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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 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 study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ 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 C1-regularity. [ABSTRACT FROM AUTHOR]

We are interested in the following semilinear elliptic problem: - Δ u + λ u = u p - 1 , x ∈ T , u > 0 , u = 0 on ∂ T , ∫ T u 2 d x = c where T = { x ∈ R N : 1 < | x | < 2 } is an annulus in R N , N ≥ 2 , p > 1 is Sobolev-subcritical, searching for conditions (about c, N and p) for the existence of positive radial solutions. We analyze the asymptotic behavior of c as λ → + ∞ and λ → - λ 1 to get the existence, non-existence and multiplicity of normalized solutions. Additionally, based on the properties of these solutions, we extend the results obtained in Pierotti et al. in Calc Var Partial Differ Equ 56:1–27, 2017. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained when N ≥ 3 or if N = 2 and p < 6 . Our paper also includes the demonstration of orbital stability/instability results. [ABSTRACT FROM AUTHOR]

In the present paper, we first establish a version of the abstract lower and upper-solution method for our class of operators. In this sense, we investigated the main objective of this paper, that is, the existence of a positive solution for a new class of fractional systems of the Kirchhoff type with ψ -Hilfer operators via the method of sub and supersolutions in ψ -fractional space H p α , β ; ψ (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of weak nonzero solutions for the following fractional Hamiltonian systems: t D ∞ α - ∞ D t α u (t) - L (t) u + λ u + ∇ W (t , u) = 0 , u ∈ H α (R , R N) , t ∈ R , where α ∈ (1 2 , 1 ] , λ ∈ R , - ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order α on real line R , the matrix L(t) is not necessarily coercive nor uniformly positive definite and W : R × R N → R satisfies some new general and weak conditions. Our results are proved using new symmetric mountain pass theorem established by Kajikia. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results. [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]

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]

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]

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type - d i v ∇ u 1 + | ∇ u | 2 + A (ϵ x , y) V ′ (u) = 0 in R 2 , where ϵ > 0 and V is a double-well potential with minima at t = α and t = β with α < β . Here, we consider some class of functions A(x, y) that are oscillatory in the variable y and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity. [ABSTRACT FROM AUTHOR]

In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including the one-Laplacian, and is perturbed by a p-Laplacian-type diffusion operator with 1 < p < ∞ . This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [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 study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K. [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 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]

Image denoising and edge extraction are two main tasks in image processing. In this paper, a game model is proposed to solve the image denoising and edge extraction, which combines an adaptive improved total variation (AdITV) model for image denoising and a global sparse gradient (GSG) model for edge extraction. The AdITV model is a forward-and-backward diffusion model. In fact, forward diffusion is applied to the homogeneous region to denoise, and backward diffusion is applied to the edge region to enhance the edge. A unified explicit discrete scheme is established in this paper to solve the AdITV model, which is compatible to forward diffusion and backward diffusion. The stability of the scheme is proved. On the other hand, GSG is a functional model based on sparse representation, which is robust to extract edges under the influence of noise. AdITV and GSG are chosen as two components of the game model. The alternate iteration method is used to solve the game problem. The convergence of the algorithm is proved and numerical experiments show the effectiveness of the model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave. [ABSTRACT FROM AUTHOR]

In this paper, we give the detailed proof of some density properties of smooth and compact functions in fractional Orlicz Sobolev spaces and a modifications of them. These spaces are crucial in solving variational problems. Our approach involves a basic convolution technique joined with a cut-off, with some care needed in order not to exceed the original support. [ABSTRACT FROM AUTHOR]

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem for the L 2 supercritical case. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2 , which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in R N. In this paper, we will show that in the exterior domain they are strongly stable. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problems for the Dirichlet fractional (p, q)-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational together with the sub- and supersolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when W 0 s 1 , p (Ω) ↪ L ∞ Ω failing, we establish ‖ u ‖ L ∞ Ω ≤ C [ u ] s 1 , p (for some C > 0 ), when u is a solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following nonlinear Choquard equation with prescribed L 2 -norm constraint - Δ u = λ u + (| x | - 1 ∗ | u | 2) u in Ω , u = 0 on ∂ Ω , ∫ Ω | u | 2 d x = a 2 , where a > 0 , λ ∈ R appears as an unknown Lagrange multiplier and Ω ⊂ R 3 is an exterior domain with smooth boundary ∂ Ω ≠ ∅ such that R 3 \ Ω is bounded. By using the splitting lemma for the unconstrained problem in exterior domains, we prove the compactness of Palais–Smale sequences corresponding to the above problem at higher energy levels. Then combining the barycentric function and Brouwer degree theory, we establish the existence of positive normalized bound states for any a > 0 provided that R 3 \ Ω is contained in a small ball and explain that the restriction on domain Ω can be equivalently transferred to a. In addition, under the radial setting of domain Ω , we use genus theory to obtain the existence and multiplicity of radial normalized solutions for any a > 0 . Finally, we point out that the main results can be extended to a general mass subcritical Choquard equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the following n-Laplacian equation with singular and exponential nonlinearities - Δ n u = λ u - q + u p - 1 e u β | x | α in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R n with smooth boundary ∂ Ω , n ≥ 2 , 0 < q < 1 , p > 2 n , β ∈ 1 , n n - 1 , 0 < α < n and λ > 0 is a parameter. By analyzing the energy functional over the suitable subsets of Nehari manifold, two distinct solutions are obtained. [ABSTRACT FROM AUTHOR]

The main result of this paper is to investigate the existence of a solution of a class of fractional problems involving the operator p-Laplacian with periodic potential and supercritical growth via the Mountain Pass theorem-Cerami version. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions for a p–Kirchhoff type problem driven by a nonlocal integro-differential operator. As a particular case, we consider the following problem: where is the fractional p–Laplacian, with sp

In this paper, we prove the existence of infinitely many solutions for the following nonperiodic fractional Hamiltonian system t D ∞ α (- ∞ D t α u) (t) + L (t) u (t) = ∇ W (t , u (t)) , t ∈ R u ∈ H α (R) , where - ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order 1 2 < α < 1 on the whole axis, respectively, L ∈ C (R , R N 2) is a symmetric matrix valued function unnecessary coercive and W (t , x) ∈ C 1 (R × R N , R) is only locally defined and superquadratic near the origin with respect to x. Our results extend and improve some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the semilinear subelliptic equation where is the self-adjoint Hörmander operator associated with the vector fields X = (Xl, X2, ..., Xm) satisfying the Hörmander's condition, is a Carathéodory function on Ω × ℝ, and Ω is an open bounded domain in ℝn with smooth boundary. Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values, we obtain two kinds of existence results for multiple weak solutions to the problem above. Furthermore, we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations. Compared with the classical elliptic cases, both approaches here have their own strengths in the degenerate cases. This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following logarithmic Schrödinger–Poisson system - Δ u + V (x) u + λ K (x) ϕ u = f (u) + u log u 2 , x ∈ R 3 , - Δ ϕ - ε 4 Δ 4 ϕ = λ K (x) u 2 , x ∈ R 3 , which has increasingly received interest due to the indefiniteness of the energy functional and fourth-order term in Poisson equation. By using variational method, we prove the existence and multiplicity of positive solutions. Finally, we obtain the asymptotic behavior of positive solutions as ε → 0 + and λ → 0 + , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of κ (ξ) -Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space ψ -fractional H κ (ξ) μ , ν ; ψ (Λ) . In this sense, we present a few special cases and remark on the outcomes explored. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of normalized ground states for the following nonlinear fractional Choquard equations with Hardy–Littlewood–Sobolev upper critical exponent: (- Δ) s u = λ u + μ (I α ∗ | u | p ) | u | p - 2 u + (I α ∗ | u | 2 α , s ∗ ) | u | 2 α , s ∗ - 2 u , x ∈ R N , having prescribed mass ∫ R N | u | 2 d x = a > 0 , where N > 2 s , s ∈ (0 , 1) , α ∈ (0 , N) , p ̲ : = N + α N < p < p ¯ : = N + 2 s + α N < 2 α , s ∗ , 2 α , s ∗ : = N + α N - 2 s is the upper Hardy–Littlewood–Sobolev critical exponent, μ > 0 and λ ∈ R . Furthermore, the qualitative behavior of the ground states as μ → 0 + is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of multiple solutions to the nonlinear p-Laplacian equation - Δ p u + h (x) | u | p - 2 u = f (x , u) + g (x) on the locally finite graph G, where Δ p is the discrete p-Laplacian on graphs, p ≥ 2 . Under more general conditions, we prove that the p-Laplacian equation admits at least two nontrivial different solutions by using the variational methods and the new analytical techniques. Our results extend some related works. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a solution to Schrödinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with the Trudinger–Moser nonlinearity. As a particular case, we consider the following fractional problem: M (∬ R 2 N | u (x) − u (y) | p | x − y | N + s p d x d y + ∫ R N V (x) | u (x) | p d x) ((− Δ) p s u (x) + V (x) | u | p − 2 u) = K (x) f (x , u) , where M : R 0 + → R 0 + is a continuous function with some appropriate assumptions, (− Δ) p s is the fractional p-Laplacian, 0 < s < 1 < p < ∞ with sp = N, K, V are positive continuous functions satisfying some additional conditions, f is a continuous function on R N × R with exponential growth. By using the mountain pass theorem, we obtain the existence of solutions to the above problem in suitable Sobolev space. A novel feature of our paper is that the above problem may be degenerate, that is, the Kirchhoff function M (0) = 0. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of the fractional Schrödinger equations involving logarithmic and critical nonlinearities. By using the Nehari manifold method and the concentration compactness principle, we show that the above problem admits at least one ground state solution and one ground state sign-changing solution. Moreover, by using variational methods, we prove that how the coefficient function of the critical nonlinearity affects the number of positive solutions. The main feature which distinguishes this paper from other related works lies in the fact that it is the first attempt to study the existence and multiplicity for the above problem involving both logarithmic and critical nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, by a new variational principle established by Moameni, we establish the existence of solutions to the following fractional elliptic equation with the Hardy potential and concave–convex nonlinearities, (- Δ) s u (x) - λ u (x) | x | 2 s = u (x) p - 1 + μ u (x) q - 1 , x ∈ Ω , u (x) > 0 , x ∈ Ω , u (x) = 0 , x ∈ R N \ Ω , where Ω ⊂ R N is a bounded Lipschitz domain with 0 ∈ Ω , (- Δ) s is the restricted fractional Laplacian, 0 < s < 1 , N > 2 s , 0 < λ < Λ N , s , Λ N , s is the sharp constant of the Hardy–Sobolev inequality, 1 < q < 2 < p < 2 s (N - 2 s ∗ γ) γ N + 2 s ∗ , here 2 s ∗ = 2 N N - 2 s is the critical fractional Sobolev exponent and 0 < γ < min { (N - 2 s) / 2 , N / p } . The main contribution of this paper is giving the explicit expression threshold of p. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following Schrödinger–Poisson system { − Δ u + u + ϕ u = | u | p − 1 u + | u | 4 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where 3