In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-Δ)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-Δ)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Bañuelos and Kulczycki.
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on star-shaped domains, where the bifurcation parameter is introduced by shrinking the domain. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
In this paper we study the quasilinear equation $- \ep^2 Δu-Δ_p u=f(u)$ in a smooth bounded domain $Ω$ with Dirichlet boundary condition. For $\ep \geq 0$, we review existence of a least energy nodal solution and then present information about the Morse Index of least nodal energy solutions this BVP. In particular we provide Morse Index information for the case $\ep =0$.
In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -Δu = ρ^2 (e^{u} - e^{-u}) & \text{ in } Ω\\ u=0 & \text{ on } \partial Ω, \end{cases} \end{equation} where $ρ$ is a small parameter and $Ω$ is a $C^2$ bounded domain in $\mathbb{R}^2$. [1] proves the existence of a $m$-point blow-up solution $u_ρ$ jointly with its asymptotic behaviour. we compute the Morse index of $u_ρ$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.
Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $Ω\subset \mathbb{R}^d$ is deformed. For a family of domains $\{Ω_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $Ω_a$ differs from the Morse index of $L$ on $Ω_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $Ω$. This is particularly useful when $Ω_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $Ω_b$) and the …
We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multipl…
In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions as $L^{\infty}$-bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a $C^0$-topology weighted with a suitable power of t…
We consider the Dirichlet problem for the Schrödinger-Hénon system $$ -Δu + μ_1 u = |x|^α\partial_u F(u,v),\quad \qquad -Δv + μ_2 v = |x|^α\partial_v F(u,v) $$ in the unit ball $Ω\subset \mathbb{R}^N, N\geq 2$, where $α>-1$ is a parameter and $F: \mathbb{R}^2 \to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v) \not = (0,0)$. We show that, as $α\to \infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar Hénon equation and extends a previou…
We study a class of $p$-Laplacian Dirichlet problems with weights that are possibly singular on the boundary of the domain, and obtain nontrivial solutions using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups.
In this paper we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent $p$ is close to $1$. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle $2π/n$. By considerations based on the Morse index we see that, depending on the values of $α$ and $n$, such least energy solutions can be radial, or nonradial and…
In this work, a Timoshenko system of type III of thermoelasticity with frictional versus viscoelastic under Dirichlet-Dirichlet-Neumann boundary conditions was considered. By exploiting energy method to produce a suitable Lyapunov functional, we establish the global existence, exponential decay of Type-III case.
For a bounded domain $Ω\subset\mathbb{R}^n$ let $H_Ω:Ω\timesΩ\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $λ_1,\dots,λ_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_Ω:{\mathcal D}\capΩ^N\to\mathbb{R}$ defined as \[ f_Ω(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N λ_jλ_k H_Ω(x_j,x_k). \] We prove that $f_Ω$ is a Morse function for most domains $Ω$ of class ${\mathc…
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.
We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.
We consider nonlinear second order elliptic problems of the type \[ -Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω, \] where $Ω$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen-Cahn type $f(s)=λ(s-|s|^{p-1}s)$ with $p>1$ and $λ>λ_2(Ω)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit e…
We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.
In this paper we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the epsilon-Dirichlet problem obtained as a variational problem for the SU(2)-epsilon-Yang Mills functional. This is equivalent to proving the existence of multiple solutions to the Dirichlet problem for the SU(2)-Yang Mills functional with small boundary data. In the first paper of this series this non-compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function J(q), which is essentially the Yang Mills functiona…
The Dirichlet problem $$ \begin{cases} Δ_{\infty}u-|Du|^2=0 \quad \text{on $Ω\subset \Rset ^n$} u|_{\partial Ω}=g \end{cases} $$ might have many solutions, where $Δ_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}$. In this paper, we prove that the maximal solution is the unique absolute minimizer for $H(p,z)={1\over 2}|p|^2-z$ from calculus of variations in $L^{\infty}$ and the minimal solution is the continuum value function from the "tug-of-war" game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting …
We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration ten…
We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. Thus we focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably t…
We carry out an investigation of the existence of infinitely many solutions to a fractional $p$-Kirchhoff type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A {\it `$C^1$ versus $W_0^{s,p}$'} analysis has also been discussed.
In this paper (Part I) and its sequels (Part II and Part III), we analyze the structure of the space of solutions to the epsilon-Dirichlet problem for the Yang-Mills equations on the 4-dimensional disk, for small values of the coupling constant epsilon. These are in one-to-one correspondence with solutions to the Dirichlet problem for the Yang Mills equations, for small boundary data. We prove the existence of multiple solutions, and, in particular, non minimal ones, and establish a Morse Theory for this non-compact variational problem. In part I, we describe the problem, state the main theore…
