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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]

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

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 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]

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 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]

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]

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 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 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]

In this paper, we consider a class of nonlocal problem with strong singularity and sublinear term in the nonlinearity. The nonlocal term may change sign. Combining the variational method, Nehari manifold and some estimates, a necessary and sufficient condition that shows the existence of positive solution is obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following one-dimensional fractional Choquard equation ε (- Δ) 1 / 2 u + V (x) u = ε μ I μ ∗ F (u) f (u) , x ∈ R , where (- Δ) 1 / 2 denotes the 1/2-Laplacian opertor, I μ is the Riesz potential with 0 < μ < 1 , V is a continuous real function satisfying some mild assumptions and f ∈ C (R , R) is a nonlinearity with exponential critical growth. The present paper has three typical features. Firstly, using a weaker assumption on f, we establish the energy inequality to recover the compactness. Second, without the strictly monotone condition and by establishing some new tricks, we obtain the existence of ground state solutions when ε is small enough. Finally, we take advantage of some refined analysis techniques to get over the difficulty carried by the nonlocality of the 1/2-Laplacian and prove the concentration of the ground state solutions when ε → 0 . Our results extend and complement the results of Alves et al. (J Differ Equ 261:1933–1972, 2016) and Clemente et al. (Z Angew Math Phys 72:16, 2021]. [ABSTRACT FROM AUTHOR]

In this paper, we study the critical fractional Choquard equation with a local perturbation (- Δ) s u = λ u + μ | u | q - 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 2 , where I α (x) is the Riesz potential, s ∈ (0 , 1) , N > 2 s , 0 < α < min { N , 4 s } , 2 < q < 2 s ∗ = 2 N N - 2 s is the fractional critical Sobolev exponent, and 2 α , s ∗ = 2 N - α N - 2 s is the fractional Hardy–Littlewood–Sobolev critical exponent, a > 0 , μ ∈ R. . Under some L 2 -subcritical, L 2 -critical and L 2 -supercritical perturbation μ | u | q - 2 u , respectively, we prove several existence and non-existence results. The qualitative behavior of the ground states as μ → 0 + is also studied. The mathematical analysis carried out in this paper can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions for fractional Choquard equation. In this framework, several related results are extended and improved. [ABSTRACT FROM AUTHOR]

This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case (see Auchmuty and Beals (1971)) to the non-isentropic case. Compared with the previous result of variational rotating solutions in the non-isentropic case (see Wu (2015)), to keep the constraint of prescribed finite total mass, we establish a new variational structure of the non-isentropic Euler-Poisson equations. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider the following nonlinear Schrödinger equation with an L 2 -constraint: - Δ u = λ u + μ | u | q - 2 u + | u | p - 2 u in R N , ∫ R N | u | 2 d x = a 2 , u ∈ H 1 (R N) , where N ≥ 3 , a , μ > 0 , 2 < q < 2 + 4 N < p < 2 ∗ , 2 q + 2 N - p N < 0 and λ ∈ R arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the L 2 sphere, and prove the existence of infinitely solutions with positive energy levels. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Choquard system (- Δ) 1 / 2 u = λ 1 u + (I μ ∗ F (u , v)) F u (u , v) , in R , (- Δ) 1 / 2 v = λ 2 v + (I μ ∗ F (u , v)) F v (u , v) , in R , ∫ R | u | 2 d x = a 2 , ∫ R | v | 2 d x = b 2 , u , v ∈ H 1 / 2 (R) , where (- Δ) 1 / 2 denotes the 1/2-Laplacian operator, a , b > 0 are prescribed, λ 1 , λ 2 ∈ R , I μ (x) = 1 | x | μ with μ ∈ (0 , 1) , F u , F v are partial derivatives of F and F u , F v have exponential critical growth in R . By using a minimax principle and analyzing the monotonicity of the ground state energy with respect to the prescribed masses, we obtain at least one normalized ground state solution for the above system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following critical Schrödinger–Poisson system - Δ u + V (x) u + K (x) ϕ u = f (u) + | u | 4 u , x ∈ R 3 , - Δ ϕ = K (x) u 2 , x ∈ R 3 , where V(x) is a (possible) sign-changing potential, K(x) is a nonnegative function and the nonlinearity f ∈ C (R , R) . By using variational methods with a more general global compactness lemma, we obtain a positive least energy solution and a least energy sign-changing solution with exactly two nodal domains, and we also prove that the energy of least energy sign-changing solution is strictly larger than twice that of least energy solutions. Moreover, this paper further analyzes the exponential decay of the positive least energy solution given by Liu, Liao and Tang (Nonlinearity30 (2017), 899–911), and can be regarded as the supplementary work of it. [ABSTRACT FROM AUTHOR]

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems. [ABSTRACT FROM AUTHOR]

In this paper, motivated by some recent progress in the development of Partial Newton-Correction Method for finding multiple solutions to nonlinear PDEs and by a closer observation to the implementation of the Local Min-Orthogonal (LMO) method developed in 2004, first a new L- ⊥ selection expression is introduced, then the (strict) separation condition and the continuity condition used in the mathematical framework of the LMO method are successively improved or weakened so that they are not only closer to the real algorithm's implementation but also able to improve the relevant analysis. A new step-size rule and a new local characterization on saddle points are then established, based on which an improved LMO method is developed. The results in the paper can be further used to improve analysis of relevant numerical methods for solving multiple-solution problems in nonlinear PDEs. Finally, two numerical examples are carried out to illustrate the effectiveness of the new method and some new numerical findings are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Neumann elliptic problem (P ε) : − Δ u + μ u = u ((n + 2) / (n − 2)) + ε , u > 0 in Ω, ∂ u / ∂ ν = 0 on ∂Ω, where Ω is a smooth bounded domain in R n , n ≥ 4 , ε is a small positive real, and μ is a fixed positive number. We show that, in contrast with the three dimensional case, (P ε) has no solution blowing up at only interior points as ε goes to zero. The proof strategy consists in testing the equation by appropriate vector fields and then using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results. [ABSTRACT FROM AUTHOR]

In this paper, we use variational tools in order to study the existence and the multiplicity of solutions for a nonlinear elliptic problem involving fractional operators. Precisely, we use the Mountain pass theorem to prove the existence of a nontrivial solution. Also, by combining this with the Ekeland variational principle the existence of multiple solutions is proved. Moreover, the existence of infinitely many solutions is given by using the Z 2 -symmetric version for even functionals of the Mountain pass Theorem. To illustrate the usefulness of the main results an illustrative example is presented. [ABSTRACT FROM AUTHOR]

In this paper, we use a variational approach to show the analyticity of solitons (traveling waves of finite energy) and the existence of x-periodic traveling waves for a certain sixth order dispersive equation that arises in the study of the deformations of a hyperelastic compressible plate. We also establish that x-periodic traveling waves have almost the same shape of solitons as the period tends to infinity, by showing that a special sequence of periodic traveling wave solutions parameterized by the period converges to a soliton in a appropriate sense. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence and multiplicity of solutions for a class of singular anisotropic problems involving a weight and a term that may change sign. The approach is based on sub-supersolutions and the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and concentration of solutions for the following 1-biharmonic Choquard equation with steep potential well Δ 1 2 - Δ 1 u + (1 + λ V (x)) u | u | = I μ ∗ F (u) f (u) in R N , u ∈ BL (R N) , where N ≥ 3 , λ > 0 is a positive parameter, V : R N → R , f : R → R are continuous functions verifying further conditions, Ω = int (V - 1 ({ 0 })) has nonempty interior and I μ : R N → R is the Riesz potential of order μ ∈ (N - 1 , N) . For λ > 0 large enough, we prove the existence of a nontrivial solution u λ of the problem above via variational methods and the concentration behavior of u λ which is explored on the set Ω . [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the following Kirchhoff equation where Ω ⊂ R 3 is a bounded connected domain and ∫ Ω | u | 2 d x = 1 . The results of existence and nonexistence on L 2 -norm solutions are given. Our argument shows that the blow-up behavior of L 2 -norm solution occurs, and the mass concentrates at an inner point of Ω , or the neighborhood of some boundary point. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following class of elliptic problems Δ 2 u = λ u + μ | u | p - 2 u + | u | 4 ∗ - 2 u in R N , ∫ R N u 2 d x = c , where N ≥ 5 , μ , c > 0 , 2 + 8 N < p < 4 ∗ : = 2 N N - 4 and λ ∈ R is a Lagrange multiplier. We show the existence of normalized solutions for μ large enough by verifying the (PS) condition at the corresponding mountain-pass level. In this sense, we extend the recent results obtained by Ma and Chang (Appl Math. Lett 135:108388, 2022) to the L 2 -supercritical perturbation. Moreover, we discuss the asymptotic behavior of the energy to the mountain pass solution when c → + ∞ . [ABSTRACT FROM AUTHOR]

In this present paper, we investigate a new class of singular double phase p-Laplacian equation problems with a ψ -Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold. [ABSTRACT FROM AUTHOR]

In this paper we present some existence and multiplicity results for a class of anisotropic p⃗-Laplacian problems with Dirichlet boundary conditions. In particular, the existence of three solutions is pointed out. The approach is based on variational methods and our main tool is a three critical point theorem. [ABSTRACT FROM AUTHOR]