Localized vertex couplings shift nonlinear Schrödinger standing-wave folds on compact metric graphs. We derive the shift through second order near a positive constant profile at a nondegenerate fold. For a C^4 nonlinearity the fold frequency satisfies ω _f(α )=ω _*+|𝒢|^-1∑ _jα _j+Q_𝒢(α )+O(|α |^3) : the tangent depends only on total length and total coupling, whereas the quadratic term resolves graph geometry and defect placement. We express Q_𝒢 through the zero-mean Kirchhoff Green kernel and its L^2 Gram matrix, and obtain closed formulas on lollipop and figure-eight graphs. Equal-length lollipops can have quadratic shifts of opposite sign. A reduction centred at the perturbed fold gives the exact local two-solution/no-solution alternative. Under the stated flow assumptions both nearby branches are orbitally stable; their Morse indices and mass slopes reverse when the sign of f”(ϕ _*) reverses. Numerical fold solves reproduce the coefficients and their graph dependence. For a cubic–quintic Lugiato–Lefever model, a conditional stationary fold reduction and refined computations support the analogous detuning shift, while a separate spectral calculation identifies nonconstant unstable modes. Changed vertex conditions or magnetic holonomy can exclude the constant reference mode, but do not by themselves establish disappearance of nearby folds.
In this paper we investigate a class of nonlinear elliptic systems of fourth order on closed Riemannian manifolds, involving a Paneitz–Branson type operator and nonlinearities with critical Sobolev growth. Such systems combine the difficulties of higher order operators, strong coupling effects, and critical nonlinear behavior, and arise naturally in problems motivated by geometric analysis. By means of variational methods, we establish the existence of weak solutions under suitable structural assumptions on the coefficients of the operator and on the interaction between the lower order terms and the geometry of the manifold. The approach relies on a careful analysis of the associated energy functional and on comparisons with the classical Paneitz–Branson operator. In addition, by standard elliptic regularity theory, we show that the weak solutions obtained are in fact classical.
We study spatial decay estimates for a nonlinear wave equation with viscoelastic-type dissipation, where the nonlinear dissipative function satisfies suitable growth conditions. For solutions defined on semi-infinite cylinders and cone-like regions, we establish Phragmén–Lindelöf alternatives: either the energy grows exponentially or polynomially with distance, or it decays exponentially or polynomially to zero. The results extend previous contributions on spatial decay in nonlinear dissipative systems.
In this paper, we are concerned with qualitative properties of peak solutions to the following nonlinear Schrödinger equations -Δ u+V(x)u= u^p-ε, u>0 in ℝ^N, where V(x)∈ C^1(ℝ^N) is a nonnegative bounded function, ε >0 , p=N+2/N-2 , N≥ 6 . Precisely, we use the blow-up analysis based on Pohozaev identities to establish the local uniqueness and Morse index of the peak solutions obtained by Cao et al. (Calc. Var. Partial Differential Equations, 64: 139, 2025), provided that V(x) possesses non-degenerate critical points.
We study the existence and multiplicity of positive weak solutions for nonhomogeneous Dirichlet problems driven by the p-Laplacian. The nonlinearity is assumed to be locally superlinear at infinity, possibly sign-changing with respect to the spatial variable, and is not required to satisfy the Ambrosetti–Rabinowitz condition. When p 2 , although a nonhomogeneous boundary datum can be removed by an affine substitution, the resulting principal part is no longer p-homogeneous in the translated variable, and the classical homogeneous AR-free compactness arguments break down. To overcome this obstruction, we introduce two structural hypotheses tailored to the nonhomogeneous setting: a localized superlinearity condition and a shifted version of the function H(x,t)=tf(x,t)-pF(x,t) . The latter compensates for the loss of admissible test functions caused by the nontrivial boundary datum. Combining the method of upper and lower solutions with Schechter’s variant of the Mountain Pass Theorem, we prove the existence of at least one positive weak solution and, under additional assumptions, the existence of a second positive weak solution. All solutions obtained are of class C^1,γ .
