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 .
We establish an explicit a priori bound for solutions to the Dirichlet problem associated with the p-Laplacian ( p>1 ), where the constant depends on both p and the geometry of the domain. From this result, we derive two main applications. First, we prove an existence theorem for nonlinear boundary value problems of the form -Δ _p u = λ f(u) in Ω , u=0 on ∂Ω , with f nonnegative, continuous, and nondecreasing. An interesting consequence is the emergence of a stabilization phenomenon: for every such nonlinearity, there exists a threshold p_0 := p_0(f,λ ,Ω ) such that for all p ≥ p_0 , a solution exists. To our knowledge, this stabilization effect with respect to p, in this generality, has apparently not been recorded in the literature before. Second, we obtain, in an elementary fashion, a lower bound for the first eigenvalue of the p-Laplacian, which is asymptotically sharp as p→∞ for a large family of domains.
In this paper we study a critical exponent equation on the Heisenberg group ℍ^n arising from the prescribed Webster scalar curvature problem on the CR sphere. Without assuming cylindrical-type symmetry, we construct multi-bubbling solutions by means of a modified finite-dimensional reduction method. This extends the existence theory of Cao, Peng and Yan (J. Geom. Anal. 23:1674–1702, 2013) for the CR sphere with cylindrical symmetry, and the results of Cao, Noussair and Yan (Calc. Var. Partial Differ. Equ. 15:403–419, 2002).
This article studies the Cauchy problem for the scalar conservation law ∂ _t u + ∂ _t w + ∂ _x f(u) = 0, where w(x,t) = [ℱ(u)(x,t)] is the output of a specific hysteresis operator, namely the Play hysteresis operator, and f is a 𝐂^2 convex flux function. The hysteresis operator models a rate-independent memory effect, introducing a specific non-local feature into the partial differential equation. We consider a suitable notion of entropy weak solution and analyse in detail the Riemann problem. Furthermore, a Godunov-type finite volume numerical scheme is developed to compute approximate solutions. The convergence of the scheme for BV initial data provides the existence of an entropy weak solution. Finally, a stability estimate is established, implying the uniqueness and overall well-posedness of the entropy weak solution.
