_{\kappa }) with respect to κ\kappa .

We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation −Δu=f(u),u∈D1,2(RN), $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c 0 if N = 5, 6 and smaller than 10c 0 if N ≥ 7, where c 0 is the ground state energy.

The existence of L 2–normalized solutions is studied for the equation −Δu+μu=f(x,u) inRN,∫RNu2dx=m. $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

Let Ω⊂Rn\Omega \subset {{\bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[q\in ]0,1{[}, α∈L∞(Ω)\alpha \in {L}^{\infty }\left(\Omega ), with α>0\alpha \gt 0, and k∈Nk\in {\bf{N}}. Then, the problem −tan∫Ω∣∇u(x)∣2dxΔu=α(x)uqinΩu>0inΩu=0on∂Ω(k−1)π