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The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like | u | α or | u | α − 1 u (α > 1) and the initial data u 0 belongs to the Sobolev spaces L s p (G) for 1 < p < ∞ and 0 < s < N / p. Since stratified Lie groups G include the Euclidean space R n as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on R n to G. It should be noted that our proof is very different from it given by Ribaud on R n. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on R n. By using the generalized fractional chain rule on G , we can avoid the discussion of Fourier analysis on G and make the proof more simple. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and decay rate of the global solution to the coupled system of semilinear structurally damped σ$\sigma$‐evolution equations with time‐dependent damping in the so‐called effective cases utt+(−Δ)σ1u+b1(t)(−Δ)δ1ut=|vt|p,vtt+(−Δ)σ2v+b2(t)(−Δ)δ2vt=|ut|q.$$\begin{equation*} \hspace*{7pc}{\begin{cases} u_{t t}+(-\Delta)^{\sigma _1} u+b_1(t) (-\Delta)^{\delta _1} u_t=|v_t|^p, \\ v_{t t}+(-\Delta)^{\sigma _2} v+b_2(t) (-\Delta)^{\delta _2} v_t=|u_t|^q. \end{cases}} \end{equation*}$$We obtain conditions for the existence and the decay estimates of the global (in time) solution that are expressed in terms of the decay character of the initial data u0(x)=u(0,x),v0(x)=v(0,x)$u_0(x)=u(0, x), \nobreakspace v_0(x)=v(0, x)$ and u1(x)=ut(0,x),v1(x)=vt(0,x)$u_1(x)=u_t(0, x),\nobreakspace v_1(x)=v_t(0, x)$. Furthermore, the blow‐up results for solutions to the semilinear problem also presented. [ABSTRACT FROM AUTHOR]

A class of semilinear elliptic differential equations was investigated in this study. By constructing the inverse function, using the method of upper and lower solutions and the principle of comparison, the existence of the maximum positive solution and the minimum positive solution was explored. Furthermore, the uniqueness of the positive solution and its asymptotic estimation at the origin were evaluated. The results show that the asymptotic estimation is similar to that of the corresponding boundary blowup problems. Compared with the conclusions of Wei's work in 2017, the asymptotic behavior of the solution only depends on the asymptotic behavior of b (x) at the origin and the asymptotic behavior of g at infinity. [ABSTRACT FROM AUTHOR]

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

We study the global bifurcation diagram of the positive solutions to the problem { Δ u + λ f (u) = 0 in B , u = 0 on ∂ B , where B is the unit ball in R N with N ≥ 3. Under general supercritical growth conditions on f (u) , we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f (u) , and we exhibit that a bifurcation curve for Δ u + λ f (u) = 0 has the same qualitative property as a classical Gel'fand problem Δ u + λ e u = 0 for N ≥ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type − div (M (x) ∇ u) = − div (h (u) E (x)) + f (x) , where M is a bounded elliptic matrix, the vector field E and the function f belong to suitable Lebesgue spaces, and the function s → h (s) features a superlinear growth at infinity. We provide some existence and non existence results for solutions to the associated Dirichlet problem and a comparison principle. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a locally finite tree, Δ be the normalized Laplacian. In this paper, we consider the following semilinear equation on G (0.1) Δ u + f (u) = 0. We first establish the existence and nonexistence of positive solutions to (0.1) with a general assumption on f , and then find the critical exponent for (0.1) on a regular tree. Moreover, we prove some interesting properties of radial solutions and the asymptotic behaviors of radial solutions under a more general condition on f. Finally, the nonexistence results can be generalized to the elliptic system on a weighted tree. [ABSTRACT FROM AUTHOR]

We study semilinear third‐order (in time) evolution equations with fractional Laplacian (−Δ)σ$$ {\left(-\Delta \right)}^{\sigma } $$ and power nonlinearity |u|p$$ {\left|u\right|}^p $$, which was proposed by Bezerra–Carvalho–Santos (J. Evol. Equ. 2022) recently. In this manuscript, we obtain a new critical exponent p=pcrit(n,σ):=1+6σmax{3n−4σ,0}$$ p={p}_{\mathrm{crit}}\left(n,\sigma \right):= 1+\frac{6\sigma }{\max \left\{3n-4\sigma, 0\right\}} $$ for n⩽103σ$$ n\leqslant \frac{10}{3}\sigma $$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case p>pcrit(n,σ)$$ p>{p}_{\mathrm{crit}}\left(n,\sigma \right) $$, and energy solutions blow up in finite time even for small data if 1

In this paper we define a new operator J for the study of Δ u + f (u) = 0 , x ∈ R N , N > 2. Using J we can easily see some qualitative properties of the solutions, for example we can determine the values where u changes sign, has a local maxima or minima, or changes concavity. We also use this functional to construct, for any k ∈ N , nonlinearities f such that this problem has at least two bound state solutions that change sign j times, for j = 0 , 1 , ... , k − 1. And another f such that this problem has a unique ground state solution, and at least two bound state solutions that change sign one time. [ABSTRACT FROM AUTHOR]

In this article we consider the following fractional semilinear elliptic equation (-Δ)s u + |x|² u = ωu + |u|²σ in RN, where s ∈ (0, 1), N > 2s, σ ∈ (0, 2s/N-2s) and ω ∈ (0, λ1). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result. [ABSTRACT FROM AUTHOR]

We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of asymptotic behavior of solutions near an isolated singularity of some semilinear elliptic equations, such as the Yamabe equation, Hardy-Hénon equations, etc. [ABSTRACT FROM AUTHOR]

In this paper, we investigate virtual element discretization of a semilinear elliptic optimal control problem with pointwise Neumann boundary control constraint. The virtual element discrete scheme is constructed based on virtual element discretization of the state equation and variational discretization of the control variable. By utilizing first-order and second-order optimality conditions along with auxiliary problems, we derive a priori error estimates for the state, adjoint state, and control variable in H 1 and L 2 norms. Theoretical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

We study the numerical solutions for a class of quasilinear Schrödinger equations arising from the self-channeling of high-power ultra short lasers in matter, which are associated with energy functionals with nonlinear principle parts so that the classical algorithm cannot directly be used. By the method of variable replacement to transform the quasilinear Schrödinger equation into an semilinear elliptic equation, the numerical mountain pass algorithm is then applied. Some numerical experiments are also performed, including zero potential, nonzero constant potential and singular problem. [ABSTRACT FROM AUTHOR]

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in \mathbb {R}^{n} with sign-changing nonlinearity f. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of f at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer. [ABSTRACT FROM AUTHOR]

We are concerned with a third order in time equation in the presence of viscoelastic effects given by the memory term and with a semi linear source term, posed on a bounded domain Ω ⊂ R 3 . Considering three different types of memory in the past history framework, we prove the well-posedness of its solutions as well as the exponential stability of the energy functional. Relaxing some hypotheses on the memory kernel, we improve and extend the results established in the existing literature. [ABSTRACT FROM AUTHOR]

In the present paper, by the variational method coupled with mountain pass type theorems, we study nontrivial homoclinic solutions of nonlinear $ (p, q) $-Laplacian partial difference equations with a parameter $ \lambda>0 $:$ \begin{equation*} \begin{split} &\Delta^2_1(\phi_{p_2}(\Delta^2_1u(k-2, l)))+\Delta^2_2(\phi_{p_2}(\Delta^2_2u(k, l-2)))-a[\Delta_1(\phi_{p_1}(\Delta_1u(k-1, l)))\\ &+\Delta_2(\phi_{p_1}(\Delta_2u(k, l-1)))]+V(k, l)\phi_q(u(k, l))\\& = \lambda f((k, l), u(k, l)). \end{split} \end{equation*} $We prove that the above equation admits two nontrivial homoclinic solutions if $ \lambda $ is sufficiently large and one nontrivial homoclinic solutions if $ \lambda>0 $. Our obtained results generalize and improve some existing ones. [ABSTRACT FROM AUTHOR]

We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a positive radially symmetric weight in the unit ball. When the weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension 3 ≤ N ≤ 9 , and it has no turning points if N ≥ 10. In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when 3 ≤ N ≤ 9. Moreover, we find a one-parameter family of radial singular solutions for a parametrized weight and study the Morse index of the singular solution. As a result, we prove that the perturbation affects the bifurcation structure in the critical dimension N = 10. Moreover, we give a classification of the bifurcation diagrams in the critical dimension. [ABSTRACT FROM AUTHOR]

We study difference approximations of an optimal control problem with a boundary observation of the conormal derivative of the state described by the Dirichlet problem for semilinear elliptic equations with controls involved in coefficients of the convective transport operator and in the nonlinear term of the equation. The well-posedness of the optimal control problem is examined. Difference approximations for the optimal control problem are constructed. The convergence of the approximations with respect to the functional and control is analyzed. A regularization of the approximations is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we consider solutions to the following nonlinear Schrödinger equation with competing Hartree‐type nonlinearities, −Δu+λu=|x|−γ1*|u|2u−|x|−γ2*|u|2uinRN,$$\begin{equation*} -\Delta u + \lambda u={\left(|x|^{-\gamma _1} \ast|u|^2\right)} u - {\left(|x|^{-\gamma _2} \ast|u|^2\right)} u\quad \mbox{in} \,\, \mathbb {R}^N, \end{equation*}$$under the L2$L^2$‐norm constraint ∫RN|u|2dx=c>0,$$\begin{equation*} \int _{\mathbb {R}^N}|u|^2 \, dx=c>0, \end{equation*}$$where N≥1$N \ge 1$, 0<γ2<γ1

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The P S c sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators \[ (-\Delta +m^2)^s u(x) = f(x,u(x)) , \] (− Δ + m 2) s u (x) = f (x , u (x)) , where 00. The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator $ \sqrt {-\Delta +m^2} $ − Δ + m 2 . When $ m\rightarrow 0^+ $ m → 0 + , the nonlocal pseudo-relativistic operator $ (-\Delta +m^2)^s $ (− Δ + m 2) s is also closely related to the fractional Laplacian operator $ (-\Delta)^s $ (− Δ) s . But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions. [ABSTRACT FROM AUTHOR]

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones. [ABSTRACT FROM AUTHOR]

Second order semilinear parabolic equation ∂u/ ∂t = div(A(x, t)∇u) + h(x, t, u) with time-periodic coefficients is considered in domain Ω × (−∞, +∞), where Ω is the exterior of a compact set in Rn x. Depending on the behavior of the function h(x, t, u) at infinity, the conditions are found under which the positive periodic solution does not exist. [ABSTRACT FROM AUTHOR]

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point. [ABSTRACT FROM AUTHOR]

In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton's method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed. [ABSTRACT FROM AUTHOR]

We prove the generalized weak Harnack inequality for non-negative super-solutions to quasi- linear elliptic equations with the generalized Orlicz growth div Φ x ∇ u ∇ u ∇ u 2 = b x f u , b x ≥ 0. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying in R n × (0 , T) (n ≥ 1) the Cauchy problem for a semilinear heat equation with source term f (x) non-rarefied at infinity, subjected to zero initial data. To such a problem, the novelty of this work lies in: (a) similar to no source case, non-rarefaction of f at infinity brings about its classical solution blowing up in a finite time T ⁎ > 0 ; (b) we derive the optimal upper and lower bound on the life span T ⁎. This may throw some light on studies of life span. [ABSTRACT FROM AUTHOR]

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments. [ABSTRACT FROM AUTHOR]

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

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280]. [ABSTRACT FROM AUTHOR]

We study the behavior near the boundary corner point of weak bounded solutions to the oblique derivative problem for singular p-Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector. However, no work has been done for investigating the behavior of solutions to the oblique problem for singular p-Laplacian equation in a corner. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for an elliptic system. More precisely, we combine the potential theory with the sub-super solution method and use the properties of the well-known Kato class to justify our existence results. The novelty of our study is that we consider either the bounded or the exterior domain; Also, the nonlinearities may be singular near the boundary. Some examples are presented to validate our main results. [ABSTRACT FROM AUTHOR]

We consider the problem of maximal regularity for semilinear non-autonomous fractional equations ∑ i = 1 n λ i ∂ α i (u - u 0) (t) + A (t) u (t) = F (t , u (t)) t -a.e. , λ i ∈ C , n ∈ N. Here, ∂ α i denotes the Riemann–Liouville fractional derivative of order α i ∈ (0 , 1) w.r.t. time and each operator A (t) arises from a time depending sesquilinear form a (t) on a Hilbert space H with constant domain V , such that V is continuously and densely embedded into H. We prove non-autonomous maximal L p -regularity results on V ′ and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the forms, the initial data u 0 and the inhomogeneous term F. [ABSTRACT FROM AUTHOR]

In this paper, we investigate variational problems in R 2 with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting. [ABSTRACT FROM AUTHOR]

In this article, we consider the following Kirchhoff equation involving fractional Laplacian a + b [ u ] s 2 (- Δ) s u + u = | u | p - 2 u in Ω , u = 0 on R 3 \ Ω , where a , b > 0 are constants, 3 4 < s < 1 , [ u ] s is the so-called Gagliardo (semi)norm of u, 4 < p < 2 s ∗ = 6 3 - 2 s and Ω ⊂ R 3 is an exterior domain with smooth boundary ∂ Ω ≠ ∅. By establishing a global compactness lemma of the fractional Kirchhoff equation in exterior domains, we verify the compactness of Palais–Smale sequences corresponding to above problem at higher energy level interval. Then combining some crucial estimates and barycentric function, we determine the existence of positive bound state solutions provided that R 3 \ Ω is contained in a small ball. In addition, we point out that the main result can be extended to fractional Sobolev critical case with small parameter. [ABSTRACT FROM AUTHOR]

We consider semilinear elliptic equations of the form Δu+f(|x|,u)=0 on RN with f(|x|,u)=q(|x|)g(u). These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity f, there exists an oscillating radial solution converging to a zero of the function g. We also study the oscillating and limiting behavior of this solution. [ABSTRACT FROM AUTHOR]

In this paper we consider a semilinear second order in time fractional wave equation governed by bi-harmonic operator with clamped boundary conditions in a bounded smooth domain that models flight structures. We address the problem of the local and global well-posedness, and we establish the existence of a compact global attractor for the associated evolution semigroup and show that the family of global attractors is upper semicontinuous. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear Schrödinger system: { − Δ u + α u = | u | p − 1 u + 2 q + 1 λ | u | p − 3 2 u | v | q + 1 2 , x ∈ R 3 , − Δ v + β v = | v | q − 1 v + 2 p + 1 λ | u | p + 1 2 | v | q − 3 2 v , x ∈ R 3 , u (x) → 0 , v (x) → 0 , as | x | → ∞ , where 3 ≤ p , q < 5 , α, β are positive parameters. We show that there exists λ k > 0 such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each k ∈ N and λ ∈ (0 , λ k) . Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each λ ∈ (0 , λ 0) where λ 0 ∈ (0 , λ 1 ] . [ABSTRACT FROM AUTHOR]

We study the semilinear equation-Δgu+V(σ)u = f(u) on a Cartan-Hadamard manifold Μ of dimension N ≥ 3, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function V ε C(Μ). In particular, the decay of V at infinity is allowed, with some restrictions related to the geometry of Μ. We generalize some results proved in RN by Alves et al. [ABSTRACT FROM AUTHOR]

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in $ \mathbf{R}^N $ R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1]. [ABSTRACT FROM AUTHOR]

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under s -Gevrey assumptions on the residual equation, we establish s -Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input. [ABSTRACT FROM AUTHOR]

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation. [ABSTRACT FROM AUTHOR]

Unfortunately, there is a flaw in the numerical analysis of the published version [ IMA J. Numer. Anal. , DOI:10.1093/imanum/drad039 ], which is corrected here. Neither the algorithm nor the results are affected, but constants have to be adjusted. [ABSTRACT FROM AUTHOR]

We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e. the total computational time. Numerical experiments underline the theory. [ABSTRACT FROM AUTHOR]

We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical p -Laplacian problems, p -Laplacian problems with critical Hardy-Sobolev exponents, critical fractional p -Laplacian problems, and critical (p , q) -Laplacian problems are given. Our results are new even in the semilinear case p = 2. [ABSTRACT FROM AUTHOR]

Let |$2\le n\le 9$|⁠. Suppose that |$f:{{\mathbb {R}}}\to {{\mathbb {R}}}$| is locally Lipschitz function satisfying |$f(t)\ge A\min \{0,t\}-K$| for all |$t\in {{\mathbb {R}}}$| with some constant |$A\ge 0$| and |$K\ge 0$|⁠. We establish an a priori interior Hölder regularity of |$C^{2}$| -stable solutions to the semilinear elliptic equation |$-\Delta u=f(u)$|⁠. If, in addition, |$f$| is nondecreasing and convex, we obtain the interior Hölder regularity of |$W^{1,2}$| -stable solutions. Note that the dimension |$n\le 9$| is optimal. [ABSTRACT FROM AUTHOR]

This article explores the solvability of semilinear multi-term time-fractional diffusion equations with memory, specifically focusing on modeling the diffusion of oxygen through capillaries. The study utilizes a continuation method and a priori estimates to prove the classical and strong solvability of these problems within a finite time interval. The article discusses suitable conditions on the given data and coefficients to ensure solvability, as well as different assumptions and approaches for one-dimensional and multidimensional cases. The research also suggests potential applications of the technique to analyze inverse problems and fully nonlinear equations. [Extracted from the article]