In a recent article by Fujishima et al. (J Funct Anal 289:110922, 2025), it was shown that wide classes of semilinear elliptic equations with exponential-type nonlinearities admit singular radial solutions U on the punctured disk in ℝ^2 which are also distributional solutions on the whole disk. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: u_s(t,x) ≡ U(x) is a stationary solution, and there exists also a solution u_r(t,x) departing from U which is bounded for t > 0 . While such non-uniqueness results have been known in higher dimensions by Ni and Sacks (Trans Am Math Soc 287:657–671, 1985), Terraneo (Commun Partial Differ Equ 27:185–218, 2002) and Galaktionov–Vazquez (Commun Pure Appl Math 50:1–67, 1997), only two very specific results have recently been obtained in two dimensions by Ioku–Ruf–Terraneo (Ann Inst H Poincaré C Anal. Non Linéaire 36:2027–2051, 2019) and Ibrahim–Kikuchi–Nakanishi–Wei (Math Ann 380:317–348, 2021).

This articles is concerned with the interaction between nonlinearities and the spectrum of linear differential operators in elliptic boundary value problems. The focus is on how solution multiplicity arises from spectral interaction and how singularity theory provides a framework for understanding this phenomenon. Starting with the classic Ambrosetti-Pro di theorem as a paradigm of a global fold, we progress to cusps, swallowtails, and butterfly singularities that appear when convexity assumptions are relaxed and when different types of nonlinearities are considered. The classical results in singularity theory of H. Whitney, R. Thom, V.I. Arnold, and J. Mather are presented, as well as their extension to the infinite-dimensional setting in Banach spaces. Detailed examples, including cubic nonlinearities and models with local eigenvalue crossing, illustrate the rich geometric structure of solution sets. Several open problems and recent developments are discussed.

In this paper, we investigate the existence and decay properties of solutions to the following elliptic systems, which arise in the context of Bose-Einstein condensation: -Delta u(i )+ V-i(x)u(i )=& sum;(m)(j=1)g (ij)(x)|u(j)|(p)|u( i)|(p-2)u( i )+ mu(i)|u(i)|(2 & lowast;-2)u( i),i,j = 1, 2,& mldr;,m. To analyze the decay behavior of solutions, we use a variant of Moser's iteration technique, with general conditions imposed on the potentials V-i(x) and g(ij)(x). Our results extend recent findings by Angeles, Clapp and Salda & ntilde;a [Exponential decay of the solutions to nonlinear Schr & ouml;dinger systems, Calc. Var. Partial Differential Equations 62(5) (2023) 160]. We pay particular attention to potential classes that either vanish or are unbounded at infinity. The existence of ground states is established under these assumptions, and the derived decay estimates are also used to show that weak solutions exhibit either exponential or polynomial decay.

We study the Hamiltonian elliptic system 0.1 { -Δ u = λ |v|^r-1v +|v|^p-1v in Ω , -Δ v = μ |u|^s-1u +|u|^q-1u in Ω , u >0, v>0 in Ω , u =v = 0 on ∂Ω , . where Ω⊂ℝ^N is a smooth bounded domain, λ and μ are nonnegative parameters and r,s,p,q>0 . Our study includes the case in which the nonlinearities in (0.1) are concave near the origin and convex near infinity, and we focus on the region of non-negative pairs of parameters (λ ,μ ) that guarantee existence and multiplicity of solutions of (0.1). In particular, we show the existence of a strictly decreasing curve λ _*(μ ) on an interval [0, μ ] with λ _*(0)> 0, λ _*(μ ) = 0 and such that the system has two solutions for (λ ,μ ) below the curve, one solution for (λ , μ ) on the curve and no solution for (λ , μ ) above the curve. A similar statement holds reversing λ and μ .

By introducing a new classification of the growth rate of exponential functions, singular solutions for semilinear elliptic equations in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity which admits an explicit singular solution. Then, using a transformation as in [8], one obtains an approximate singular solution, and then one concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way. As a special case, our result contains a pioneering contribution by Ibrahim--Kikuchi--Nakanishi--Wei [15] for the Moser--Trudinger type nonlinearity.

We study the local well-posedness in the framework of the Sobolev space H-1 (R-N), N >= 3, for a semilinear parabolic equation with asymptotically polynomial nonlinearity up to the critical Sobolev growth. Then we establish the dichotomy between blow-up and global existence for solutions with small energy by means of variational methods and the so-called potential well argument.

We prove the following Limiting Bliss inequalities sup_v(0) = 0, ∫_0^1|v'|^Ndx=1 ∫_0^1 e^β(loge/s)v^N(s)/s^N-1ds≤ C(N,β), for β≤ 1 The inequalities are optimal with respect to β≤ 1; there is compactness for β<1, and along the infinitesimal Moser sequence for β = 1. Moreover, we show that the improved inequalities sup_v(0) = 0, ∫_0^1|v'|^Ndx=1 ∫_0^1 e^(loge/s+γlogloge/s)v^N(s)/s^N-1ds≤ C(N,γ) hold for γ≤1, and for γ=1 the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from N=2 to general dimensions N≥2.

In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear eigenvalue problem. We also show the existence of multiple solutions of the nonlinear problem using variational methods.

In this paper, we consider the semilinear elliptic equations [Formula: see text] where [Formula: see text] is a Riesz potential, [Formula: see text], [Formula: see text] and [Formula: see text] is continuous periodic. We assume that [Formula: see text] lies in the spectral gap [Formula: see text] of [Formula: see text]. We prove the existence of infinitely many geometrically distinct solutions in [Formula: see text] for each [Formula: see text], which bifurcate from [Formula: see text] if [Formula: see text]. Moreover, [Formula: see text] is the unique gap-bifurcation point (from zero) in [Formula: see text]. When [Formula: see text], we find infinitely many geometrically distinct solutions in [Formula: see text]. Final remarks are given about the eventual occurrence of a bifurcation from infinity in [Formula: see text].

We discuss the existence and nonexistence of positive solutions for Hamiltonian elliptic systems of two equations. This work is motivated by the results by Ambrosetti, Brezis and Cerami for a single equation featuring nonlinearities that are concave (near the origin) and convex (at infinity), see [2]. In this review we first recall the results in [2]. Then we discuss systems of equations where each non-linearity has the behavior described in [2]. As a part of this discussion we revisit the results in [13] by Dos Santos and we present some recent existence and nonexistence results that are in the lines of those by Ambrosetti, Brezis and Cerami.

We consider eigenvalue problems and bifurcation of positive solutions for elliptic equations with indefinite weights and with Neumann boundary conditions. We give complete results concerning the existence and non-existence of positive solutions for the superlinear coercive and non-coercive problems, showing a surprising complementarity of the respective results. For more information see https://ejde.math.txstate.edu/special/01/c4/abstr.html

We consider a one-dimensional integral inequality of Moser type: set J_c(v) = ∫ _0^1 e^c(s) v^2(s) ds and consider sup _{∫ _0^1 |v'|^2 = 1, v(0) = 0} J_c(v) We show that the supremum remains finite up to the optimal coefficient c_1(s) = 1/s(loge/s + logloge/s) . Indeed, for c_γ = 1/s(loge/s + γlogloge/s) , with γ > 1 , the supremum is infinite. For c_1 the inequality is critical with loss of compactness: the functional J_c_1 fails to be weakly continuous along the infinitesimal Moser sequence w_n(t):= t√(n) (0 ≤ t ≤1/n) w_n(t) = 1/√(n) (1/n≤ t ≤ 1) . Since w'(t) = √(n) (0 ≤ t ≤1/n) , one may say that w_n develops an infinitesimal shock at the origin.

The Trudinger-Moser inequality is the borderline inequality for the Sobolev inequalities in the case p = N. The famous Adams inequality concerns the generalization of the Trudinger-Moser inequality to mth-order derivatives in the case mp = N. Recently, in [11] a borderline inequality was derived for the Bliss-inequalities [5]. In this note, we give a second-order generalization of this inequality, which may be called of Bliss-Adams type.

. In this work we study a Hamiltonian elliptic system of equations with Dirichlet boundary condition and with non-linearities that are concave near the origin and are convex and superlinear at infinity. The concavity of the non-linearities depends on non-negative parameters lambda and mu and we provide regions for the pairs (lambda, mu) guaranteeing existence and non-existence of nonnegative solutions. This work is inspired by the seminal work for the single equation done by Ambrosetti, Brezis and Cerami in [1].

The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Ω =Ω '× (0,a)⊂ℝ^n , where Ω '⊂ℝ^n-1 is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω with another superlinear (or linear) equation defined at the bottom of the cylinder Ω '×{0} . Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L^∞ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.

We consider singularly perturbed equations of the form{epsilon(2)delta u - u + u(p) = 0 in A subset of R-N,u > 0 in A, u = 0 on & part; A,where A is a annulus and p > 1. It has been conjectured for a long time that such problems possess solutions having m-dimensional concentration sets for every 0 <= m <= N - 1. For N = 3 solutions with 2-dimensional and 0-dimensional concentration sets are known, while no result was available for 1-dimensional concentration sets. We answer positively this conjecture, proving the existence of solutions which concentrate on circles S-1. The proof relies on work by SantraWei who proved the existence of solutions concentrating on a Clifford torus S-1 x S-1 for an annulus in R-4. We extend this result to equations with weights. Then we use the Hopf fibration to show that these solutions give rise to the S-1 concentrating solutions in R-3. (C)2022 Elsevier Inc. All rights reserved.

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $$L^{\infty }$$ , which implies $$C^{1,\alpha }$$ smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

We show existence and multiplicity results for nonlinear Dirac-Klein-Gordon systems with two different nonlinear interaction terms. Both the Dirac field and the Klein-Gordon field considered here are nonlinear fields, different from previously considered models. Moreover, results for the regularity and the decay rate of the solutions are given.

We consider the Cauchy problem for a two space dimensional parabolic equation with square exponential nonlinearity. More precisely, $$ \left\{ \begin{aligned} & \partial_{t} u = \Delta u - u + \lambda f(u)\;\;{\text{in}}\; ( 0 ,T )\times {\mathbb{R}}^{2} , \hfill \\ & u(0,x)\; = u_{0} (x)\;\;\;\;\;\;\;\;\;\;{\text{in}}\;{\mathbb{R}}^{2} , \hfill \\ \end{aligned} \right. $$ where λ > 0, and f (u) := $$ 2\alpha_{0}ue^{\alpha_{0}u^{2}} $$, for some α0 > 0. We take into account initial data in the energy space H1($$ {\mathbb{R}}^{2} $$), i.e. u0 $$ \in $$ H1($$ {\mathbb{R}}^{2} $$), and in view of the Trudinger-Moser inequality, the nonlinearity f (which has square exponential growth at infinity) is in the energy critical regime. We look for sufficient conditions in order to predict from the initial data whether the solution blows up in finite time or the solution exists globally in time. Our main tools are energy methods, and the so-called potential well argument. $$ {\text{If}}\;0{ < }\lambda < \frac{1}{{2\alpha_{0} }} $$, we prove that for energies below the ground state level, the dichotomy between blow-up and global existence is determined by the sign of a suitable functional.