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This paper deals with global compactness and the multiplicity of positive solutions to problems of the type −Δ픹Nu − λu = a(x)|u|2∗−2u + f(x)in 픹N, u ∈ H1(픹N), where 픹N denotes the ball model of the hyperbolic space of dimension N ≥ 4, 2∗ = 2N N−2, N(N−2) 4 < λ < (N−1)2 4 and f ∈ H−1(픹N) (f≢0) is a nonnegative functional in the dual space of H1(픹N). The potential a ∈ L∞(픹N) is assumed to be strictly positive, such that limd(x,0)→∞a(x) = 1, where d(x, 0) denotes the geodesic distance. We establish profile decomposition of the functional associated with the above equation. We show that concentration occurs along two different profiles, namely, hyperbolic bubbles and localized Aubin–Talenti bubbles. For f = 0 and a ≡ 1, profile decomposition was studied in Bhakta and Sandeep [Poincaré–Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations 44 (2012) 247–269]. However, due to the presence of the potential a(.), an extension of profile decomposition to the present set-up is highly nontrivial and requires several delicate estimates and geometric arguments concerning the isometry group (Möbius group) of the hyperbolic space. Further, using the decomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin–Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: a ≥ 1 and a < 1 separately. The equation studied in this paper can be thought of as a variant of a scalar-field equation with a critical exponent in the hyperbolic space, although such a critical exponent problem in the Euclidean space ℝN has only a trivial solution when f ≡ 0, a(x) ≡ 1 and λ < 0. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian L on the Heisenberg group H n with power type nonlinearity | u | p and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space H ˙ L - γ (H n) , γ > 0 , on H n . In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent p crit (Q , γ) = 1 + 4 Q + 2 γ , for γ ∈ (0 , Q 2) , where Q : = 2 n + 2 is the homogeneous dimension of H n . More precisely, we establish A global-in-time existence of small data Sobolev solutions of lower regularity for p > p crit (Q , γ) in the energy evolution space; A finite time blow-up of weak solutions for 1 < p < p crit (Q , γ) under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case. [ABSTRACT FROM AUTHOR]

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: − M ∫ Ω ∇ u p d x Δ p u = u p ∗ − 2 u + λ u p − 2 u − u p − 2 u ln u 2           x ∈ Ω ,                                                                         u = 0                                                                                   x ∈ ∂ Ω ,                                   where Ω ⊂ ℝ N is a bounded domain with a smooth boundary, 2 < p < p ∗ < N , and both p and N are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial-boundary value problem of the doubly dispersive quasilinear wave equation u tt - div (| ∇ u | p - 2 ∇ u) + Δ 2 u - Δ u tt = | u | q - 2 u log | u | in Ω × (0 , T max) , where Ω is an open bounded domain in R n with smooth boundary; T max (≤ + ∞) denotes the maximal existence time; p , q > 2 are constants. We denote q = p the critical exponent for blow-up solutions. For q < p , we prove that all the weak solutions are globally bounded even if the initial energy is negative. For q ≥ p , we obtain the optimal classification of initial data on the existence of global and blow-up solutions, which is divided into the subcritical, critical, and super critical initial energy in the framework of potential well. By constructing new auxiliary functions, we obtain the upper bounds of blow-up time for different norms. [ABSTRACT FROM AUTHOR]

This paper dealt with a class of Kirchhoff type equations involving singular nonlinearity in a closed Riemannian manifold (M; g) of dimension n ≥ 3. Existence and uniqueness of a positive weak solution were obtained under certain assumptions with the help of the variation methods and some analysis techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new kind of electromagnetostatic Schrödinger–Maxwell system with critical exponent and prove the existence of vortex ground state solutions with cylindrically symmetry using the Nehari manifold approach combining with the Brezis–Nirenberg technique. [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]

In this paper, we establish the existence of a nonnegative nontrivial weak solution for a fractional critical (p, q)-Laplacian problem with discontinuous nonlinearity. The approach is based on suitable variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of positive solutions for the following coupled elliptic system with critical exponent and logarithmic terms: - Δ u = λ 1 u + μ 1 | u | 2 u + β | v | 2 u + θ 1 u log u 2 , x ∈ Ω , - Δ v = λ 2 v + μ 2 | v | 2 v + β | u | 2 v + θ 2 v log v 2 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , where Ω ⊂ R 4 is a bounded smooth domain, the parameters λ 1 , λ 2 , θ 1 , θ 2 ∈ R , μ 1 , μ 2 > 0 and β ≠ 0 is a coupling constant. Note that the logarithmic term s log s 2 has special properties, which makes the problem more complicated. We show that this system has a positive least energy solution for | β | small and positive large β if λ 1 , λ 2 ∈ R and θ 1 , θ 2 > 0 . While the situations for the case θ 1 , θ 2 < 0 are quite thorny, in this challenging setting we establish the existence result of positive local minimum solutions and nonnegative solutions under various conditions on the parameters. Besides, under some further assumptions, we obtain the nonexistence of positive solutions for both the case where θ 1 , θ 2 are negative and the case where they have opposite signs. Comparing our results with those of Chen and Zou (Arch. Ration. Mech. Anal. 205:515–551, 2012), the logarithmic term s log s 2 introduces some new interesting phenomenon. Moreover, its presence brings major challenges and make it difficult to use the comparison theorem used in the work of Chen and Zou without new ideas and innovative techniques. To the best of our knowledge, our paper is the first to give a rather complete picture for the existence and nonexistence results to the coupled elliptic system with critical exponent and logarithmic terms. Also, we consider the related single equation - Δ u = λ u + μ | u | 2 u + θ u log u 2 , u ∈ H 0 1 (Ω) with μ > 0 , θ < 0 , λ ∈ R or λ ∈ [ 0 , λ 1 (Ω)) and prove the existence of the positive solution under some further suitable assumptions, which is the type of a local minimum or a least energy solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $$\begin{align*} \left \{ \begin{array}{@{}l@{}} \displaystyle -\left(a+b\int_{R^{N}}|\nabla u|^{2}\,{\rm d}x\right)\Delta u + V(x)u =P(x)|u|^{2^*-2}u+\mu|u|^{q-2}u, \ {\rm in}\ \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N) \end{array} \right. \end{align*}$$ { − (a + b ∫ R N | ∇u | 2 d x) Δu + V (x) u = P (x) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 (R N) where $ a, b, \mu \gt 0 $ a , b , μ > 0 , $ N\geq 3 $ N ≥ 3 , $ \max \{2^*-1, 2\} \lt q \lt 2^*=\frac {2N}{N-2} $ max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , $ V(x) \gt 0 $ V (x) > 0 and $ P(x)\geq 0 $ P (x) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P). [ABSTRACT FROM AUTHOR]

In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms: - Δ u = λ 1 u + μ 1 | u | 2 p - 2 u + β | u | p - 2 | v | p u + θ 1 u log u 2 , x ∈ Ω , - Δ v = λ 2 v + μ 2 | v | 2 p - 2 v + β | u | p | v | p - 2 v + θ 2 v log v 2 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, 2 p = 2 ∗ = 2 N N - 2 is the Sobolev critical exponent. When N ≥ 5 , for different ranges of β , λ i , μ i , θ i , i = 1 , 2 , we obtain existence and nonexistence results of positive solutions via variational methods. The special case N = 4 was studied by Hajaiej et al. (Positive solution for an elliptic system with critical exponent and logarithmic terms, arXiv:2304.13822, 2023). Note that for N ≥ 5 , the critical exponent is given by 2 p ∈ 2 , 4 ; whereas for N = 4 , it is 2 p = 4 . In the higher-dimensional cases N ≥ 5 brings new difficulties, and requires new ideas. Besides, we also study the Brézis–Nirenberg problem with logarithmic perturbation - Δ u = λ u + μ | u | 2 p - 2 u + θ u log u 2 in Ω , where μ > 0 , θ < 0 , λ ∈ R , and obtain the existence of positive local minimum and least energy solution under some certain assumptions. [ABSTRACT FROM AUTHOR]

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α , i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we would like to consider the Cauchy problem for semilinear σ$$ \sigma $$‐evolution equations with time‐dependent damping for any σ≥1$$ \sigma \ge 1 $$. Motivated strongly by the classification of damping terms in some previous papers, the first main goal of the present work is to make some generalizations from σ=1$$ \sigma &#x0003D;1 $$ to σ>1$$ \sigma >1 $$ and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so‐called effective damping cases. For the next main goals, we are going not only to prove the global well‐posedness property of small data solutions but also to indicate blow‐up results for solutions to the semilinear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow‐up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ$$ \sigma $$ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following elliptic system involving critical Sobolev and Hardy-Sobolev exponents - Δ u = 2 α 2 ∗ | u | α - 2 u | v | β + | u | 2 ∗ (s) - 2 u | x | s + a (x) | u | q - 2 u in Ω , - Δ v = 2 β 2 ∗ | u | α | v | β - 2 v + | v | 2 ∗ (s) - 2 v | x | s + b (x) | v | q - 2 v in Ω , u = v = 0 on ∂ Ω , where Ω is a bounded domains in R N satisfying some geometric condition, N ≥ 3 , 0 < s < 2 , 1 < q < 2 , 2 ∗ (s) : = 2 (N - s) (N - 2) and α , β > 1 such that α + β = 2 ∗ : = 2 ∗ (0) . Our main result asserts that, if N > max (4 , ⌊ 2 s ⌋ + 2) , then our problem has two disjoint and infinite sets of solutions. The present work may be seen as a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in Ambrosetti et al. (J Funct Anal 122:519–543, 1994) for an elliptic systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the multiplicity of solutions for the fractional Kirchhoff type equations with a critical nonlocal term M ‖ u ‖ Z 2 L K u = λ f (x , u) + ∫ Ω | u (y) | 2 α , s ∗ | x - y | α d y | u | 2 α , s ∗ - 2 u , x ∈ Ω , u (x) = 0 , x ∈ R N \ Ω , where L K is an integrodifferential operator with kernel K , Ω is an open bounded subset of R N , M and f are continuous functions. ‖ · ‖ Z is a functional norm and 2 α , s ∗ = 2 N - α N - 2 s is the Hardy-Littlewood-Sobolev critical exponent with N > 2 s , s ∈ (0 , 1) and α ∈ (0 , N) . We first deal with the case that, function f satisfies some superlinear and quasi-critical growth conditions, in which case we prove that, for any given k ∈ N , there exists at least k pairs of nontrivial solutions for λ large enough; while in the case that f is sublinear growth, we obtain the existence of infinitely many solutions which tend to zero under a suitable small value of λ . To establish these multiplicity results, we use a new concentration-compactness principle dealing with nonlocal critical problems from the paper (X. He, X. Zhao, W. Zou, The Benci-Cerami problem for the fractional Choquard equation with critical exponent, Manuscript Math. 2022), together with Kajikiya's new version of the symmetric mountain pass lemma and the truncation skill. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of positive solutions for a p(x)-Kirchhoff problem with singularity and critical exponents. Based on the variational methods and perturbation technique, we prove the existence of at least two positive solutions for above problem. The main feature and difficulty of this paper is the presence of the singular and critical terms. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

In this paper, we derive suitable optimal L p - L q decay estimates, 1 ≤ p ≤ 2 ≤ q ≤ ∞ , for the solutions to the σ -evolution equation, σ > 1 , with scale-invariant time-dependent damping and power nonlinearity | u | p , u tt + (- Δ) σ u + μ 1 + t u t = | u | p , t ≥ 0 , x ∈ R n , where μ > 0 , p > 1 . The critical exponent p = p c for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly μ ∈ (0 , 1) or μ > 1 . Under the assumption of small initial data in L m (R n) ∩ L 2 (R n) , m = 1 , 2 , we find the critical exponent at low space dimension n with respect to σ , namely, p c = max p ¯ (γ m) , p ¯ (γ m + μ - 1) , γ m : = n m σ , μ > 1 - γ m , where p ¯ (γ) : = 1 + 2 γ is the well known Fujita exponent. Hence, p c = p ¯ (γ m) if μ > 1 , whereas p c = p ¯ (γ m + μ - 1) is a shift of Fujita type exponent if μ ∈ (0 , 1) . [ABSTRACT FROM AUTHOR]

The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen über Mechanik, Leipzig, Teubner, 1883] as an extension of the classical D'Alembert's wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions ["On some questions in boundary value problems of mathematical physics," in Contemporary Developments in Continuum Mechanics and PDE's, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as ∂ 2 u ∂ t 2 + a + b ∫ Ω | ∇ u | 2 d x Δ u = f (x , u) , where Δ = − ∑ ∂ 2 ∂ x i 2 is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck. [ABSTRACT FROM AUTHOR]

The Cauchy problem considered in this paper is the following: 1ut=Δu+uαM0−∫Rnu(x,t)dx,x∈Rn,t>0,u(x,0)=u0(x)≥0,x∈Rn,$$\begin{align} \hspace*{4pc}\left\{ \def\eqcellsep{&}\begin{array}{ll} u_t=\Delta u+u^\alpha {\left(M_0- \int _{{\mathbb {R}}^n} u(x,t)dx\right)},\quad & x \in {\mathbb {R}}^n, t>0, \\[3pt] u(x,0)=u_0(x)\ge 0,\quad & x \in {\mathbb {R}}^n, \end{array} \right.\hspace*{-4pc} \end{align}$$where M0>0,α>1,n≥3$M_0>0, \alpha >1, n \ge 3$. When the coefficient M0−∫Rnu(x,t)dx$M_0-\int _{{\mathbb {R}}^n} u(x,t) dx$ remains positive, (1) is analogous to 2ut=Δu+uα,x∈Rn,t>0,u(x,0)=u0(x)≥0,x∈Rn.$$\begin{align} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{ll}u_t=\Delta u+u^\alpha ,\quad & x \in {\mathbb {R}}^n, t>0, \\[3pt] u(x,0)=u_0(x)\ge 0,\quad & x \in {\mathbb {R}}^n. \end{array} \right.} \hspace*{-6pc}\end{align}$$It is well known that when 1<α≤1+2/n$1<\alpha \le 1+2/n$, the local solution of (2) blows up in finite time as long as the initial value is nontrivial. The main aim of this paper is to obtain sufficient conditions for global existence of solutions to (1) for 1<α≤1+2/n$1<\alpha \le 1+2/n$ and thus it forms a contrast to (2). The exponent α=1+2/n$\alpha =1+2/n$ produces an exact balance in the mass‐invariant scaling of diffusion and growth in (1). In particular, we prove that if n≥3,α=1+2/n,∥u0∥L1(Rn)

In this paper, we study the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities in R 4 . In particular, in the case of N = 4 , there is a new mass-energy doubly critical phenomenon for Kirchhoff equation with combined nonlinearities that the mass critical exponent 2 + 8 N is equal to the energy critical exponent 2 N N − 2 , which remains unsolved in the existing literature. To deal with the special difficulties created by the nonlocal term and doubly critical term, we develop a perturbed Pohožaev constraint method based on the splitting properties of the Brézis-Lieb lemma, and make some subtle energy estimates. By decomposing Pohožaev manifold and constructing fiber map, we prove the existence of a positive normalized ground state. Moreover, we also explore the asymptotic behavior of the obtained normalized solutions. These conclusions extend some known ones in previous papers. [ABSTRACT FROM AUTHOR]

In the present paper, we are interested in the critical Kirchhoff-type fractional Laplacian problem involving strong singularity as shown below: a + b ‖ u ‖ 2 m - 2 - Δ s u = f (x) u - γ - h (x) u 2 s ∗ - 1 , in Ω , u > 0 , in Ω , u = 0 , in R N \ Ω , where Ω ⊂ R N is a bounded smooth domain, - Δ s is the fractional Laplace operator, s ∈ (0 , 1) , N > 2 s , a , b ⩾ 0 , a + b > 0 , m ⩾ 1 , γ > 1 , h ∈ L ∞ (Ω) is a nonnegative function, 2 s ∗ = 2 N / (N - 2 s) is the critical Sobolev exponent, and f ∈ L 1 (Ω) is positive almost everywhere in Ω . By the Nehari method and Ekeland’s variational principle, we overcome the shortage of compactness due to the critical nonlinearity and establish the existence and uniqueness of weak solution for the above problem. The novelties of our paper are that the Kirchhoff term M may vanish at zero and the considered fractional elliptic problem involves strong singularity and the critical exponent. [ABSTRACT FROM AUTHOR]

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Firstly, we establish optimal Liouville type theorems for nonnegative or positive supersolutions of the fractional parabolic equation u t + (- Δ) s u = u p on the whole space R N × R , where p ∈ R and 0 < s < 1 . Secondly, we study similar questions for the fractional parabolic system u t + (- Δ) s u = v p v t + (- Δ) s v = u q , where p and q are real numbers. [ABSTRACT FROM AUTHOR]

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This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents (S ± ε) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε , u > 0 on S n , where n ≥ 5 , ε is a small positive parameter and K is a smooth positive function on S n . We construct some solutions of (S − ε) that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation (S + ε) . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solution for a class of variational inequality in whole R N where the nonlinearity has a critical growth for N ≥ 2. By combining a penalization scheme found in del Pino and Felmer (Calc Var 4:121–137, 1996) with a penalization method due to Bensoussan and Lions (Applications des inéquations variationelles en contrôle stochastique. Dunod, Paris, 1978), we improve a recent result by Alves et al. (J Math Anal Appl 494:124672, 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]

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: utt+(−Δ)σu+μ(−Δ)σ2ut=∫0t(t−τ)−γ|ut(τ,·)|pdτ,with σ>0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m>1. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with ground state solutions to fractional equations with Coulomb potential and critical exponent. The existence of ground state solutions is established under certain conditions by proposing a new analytical method. [ABSTRACT FROM AUTHOR]

In this paper we consider the attainability of the optimal constant $ S(N,p,s,\mu) $ associated to the fractional Hardy-Sobolev type embedding which is defined as$\begin{align*} S(N,p,s,\mu): = \inf\limits_{u\in\dot{W}^{s,p}(\mathbb{R}^{N})\backslash\{0\}}\frac{\int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}dxdy-\mu\int_{\mathbb{R}^{N}}\frac{|u|^{p}} {|x|^{ps}}dx}{\big(\int_{\mathbb{R}^{N}}|u|^{p^{\ast}_{s}}dx\big)^{\frac{p}{p^{\ast}_{s}}}}, \end{align*}$where $ s\in(0,1) $, $ p>1 $ and $ N>ps $, $ 0\leq\mu<\mu_{H} $, the latter being the best constant in the fractional Hardy inequality on $ \mathbb{R}^{N} $, $ p_{s}^{\ast} = \frac{Np}{N-ps} $ is the fractional critical Sobolev exponent. The technique that we use to prove the existence of extremals for $ S(N,p,s,\mu) $ is based on blow-up analysis argument combined with a variational method. Further, as an application of the inequality, we prove an existence result for the critical fractional $ p $-Laplacian equation with Hardy potential and involving continuous nonlinearities having quasicritical growth. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the coupled Schrödinger systems with critical exponent: { − Δ u i + λ i u i = ∑ j = 1 d β i j | u j | 3 | u i | u i in Ω , u i ∈ H 0 1 (Ω) , i = 1 , 2 ,... , d. Here, Ω ⊂ R 3 is a smooth bounded domain, d ≥ 2 , β i i > 0 for every i , and β i j = β j i for i ≠ j. We study a Brézis-Nirenberg type problem: − λ 1 (Ω) < λ 1 , ⋯ , λ d < − λ ⁎ (Ω) , where λ 1 (Ω) is the first eigenvalue of −Δ with Dirichlet boundary conditions and λ ⁎ (Ω) ∈ (0 , λ 1 (Ω)). We acquire the existence of least energy positive solutions to this system for weakly cooperative case (β i j > 0 small) and for purely competitive case (β i j ≤ 0) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case N ≥ 5. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff-Schrödinger-Poisson system: { - (a + b ∫ R 3 | ∇ u | 2) Δ u + u + ϕ u = µ Q (x) | u | q - 2 u + K (x) | u | 4 u , i n R 3 , - Δ ϕ = u 2 , i n R 3 , the nonlinear growth of ∣u∣4u reaches the Sobolev critical exponent. By combining the variational method with the concentration-compactness principle of Lions, we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions. The nonlinear term includes the nonlinearity f(u) ∼∣u∣q−2u for the well-studied case q ∈ [4, 6), and the less-studied case q ∈ (2, 3), we adopt two different strategies to handle these cases. Our result improves and extends some related works in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following critical fractional semilinear Neumann problem {(−Δ)1/2u + λu = u n+1 n−1, u > 0 in Ω, ∂νu = 0 on ∂Ω, where Ω ⊂ Rn (n ≥ 5) is a smooth bounded domain, λ > 0 and ν is the outward unit normal to ∂Ω. We prove that there exists a constant λ0 > 0 such that the above problem admits a minimal energy solution for λ < λ0. Moreover, if Ω is convex, we show that this solution is constant for sufficiently small λ. [ABSTRACT FROM AUTHOR]

In this paper, the following critical biharmonic elliptic problem { Δ 2 u = λ u + μ u ln ⁡ u 2 + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u = ∂ u ∂ ν = 0 , x ∈ ∂ Ω is considered, where Ω ⊂ R N is a bounded smooth domain with N ≥ 5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of a class of second-order evolution equations with dispersive and dissipative terms’ critical nonlinearity in locally uniform spaces. First of all, we prove the global well-posedness of solutions to the evolution equations in the locally uniform spaces H lu 1 (R N) × H lu 1 (R N) and define a strong continuous analytic semigroup. Secondly, the existence of the (H lu 1 (R N) × H lu 1 (R N) , H ρ 1 (R N) × H ρ 1 (R N)) -global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appear to be optimal and the existence of a bounded subset(in H lu 2 (R N) × H lu 2 ( R N )), which attracts exponentially every initial H lu 1 (R N) × H lu 1 (R N) -bounded set with respect to the H lu 1 (R N) × H lu 1 (R N) -norm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger-Poisson equation where is a bounded smooth domain in , and the nonlinear growth of reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical p & q -Laplacian problem with shifting subcritical perturbation nonlinearity: where Ω ⊂ R N is open bounded, k > 0 , 1 < q < p < N , p ∗ = Np N - p is the critical Sobolev exponent and h (x) ∈ C (Ω) is nonnegative. Under suitable assumptions on exponents q and r, we show the existence of mountain pass solutions u k to ( P k ) for all k ∈ (0 , + ∞) . Moreover, we also investigate the behavior of the solutions u k for varying k and show the concentration phenomena as k → + ∞ . [ABSTRACT FROM AUTHOR]

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

We consider the quasilinear Schrödinger equation involving a general nonlinearity at critical growth. By using Jeanjean's monotonicity trick and the Pohozaev identity we get the existence results that generalize an earlier work [H. Liu and L. Zhao, Existence results for quasilinear Schrödinger equations with a general nonlinearity, Commun. Pure Appl. Anal., 19(6):3429–3444, 2020] about the subcritical case to the critical case. [ABSTRACT FROM AUTHOR]

The paper deals with a parametrized $ p $-Kirchhoff type problem involving a criticalnonlinearity on a closed Riemannian manifold $ (M,{{{g}}}) $ of dimension $ n\geq 3 $. Existence and multiplicity of solutions are obtained under suitable conditions by variation methods. Our results extend and enrich the previous work of E. Hebey [Comm. Partial Differential Equations, 41 (2016), 913-924] from the specific case of $ p = 2 $ to the general case of $ 1

In this paper, we study a class of fractional Schrödinger–Poisson system with doubly critical growth and a steep potential well - Δ s u + (λ a (x) - β) u - ϕ u 2 s ∗ - 3 u = | u | 2 s ∗ - 2 u , x ∈ R 3 - Δ s ϕ = u 2 s ∗ - 1 , x ∈ R 3 where s ∈ (0 , 3 4 ] , λ , β are positive parameters, and 2 s ∗ = 6 3 - 2 s is the fractional Sobolev critical exponent, a ∈ C (R 3) such that Ω : = int a - 1 (0) is a nonempty bounded set with smooth boundary. If β > 0 is a constant such that the operator (- Δ) s + λ a (x) - β is non-degenerate, we prove the existence of ground state solutions which localize near the potential well Ω for λ large enough and also investigate the asymptotic behavior of the solutions as the parameter λ goes to infinity. Furthermore, for any 0 < β < β 1 , we shall prove the existence of multiple solutions by the Ljusternik–Schnirelmann category theory, where β 1 denotes the first eigenvalue of (- Δ) s with Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We are concerned with the following Choquard equation: −Δpu+A|x|θ|u|p−2u=Iα∗F(u)f(u),x∈RN,$$\begin{equation*} \hspace*{5pc}-\Delta _{p}u + \frac{A}{|x|^{\theta }}|u|^{p-2}u = {\left(I_{\alpha }*F(u)\right)}f(u), \, x\in \mathbb {R}^{N}, \end{equation*}$$where p∈(1,N)$p\in (1,N)$, α∈(0,N)$\alpha \in (0,N)$, θ∈[0,p)∪p,(N−1)pp−1$\theta \in [0,p)\cup \left(p,\frac{(N-1)p}{p-1}\right)$, A>0$A>0$, Δp$\Delta _{p}$ is the p‐Laplacian, Iα$I_{\alpha }$ is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle. [ABSTRACT FROM AUTHOR]

This paper considers blow-up and global existence for a semilinear space-time fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. We determine the critical exponents of the Cauchy problem when α < γ and α ≥ γ , respectively. The results obtained in this study are noteworthy extension to the results of time-fractional differential equation. The critical exponent is consistent with the corresponding Cauchy problem for the time-fractional differential equation with nonlinear memory, which illustrates that the diffusion effect of the third order term is not strong enough to change the critical exponents. [ABSTRACT FROM AUTHOR]

The paper is concerned with the p-Laplacian minimizing problem with critical Stein–Weiss type term, and a nonlocal nonlinear equation involving fractional p-Laplacian operator and critical Stein–Weiss type term. Then main difficulty is hard to get the compactness of bounded minimizing sequence (and bounded Palais–Smale sequence). By the refined Sobolev inequality with Lorentz norm, we establish a abstract theorem, which is the relatively compactness result of bounded symmetric decreasing sequence. And then, as applications, we prove the existence, decay and regular results for the problems. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying a fractional elliptic system involving (p,q)-Laplacian and critical exponent. Based upon the Nehari manifold and fibering maps, we prove a nontrivial solution for the problem under the proper conditions. [ABSTRACT FROM AUTHOR]