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We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=\mu _{1}|u|^{{2}^{*}-2}u+\lambda \alpha |u|^{\alpha -2}|v|^{\beta }u,\\ -\,\Delta v=\mu _{2}|v|^{{2}^{*}-2}v+\lambda \beta |u|^{\alpha } |v|^{\beta {-2} }v,\\ u,v\in D^{1,2}({\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}$$ where $$N\ge 4,$$ $$2^{*}:=\frac{2N}{N-2}$$ is the critical Sobolev exponent, $$\alpha ,\beta \in (1,2],$$ $$\alpha +\beta =2^{*},$$ $$\mu _{1},\mu _{2}>0,$$ and $$\lambda

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D} $ is an open bounded domain in $\mathbb{R}^N,$ $\nu$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal D,$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial\mathcal D$ as $q$ approaches from either below or above the higher critical Sobolev exponent ${2(N-h)\over N-h-2}.$, Comment: 13 pages

Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a smooth function on $M$ and assume that it has a critical point $\xi\in M$ such that $h(\xi)=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_\lambda=u_\lambda^\frac4{n-2}g$, with $u>0$, whose scalar curvature is the prescribed function $h_\lambda=\lambda^2+h$, where $\lambda$ is a small parameter. In the positive case, i.e. when the scalar curvature $R_g$ is strictly positive, we find a family of bubbling metrics $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches zero far from $\xi$ as $\lambda$ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric $g_0=u_0^\frac4{n-2}g$, with $u_0>0$, whose scalar curvature is equal to $h$, then there exists a bounded family of conformal metrics $g_{0,\lambda}=u_{0,\lambda}^\frac4{n-2}g$, with $u_{0,\lambda}>0$, which satisfies $u_{0,\lambda}\to u_0$ uniformly as $\lambda\to 0$. Here, we build a second family of bubbling metrics $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches $u_0$ far from $\xi$ as $\lambda$ goes to zero. In particular, this shows that this problem admits more than one solution., Comment: 30 pages, final version. Accepted for publication in J. Differential Equations

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is \(n\ge 7\) and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.

In this paper we deal with the nonlinear Schr\"odinger system \[ -\Delta u_i =\mu_i u_i^3 + \beta u_i \sum_{j\neq i} u_j^2 + \lambda_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(\Omega) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($\beta, Comment: 33 pages

We consider the Yamabe equation Δ u + n ( n − 2 ) 4 | u | 4 n − 2 u = 0 in R n , n ⩾ 3 . Let k ⩾ 1 and ξ j k = ( e 2 j π i k , 0 ) ∈ R n = C × R n − 2 . For all large k we find a solution of the form u k ( x ) = U ( x ) − ∑ j = 1 k μ k − n − 2 2 U × ( μ k − 1 ( x − ξ j ) ) + o ( 1 ) , where U ( x ) = ( 2 1 + | x | 2 ) n − 2 2 , μ k = c n k 2 for n ⩾ 4 , μ k = c k 2 ( log k ) 2 for n = 3 and o ( 1 ) → 0 uniformly as k → + ∞ .

Let Ω be a bounded smooth domain in R N , N ≥ 3, and let M be a compact smooth submanifold of Ω without boundary such that 1 ≤ dim M ≤ N − 2. We consider the problem where Ωϵ ≔ {x ∈ Ω : dist(x, M) > ϵ}, is the critical Sobolev exponent, and ϵ > 0 is small enough. Under some additional assumptions we obtain the multiplicity of positive and sign changing solutions.

Given (M, g) the set {(ε, h) ∈ (0, 1) × Sk : any solution u ∈ A of -ε2 Δg+h u+u = (u+)p-1 on M is non degenerate} is an open dense subset of (0, 1) × Bρ where Bρ is a suitable ball centered at 0 with radius ρ in the Banach space Sk. Here Sk is the space of all Ck symmetric covariant 2-tensors on M and A is a bounded subset of Hg 1 (M) which does not contain the constant function 1.

We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem { − e 2 Δ v + V ( x ) v = f ( v ) in Ω , v = 0 on ∂ Ω , where e is a small positive parameter, Ω is a smooth, possibly unbounded, domain, f is a superlinear and subcritical nonlinearity, V is a positive potential bounded away from zero. No symmetry on V or on the domain Ω is assumed. It is known by Kang and Wei (see [X. Kang, J. Wei, On interacting bumps of semiclassical states of nonlinear Schrodinger equations, Adv. Differential Equations 5 (2000) 899–928]) that this problem has positive clustered solutions with peaks approaching a local maximum of V. The aim of this paper is to show the existence of clustered solutions with mixed positive and negative peaks concentrating at a local minimum point, possibly degenerate, of V.

We consider a system of the form \(-\varepsilon^{2}\triangle u+u = g(\upsilon), -\varepsilon^{2}\triangle \upsilon+\upsilon = f(u)\) in Ω with Dirichlet boundary condition on \(\partial\Omega\), where Ω is a smooth bounded domain in \({\mathbb{R}}^{N}, N\geq3\) and f, g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as \(\varepsilon\) goes to zero, at a point of Ω which maximizes the distance to the boundary of Ω.

We consider the problem $-\Delta u= |u|^{\frac4{N-2}} u \mbox{ in } \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\},$ $ u = 0 \mbox{ on } \partial( \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\}),$ where $\Omega$ is a smooth bounded domain in $R^N$, $N\ge 3,$ $\xi_1,$ $\xi_2$ are different points in $\Omega$ and e is a small positive parameter. We show that, for e small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at $\xi_1$ and $\xi_2$ as e goes to zero.

We study the existence of steady states to the Keller–Segel system with linear chemotactical sensitivity function on a smooth bounded domain in \(\mathbb {R}^N,\)\(N\ge 3,\) having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable \((N-2)\)-dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.

We consider the $SU(n+1)$ Toda system $$(S_\lambda) \quad \left\{ \begin{aligned} & \Delta u_1 + 2\lambda e^{u_1} - \lambda e^{u_2}- \dots - \lambda e^{u_k} = 0\quad \hbox{in}\ \Omega,\\ & \Delta u_2 - \lambda e^{u_1} + 2\lambda e^{u_2} - \dots - \lambda e^{u_k}=0\quad \hbox{in}\ \Omega,\\ &\vdots \hskip3truecm \ddots \hskip2truecm \vdots\\ & \Delta u_k -\lambda e^{u_1}-\lambda e^{u_2}- \dots+2\lambda e^{u_k}=0\quad \hbox{in}\ \Omega, &u_1 = u_2 = \dots = u_k =0 \quad \hbox{on}\ \partial\Omega.\\ \end{aligned}\right. $$ If $0\in\Omega$ and $\Omega$ is symmetric with respect to the origin, we construct a family of solutions $({u_1}_\lambda,\dots,{u_k}_\lambda)$ to $(S_\lambda )$ such that the $i-$th component ${u_i}_\lambda$ blows-up at the origin with a mass $2^{i+1}\pi $ as $\lambda$ goes to zero., Comment: arXiv admin note: text overlap with arXiv:1210.5719

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.

Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $\xi_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_\epsilon)\qquad-\mathcal L_g u+\epsilon u=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $\epsilon_k>0$ such that for all $\epsilon\in (0, \epsilon_k)$ the problem $(P_\epsilon)$ has a symmetric solution $u_\epsilon ,$ which looks like the superposition of $k$ positive bubbles centered at the point $\xi_0$ as $\epsilon\to 0$. In particular, $\xi_0$ is a {\em towering} blow-up point.

We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.

We consider the boundary value problem { − Δ u + u = λ e u , in Ω ∂ ν u = 0 on ∂ Ω where Ω is a bounded smooth domain in R 2 , λ > 0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis. We establish the existence of a solution u λ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ → 0 . These solutions have large mass in the sense that ∫ Ω λ e u λ ∼ | log ⁡ λ | .

Let N ≥ 3 , 2 ∗ = 2 N / ( N − 2 ) . Our purpose in this paper is to consider the existence and multiplicity of solutions of the problem { − Δ u = | u | 2 ∗ − 2 u + f on R N u ∈ H 1 ( R N ) where f ∈ L 2 ∗ / ( 2 ∗ − 1 ) ( R N ) ∩ L ∞ ( R N ) , f ≥ 0 and f ≢ 0 .

We consider the boundary value problem Δ u + u p = 0 in a bounded, smooth domain Ω in R 2 with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution u p concentrating at exactly m points as p → ∞ . In particular, for a nonsimply connected domain such a solution exists for any given m ⩾ 1 .

We prove the existence of a positive solution to -Δu - λ/|x|2u = k(x)u2*-1, u ∈ script D sign1,2(IRN), which blows-up at a suitable critical point of k as the positive parameter λ goes to zero.

In this paper we consider the following problem (0.1) { − Δ u + u = u N + 2 N − 2 + ɛ in Ω , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N and N ⩾ 3 . We prove the existence of a one-spike solution to (0.1) which concentrates around a topologically non trivial critical point of the mean curvature of the boundary with positive value. Under some symmetry assumption on Ω, namely if Ω is even with respect to N − 1 variables and 0 ∈ ∂ Ω is a point with positive mean curvature, we prove existence of solutions to (0.1) which resemble the form of a super-position of spikes centered at 0.

In this paper we shall consider the critical elliptic equation \( -\triangle u + \lambda a(x) u = u^{(N+2)/(N-2)}, \ \ x\in \Bbb R^N, \\ u > 0, \quad \int_{\Bbb R^N} |\nabla u|^2 \, dx 0, N > 4\) and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions \(u_\lambda\) which blows-up and concentrates as \(\lambda \to +\infty\) at some zero point x0 of a(x) if the order of flatness of the function a(x) at x0 is \(\beta\in[2,N-4)$ and $N \geq 7\)

Let \(\tilde{\Omega}\in\mathbb{R}^{N}\), \(N\geq 3\), be a bounded domain as defined by Flucher, Garroni and Muller [6], which has a singular point \(\overline{x}\in\partial\tilde{\Omega}\) such that the Robin’s function achieves its infimum at \(\overline{x}\). Considering the elliptic problem \((P_{\varepsilon}): \ -\Delta u = u^{p-\varepsilon},\ u > 0\) in \(\tilde{\Omega}\); u = 0 on \(\partial\tilde{\Omega}\), with p = (N + 2)/(N-2), \(\varepsilon > 0\), and \(u_{\varepsilon}\) a minimizing solution of \((P_{\varepsilon})\), \(u_{\varepsilon}\) concentrates at \(\overline{x}\) as \(\varepsilon\) goes to zero.

The main purpose of this paper is to construct a family of positive solutions for both the slightly subcritical and slightly supercritical equations -Δu + V(x)u = n(n - 2)(u+)(n+2)/(n-2)±e in Rn, which blow-up and concentrate at a single point as e goes to 0, under certain conditions on the potential V.

We consider the supercritical problem $$ - \Delta v = {\left| v \right|^{p - 2}}v{\text{ in }}{\Theta _\epsilon },{\text{ }}v = 0{\text{ on }}\partial {\Theta _\epsilon }$$ , where Θ is a bounded smooth domain in ℝ N , N ≥ 3, p > 2*:= 2N/(N − 2), and Θ ∈ is obtained by deleting the ∈-neighborhood of some sphere which is embedded in Θ. We show that in some particular situations, for small enough ∈ > 0, this problem has a positive solution {itv}{in{it\te}} and that this solution concentrates and blows up along the sphere as ∈ → 0. Our approach is to reduce this problem by means of a Hopf map to a critical problem of the form $$ - \Delta u = Q(x){\left| u \right|^{4/n - 2}}u{\text{ in }}{\Omega _\epsilon },{\text{ }}u = 0{\text{ on }}\partial {\Omega _\epsilon }$$ , in a punctured domain $${\Omega _\epsilon }: = \{ x \in \Omega :\left| {x - {\xi _0}} \right| > \epsilon \} $$ of lower dimension. We show that if Ω is a bounded smooth domain in ℝ n , n ≥ 3, $${\xi _0} \in \Omega ,Q \in {C^2}(\overline \Omega )$$ is positive, and $$\nabla Q({\xi _0}) \ne 0$$ , then for small enough ∈ > 0, this problem has a positive solution u e which concentrates and blows up at Ξ 0 as ∈ → 0.

In this paper we consider the problem $$\displaystyle{ \qquad \left \{\begin{array}{ll} -\mathrm{div}(a(x)\nabla u) = c_{\alpha }a(x)\vert x -\xi \vert ^{\alpha }u^{p_{\alpha }\pm \varepsilon }&\mbox{ in }\varOmega, \\ u > 0 &\mbox{ in }\varOmega, \\ u = 0 &\mbox{ on }\partial \varOmega,\end{array} \right. }$$ (1) where Ω is a bounded smooth domain in \(\mathbb{R}^{N}\), N ≥ 3, the function \(a \in C^{2}(\overline{\varOmega })\) is strictly positive on \(\overline{\varOmega }\), \(\xi \in \varOmega\), \(p_{\alpha }:= \frac{N+2+2\alpha } {N-2}\), \(\varepsilon > 0\), c α : = (N +α)(N − 2) and α is a positive real number which is not an even integer. Here \(\varepsilon\) is a positive small parameter. We give some sufficient conditions on the function a which ensure existence of solutions to (1) blowing up at the point \(\xi\) as \(\varepsilon\) goes to zero.

We consider the boundary-value problemwhere Br0 is the ball of radius r0 in ℝN, N ≥ 2, λ > 0 and ν is the outer normal derivative at ∂Br0. This problem is equivalent to the stationary Keller—Segel system from chemotaxis. We show the existence of a solution concentrating at the boundary of the ball as λ goes to 0.

In this paper we consider the study of positive solutions of¶¶\( -\varepsilon^2\Delta u+\lambda u=f(x,u)\quad {\rm on}\quad \mathbb{R}^N, \)¶¶where e is a small parameter, λ>0 and f is an appropriate function. Here we find multi-peak solutions exhibiting concentration at any prescribed "stable" set of zeroes of the field¶¶\( {\cal S}(P)=\int\limits_{\mathbb{R}^N}\left[\nabla_xf(P,U_P(y))\cdot y\right]\nabla U_P(y)dy,\quad P\in \mathbb{R}^N, \)¶¶where UP is the unique radial solution of the limit equation¶¶\( -\Delta U_P+\lambda U_P=f(P,U_P)\quad {\rm on} \quad \mathbb{R}^N. \)¶¶Conversely, we show that the points at which a sequence of multi-peak solutions concentrate must be zeroes of the field \( {\cal S} \).

We prove the existence of critical points of the $N$-vortex Hamiltonian $H_\Omega (x_1,\ldots, x_N) =\sum\limits^N_{i=1}\Gamma^2_i h(x_i) + \sum\limits_{i,j=1\atop j\not= k}^N \Gamma_i\Gamma_jG(x_i,x_j)+2\sum\limits_{i=1}^N\Gamma_i\psi_0(x_i)$ in a bounded domain $\Omega\subset{\mathbb R}^2$ which may be simply or multiply connected. Here $G$ denotes the Green function for the Dirichlet Laplace operator in $\Omega$, more generally a hydrodynamic Green function, and $h$ the Robin function. Moreover $\psi_0\in C^1(\overline\Omega)$ is a harmonic function on $\Omega$. We obtain new critical points $x=(x_1,\dots,x_N)$ for $N=3$ or $N=4$ under conditions on the vorticities $\Gamma_i\in{\mathbb R}\setminus\{0\}$. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case $\Gamma_i=(-1)^i$ of counter-rotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for $N$ vorticFity blobs near $x_1,\dots,x_N$.

We study a perturbed semilinear problem with Neumann boundary condition \[ \cases{ -\varepsilon^2\Delta u+u=u^p & {\rm in} \Omega \cr &\cr u>0 & {\rm in} \Omega\cr &\cr {{\partial u}\over{\partial\nu}}=0& {\rm in} \partial\Omega,\cr} \] where $\Omega$ is a bounded smooth domain of ${mathbb{R}}^N$ , $N\ge2$ , $\varepsilon>0$ , $1 < p < {{N+2}\over{N-2}}$ if $N\ge3$ or $p>1$ if $N=2$ and $\nu$ is the unit outward normal at the boundary of $\Omega$ . We show that for any fixed positive integer K any “suitable” critical point $(x_0^1,\dots,x_0^K)$ of the function \begin{eqnarray*} \lefteqn{\varphi_K(x^1,\dots,x^K)} &=& \min\left\{{\rm dist}(x^i,{\partial\Omega}),{|x^j-x^l|\over2} \mid i,j,l=1.\dots,K, j\ne l\right\} \end{eqnarray*} generates a family of multiple interior spike solutions, whose local maximum points $x_\varepsilon^1,\dots,x_\varepsilon^K$ tend to $x_0^1,\dots,x_0^K$ as $\varepsilon$ tends to zero.

We consider the supercritical Lane–Emden problem \( (P_{\varepsilon}) \quad -\Delta \upsilon = |\upsilon|^{P{\varepsilon}-1}\upsilon\; \rm in \; \mathcal{A}, \quad \upsilon=0 \; \rm on \; \partial \mathcal{A} \) where \( \mathrm{\mathcal{A} \ is\ an \ annulux\ in \ }\mathbb{R}^{2m}, m \geq 2 \ \rm and \ p_{\varepsilon}=\frac{(m+1+2)}{(m+1)-2}- \varepsilon, \varepsilon > 0. \) We prove the existence of positive and sign changing solutions of \( (p_{\varepsilon})\) concentrating and blowing-up \( \varepsilon \ \rightarrow \ 0, \rm{on}(m-1)- \) dimensional spheres. Using a reduction method ([18, 14]) we transform problem \( (p_{\varepsilon})\) into a nonhomogeneous problem in an annulus \( \mathcal{D}\subset \mathbb{R}^{m+1} \) which can be solved by a Ljapunov–Schmidt finite-dimensional reduction.

Let ( M , g ) be an n-dimensional compact Riemannian manifold without boundary and Γ be a non-degenerate closed geodesic of ( M , g ) . We prove that the supercritical problem − Δ g u + h u = u n + 1 n − 3 ± ϵ , u > 0 , in ( M , g ) has a solution that concentrates along Γ as ϵ goes to zero, provided the function h and the sectional curvatures along Γ satisfy a suitable condition. A connection with the solution of a class of periodic Ordinary Differential Equations with singularity of attractive or repulsive type is established.

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold \(\left( M,g\right) \) is compact. Established in the locally conformally flat case by Schoen (Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin 1989, Surveys Pure Application and Mathematics, 52 Longman Science, Technology, pp. 311–320. Harlow 1991) and for \(n\le 24\) by Khuri–Marques–Schoen (J Differ Geom 81(1):143–196, 2009), it has revealed to be generally false for \(n\ge 25\) as shown by Brendle (J Am Math Soc 21(4):951–979, 2008) and Brendle–Marques (J Differ Geom 81(2):225–250, 2009). A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential \(\frac{n-2}{4(n-1)} {{\mathrm{Scal}}}_g,\, {{\mathrm{Scal}}}_g\) being the Scalar curvature of \(\left( M,g\right) \). We show that a-priori \(L^\infty \)–bounds fail for linear perturbations on all manifolds with \(n\ge 4\) as well as a-priori gradient \(L^2\)–bounds fail for non-locally conformally flat manifolds with \(n\ge 6\) and for locally conformally flat manifolds with \(n\ge 7\). In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of \(g\).

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior is asymmetric, and moreover one component includes also a certain global mass. The proof uses singular perturbation methods., Comment: arXiv admin note: text overlap with arXiv:1407.8407

Let ( M , g ) be an m -dimensional compact Riemannian manifold without boundary. Assume κ ∈ C 2 ( M ) is such that − Δ g + κ is coercive. We prove the existence of solutions to the supercritical problems − Δ g u + κ u = u p , u > 0 in ( M , g ) and − Δ g u + κ u = λ e u in ( M , g ) which concentrate along an ( m − 1 ) -dimensional submanifold of M as p → ∞ and λ → 0 , respectively, under suitable symmetry assumptions on the manifold M . A connection with the solutions of a class of periodic ODE’s is established.

Let (M,g) be a smooth, compact Riemannian manifold of dimension N \geq 3. We consider the almost critical problem (P_\epsilon) -\Delta_g u+ {N-2\over 4(N-1)} Scal_g u= u^{{N+2\over N-2}+\epsilon } in} M, u>0 in M, where \Delta_g denotes the Laplace-Beltrami operator, Scal_g is the scalar curvature of g and \epsilon \in R is a small parameter. It is known that problem (P_\epsilon) does not have any blowing-up solutions when \epsilon \to 0^-, at least for N \leq 24 or in the locally conformally flat case, and this is not true anymore when \epsilon \to 0^+. Indeed, we prove that, if N \geq 7 and the manifold is not locally conformally flat, then problem (P_\epsilon) does have a family of solutions which blow-up at a maximum point of the function \xi \to |Weyl_g(\xi)|_g as \epsilon \to 0^+. Here Weyl_g denotes the Weyl curvature tensor of g.

The set of points (α, β) ∈ℝ2 for which the problem − δu = αu+ − βu− in Ω, u = 0 on ∂Ω(u+ = max {u, o} and u− = min {−u, 0}) has nontrivial solutions is important for the study of certain nonlinear problems. It is shown that for ‘most’ bounded domains Ω in ℝn, such a set is locally the union of a finite number of curves.

We show a result of genericity for non degenerate critical points of the Robin function with respect to deformations of the domain.

— We construct sequences of sign changing solutions for some conformally invariant semilinear elliptic equation which is defined in Sn, when n ≥ 4. The solutions we obtain have large energy and concentrate along some special submanifolds of Sn. For example, when n ≥ 4, we obtain sequences of solutions whose energy concentrates along one great circle or finitely many great circles which are linked (they correspond to Hopf links embedded in S3 × {0} ⊂ Sn). In dimension n ≥ 5, we obtain sequences of solutions whose energy concentrates along a two dimensional torus (which corresponds to a Clifford torus embedded in S3 × {0} ⊂ Sn).

We study a class of problems which are perturbation of some critical problems, namely the Brezis-Nirenberg problem $$-\Delta{u}=|u|^{2^{*}-2}u\;+\;\epsilon{u}\;\mathrm{in}\;\Omega,\quad u=0\;\mathrm{on}\;\partial\Omega$$ the almost-critical problem $$-\Delta{u}=|u|^{2^{*}-2-\epsilon}u\;\mathrm{in}\;\Omega,\quad u=0\;\mathrm{on}\;\partial\Omega$$ the Coron problem $$-\Delta{u}=|u|^{2^{*}-2}u\;\mathrm{in}\;\Omega\;\backslash B(x_{0},\epsilon),\quad u=0\;\mathrm{on}\;\partial(\Omega\backslash B(x_{0},\epsilon)),$$

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\). The warped product \({M \times_\omega K}\) is the (m + k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.

In this work, we consider sign changing solutions to the critical elliptic problem $\Delta u + |u|^{\frac{4}{N-2}}u = 0$ in $\Omega_\varepsilon$ and $u=0$ on $\partial\Omega_\varepsilon$, where $\Omega_\varepsilon:=\Omega-\left(\bigcup_{i=1}^m (a_i+\varepsilon\Omega_i)\right)$ for small parameter $\varepsilon>0$ is a perforated domain, $\Omega$ and $\Omega_i$ with $0\in \Omega_i$ ($\forall i=1,\cdots,m$) are bounded regular general domains without symmetry in $\mathbb{R}^N$ and $a_i$ are points in $\Omega$ for all $i=1,\cdots,m$. As $\varepsilon$ goes to zero, we construct by gluing method solutions with multiple blow up at each point $a_i$ for all $i=1,\cdots,m$.

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents., Comment: Published online in Calculus of Variations and Partial Differential Equations