Given a smooth positive function K on the standard sphere (Sn, g0), we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function K, there are arbitrarily many metrics g conformally equivalent to g0 and whose scalar curvature is given by the function K provided that the function is sufficiently close to the scalar curvature of g0. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function K. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In this paper, we study the Dirichlet elliptic problem (P-epsilon): -Delta u + Vu = u(p-epsilon), u > 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R-n (n >= 3) is a bounded domain, V is a smooth positive function on Omega & strns;, p + 1 = 2n/(n - 2) is the critical Sobolev exponent, and epsilon>0 is a small parameter. First, we show that, unlike the case of weak convergence to zero, interior blowing-up solutions with a nonzero weak limit cannot occur in low dimensions. We then treat the general setting by removing the restriction that blow-up points are confined to the interior. Using delicate asymptotic expansions of the gradient of the associated functional, we prove that in dimensions n = 4 and n = 5, a single blow-up point cannot coexist with residual mass. We further elucidate the role of the sign of the normal derivative of the potential V on the boundary: if it is positive, any single blow-up solution with residual mass must occur in the interior; if it is negative at some boundary point, boundary blow-up solutions with residual mass can be constructed. Finally, we construct both simple and non-simple interior blow-up solutions exhibiting residual mass, without any assumption on the sign of the normal derivative of V. These results provide new insights into the interaction between the potential, the geometry of the domain, and the critical nonlinearity.

In this paper, we study energy bounded solutions u epsilon converging weakly to 0 of the subcritical problem -Delta u+gu=hu(n-2)(n+2)-epsilon,u>0in Omega,u=0on partial derivative Omega, where Omega is a C(2 )bounded domain in R-n with n >= 4, g is a C(1 )positive function on Omega & strns;, h is a C-3 positive function on Omega & strns;, and epsilon is a small positive parameter. Assuming that the normal derivative of h is negative on the boundary, we prove that u epsilon must blow up in the interior of the domain. Moreover, we determine the precise location of the blow-up points and the corresponding blow-up rates. Conversely, for sufficiently small epsilon, we construct blowing-up solutions that converge weakly to zero, which allows us to obtain a multiplicity result for the problem. In contrast, when the normal derivative of h is positive at a boundary point b, we show that it is possible to construct solutions converging to zero and blowing up precisely at b.

Consider a smooth, bounded domain Omega subset of R-n with n >= 4 and a smooth positive function V. We analyze the asymptotic behavior of a sequence of positive solutions. u(epsilon) to the equation - Delta u + V(x) u = u((n + 2))/((n - 2)) - epsilon in Omega with zero Dirichlet boundary conditions, epsilon -> 0. We determine the precise blow-up rate and characterize the locations of interior concentration points in the general case of multiple blow-up, providing an exhaustive description of interior blow-up phenomena of this equation. Our result is established through a delicate analysis of the gradient of the corresponding Euler-Lagrange functional. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

We investigated the existence of boundary blow-up solutions for a slightly subcritical semilinear elliptic problem posed in a smooth bounded domain of Rnwith n E {4, 5, 6}. We constructed solutions that have a nonzero weak limit while simultaneously blowing up at a boundary point. This behavior stands in contrast to the well-known compactness properties on manifolds without boundary for small dimensions, where such weak convergence would force strong convergence. Our construction shows that, in low dimensions, the presence of a boundary allows blow-up and residual mass to coexist. Moreover, we identified the precise boundary points where this concentration occurs. The results are proved by means of delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.

In this paper, we consider the nonlinear Neumann problem (Qε):−Δu+V(x)u=un+2n−2−ε, with u>0 in Ω and ∂u/∂ν=0 on ∂Ω, where Ω is a bounded regular domain in Rn, with n≥4, ε is a small positive parameter, and V is a non-constant smooth positive function on Ω¯. Assuming the flatness of the boundary near the critical points of the restriction of the function V on the boundary, we construct boundary peak solutions with isolated bubbles, leading to a multiplicity result for (Qε). The proof of our results relies on expanding the gradient of the associated functional and testing the equation with the appropriate vector fields, which yields constraints for the concentration points and blow-up rates. A thorough analysis of these constraints leads to our results.

In this paper, we consider the semilinear Dirichlet problem (Pε):−Δu+V(x)u=un+2n−2−ε, u>0 in , u=0 on ∂, where is a bounded regular domain in Rn, n≥4, ε is a small positive parameter, and V is a non-constant positive C2-function on Ω¯. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for (Pε). The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with (Pε), along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems.

In this paper, we investigate the nonlinear problem $(P_\varepsilon): -\Delta u + V(x)u = f u^{\frac{n+2}{n-2} - \varepsilon}$, $u > 0$ in $\Omega$ and $\partial u/\partial \nu = 0$ on $\partial \Omega$, where $\Omega$ is a bounded regular domain in $\mathbb{R}^n$, with $n \geq 4$, $\varepsilon$ is a small positive parameter, $V$ and $f$ are smooth positive functions on $\overline{\Omega}$. Under certain conditions involving the function $f$ and the mean curvature of the boundary, we construct boundary blowing up solutions, leading to a multiplicity result for $(P_\varepsilon)$. The proof of these results involves expanding the gradient of the associated functional and testing the equation with suitable vector fields. This process imposes constraints on the concentration parameters, and a careful analysis of these constraints leads to the conclusions presented.

In this paper, we are concerned with the following elliptic equation $$ ( SC_\varepsilon ) \qquad \begin{cases} -Δu = |u|^{4/(n-2)}u [\ln (e+|u|)]^\varepsilon & \hbox{ in } Ω,\\ u = 0 & \hbox{ on }\partial Ω, \end{cases} $$ where $Ω$ is a smooth bounded open domain in $\mathbb{R}^n, \ n\geq 3$ and $\varepsilon >0$. In Comm. Contemp. Math. (2003), Ben Ayed et al. showed that the slightly supercritical usual elliptic problem has no single peaked solution. Here we extend their result for problem $( SC_\varepsilon )$ when $\varepsilon$ is small enough, and that by assuming a new assumption.

We investigate the existence of blowing-up solutions of the following almost-critical problem: −Δu+V(x)u=up−ε,u>0inΩ,u=0on∂Ω,-\Delta u+V\left(x)u={u}^{p-\varepsilon },\hspace{1.0em}u\gt 0\hspace{0.25em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,u=0\hspace{0.25em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.25em}\partial \Omega , where Ω\Omega is a bounded regular domain in Rn{{\mathbb{R}}}^{n}, n≥4n\ge 4, ε\varepsilon is a small positive parameter, p+1=(2n)∕(n−2)p+1=\left(2n)/\left(n-2) is the critical Soblolev exponent, and the potential VV is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε\varepsilon goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.

In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a non-zero weak limit, which necessarily constitutes a solution of the Nirenberg problem itself. Our focus lies in providing a comprehensive description of such blowing up solutions, including precise determinations of blow-up points and blow-up rates. Additionally, we compute the topological contribution of these solutions to the difference in topology between the level sets of the associated Euler-Lagrange functional. Such an analysis is intricate due to the potential degeneracy of the involved solutions. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate.

In this paper, we study the problem of prescribing Q -Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q -Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K .

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Given a smooth positive function K defined on the standard half sphere endowed with its standard metric g, we consider the problem of finding a metric g similar to conformally equivalent to g and whose scalar curvature is equal to K and the boundary mean curvature is equal to zero. Using careful study of Hopf-Poincare ' counting index formulae, we prove the existence of many of such metrics g similar to provided that K is close to the scalar curvature of g. Such Counting index formulae are related to the topological contribution of solutions at the level sets of the associated approximate variational functional. The location of the levels of these solutions are given very precisely which leads to a new type of multiplicity results on half spheres. Such multiplicity results are proved without any assumptions of symmetry or periodicity on the function K.

In this paper, we consider the Neumann elliptic problem (𝒫_ε) : -Δ u +μ u = u^((n+2)/(n-2))+ε , u>0 in Ω, ∂ u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in ℝ^n , n≥ 4 , ε is a small positive real, and μ is a fixed positive number. We show that, in contrast with the three dimensional case, (𝒫_ε) has no solution blowing up at only interior points as ε goes to zero. The proof strategy consists in testing the equation by appropriate vector fields and then using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results.

In this paper we study the following mean field type equation \begin{equation*} (MF) \qquad -\D_g u \, = \varrho ( \frac{K e^{u}}{\int_{\Sig} K e^{u} dV_g} \, - \, 1) \, \mbox{ in } \Sigma, \end{equation*} where $(\Sigma, g)$ is a closed oriented surface of unit volume $Vol_g(\Sigma)$ = 1, $K$ positive smooth function and $\varrho= 8 \pi m$, $ m \in \N$. Building on the critical points at infinity approach initiated in \cite{ABL17} we develop, under generic condition on the function $K$ and the metric $g$, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters $B_m(\Sigma)$.\\ We derive from these \emph{Morse inequalities at infinity} various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8 \pi \N$.

. In this paper, we are concerned with the following elliptic equation where Ω is a smooth bounded open domain in R n , n ≥ 3 and ε > 0. Clapp et al. in Journal of Diff. Eq. (Vol 275) proved that there exists a single-peak positive solution for small ε if n ≥ 4. Here we construct positive as well as changing sign solutions concentrated at several points at the same time.

In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical solutions having zero or nonzero weak limit. The formation of such non simple blow ups is governed by a vortex problem, unveiling an unexpected connection with Euler equations in fluid dynamic and mean fields type equations in mathematical physics.

In this paper we study a Nirenberg type problem on standard half spheres (S+, g0) consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: (P) { −∆g0u + n(n−2) 4 u = K u n+2 n−2 , u > 0 in S+, ∂u ∂ν = 0 on ∂S n +, whereK ∈ C(S+) is a positive function. We construct, under generic conditions on the function K, finite energy solutions of a subcritical approximation of (P) on half spheres of dimension n ≥ 5, which exhibit multiple blow up of cluster-type at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy subsolutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of K on ∂S+, approximate solutions of arbitrarily large energy and Morse index.

In this paper, we study the problem of prescribing a fourth-order conformal invariant on standard spheres. This problem is variational but it is noncompact due to the presence of nonconverging orbits of the gradient flow, the so called critical points at infinity. Following the method advised by Bahri we determine all such critical points at infinity and compute their contribution to the difference of topology between the level sets of the associated Euler–Lagrange functional. We then derive some existence results under pinching conditions.