Massimo Grossi

Massimo Grossi
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Massimo Grossi

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Mathematics - Analysis of PDEs (18)

Publications Authored By Massimo Grossi

We consider the following system of Liouville equations: $$\left\{\begin{array}{ll}-\Delta u_1=2e^{u_1}+\mu e^{u_2}&\text{in }\mathbb R^2\\-\Delta u_2=\mu e^{u_1}+2e^{u_2}&\text{in }\mathbb R^2\\\int_{\mathbb R^2}e^{u_1}<+\infty,\int_{\mathbb R^2}e^{u_2}<+\infty\end{array}\right.$$ We show existence of at least $n-\left[\frac{n}3\right]$ global branches of nonradial solutions bifurcating from $u_1(x)=u_2(x)=U(x)=\log\frac{64}{(2+\mu)\left(8+|x|^2\right)^2}$ at the values $\mu=-2\frac{n^2+n-2}{n^2+n+2}$ for any $n\in\mathbb N$. Read More

In this paper we establish existence of radial and nonradial solutions to the system $$ \begin{array}{ll} -\Delta u_1 = F_1(u_1,u_2) &\text{in }\mathbb{R}^N,\newline -\Delta u_2 = F_2(u_1,u_2) &\text{in }\mathbb{R}^N,\newline u_1\geq 0,\ u_2\geq 0 &\text{in }\mathbb{R}^N,\newline u_1,u_2\in D^{1,2}(\mathbb{R}^N), \end{array} $$ where $F_1,F_2$ are nonlinearities with critical behavior. Read More

In this paper we prove an existence result to the problem $$\left\{\begin{array}{ll} -\Delta u = |u|^{p-1} u \qquad & \text{in} \Omega, \\ u= 0 & \text{on} \partial\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in ${\mathbb R}^{N}$ which is a perturbation of the annulus. Then there exists a sequence $p_1Read More

In this paper we consider the non-variational system $$ \begin{cases} -\Delta u_i = \sum\limits_{j=1}^k a_{ij} u_j^{(N+2)/(N-2)} &\text{in }\mathbb R^N,\newline u_i>0 &\text{in }\mathbb R^N,\newline u_i\in D^{1,2}(\mathbb R^N). \end{cases} $$ and we give some sufficient conditions on the matrix $(a_{ij})_{i,j=1,\dotsc ,k}$ which ensure the existence of solutions bifurcating from the bubble of the critical Sobolev equation. Read More

In this paper we study the Hardy problem in R^N with N>2 and in a ball B of R^N. Using a suitable map we transform the Hardy problem into another one without the singular term. Then we obtain some bifurcation results from the radial positive solutions corresponding to some explicit values of the parameter lambda. Read More

Using the Gegenbauer polynomials and the zonal harmonics functions we give some representation formula of the Green function in the annulus. We apply this result to prove some uniqueness results for some nonlinear elliptic problems. Read More

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$. Read More

We study the following generalized $SU(3)$ Toda System $$ \left\{\begin{array}{ll} -\Delta u=2e^u+\mu e^v & \hbox{ in }\R^2\\ -\Delta v=2e^v+\mu e^u & \hbox{ in }\R^2\\ \int_{\R^2}e^u<+\infty,\ \int_{\R^2}e^v<+\infty \end{array}\right. $$ where $\mu>-2$. We prove the existence of radial solutions bifurcating from the radial solution $(\log \frac{64}{(2+\mu) (8+|x|^2)^2}, \log \frac{64}{ (2+\mu) (8+|x|^2)^2})$ at the values $\mu=\mu_n=2\frac{2-n-n^2}{2+n+n^2},\ n\in\N $. Read More

We derive a second order estimate for the first m eigenvalues and eigenfunctions of the linearized Gel'fand problem associated to solutions which blow-up at m points. This allows us to determine, in some suitable situations, some qualitative properties of the first m eigenfunctions as the number of points of concentration or the multiplicity of the eigenvalue . Read More

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth nonlinearity and $\a$, $\l$ are real numbers with $\a>0$. From a careful study of the linearized operator we compute the Morse index of some radial solutions to \eqref{i0}. Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter $\l$. Read More

In this paper we consider the problem $$ {ll} -\Delta u=(N+\a)(N-2)|x|^{\a}u^\frac{N+2+2\a}{N-2} & in R^N u>0& in R^N u\in D^{1,2}(R^N). $$ where $N\ge3$. From the characterization of the solutions of the linearized operator, we deduce the existence of nonradial solutions which bifurcate from the radial one when $\alpha$ is an even integer. Read More

In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 \Delta u + u = u^p, &\text{in}\Omega u > 0, &\text{in}\Omega \frac{\partial u}{\partial \nu} = 0,& \text{on}\partial \Omega {cases} {align*} for $\varepsilon$ small and $p$ subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed. Read More

We consider the sinh-Poisson equation $$(P)_\lambda\quad -\Delta u=\la\sinh u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\rr^2$ and $\lambda$ is a small positive parameter. If $0\in\Omega$ and $\Omega$ is symmetric with respect to the origin, for any integer $k$ if $\la$ is small enough, we construct a family of solutions to $(P)_\la$ which blows-up at the origin whose positive mass is $4\pi k(k-1)$ and negative mass is $4\pi k(k+1).$ It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Read More

Blow-up solutions to the two-dimensional Gel'fand problem are studied. It is known that the location of the blow-up points of these solutions is related to a Hamiltonian function involving the Green function of the domain. We show that this implies an equivalence between the Morse indices of the solutions and the associated critical points of the Hamiltonian. Read More

We consider the Lane-Emden Dirichlet problem -\Delta u = \abs{u}^{p-1}u, in B, u =0, on \partial B, where $p>1$ and $B$ denotes the unit ball in $\IR^2$. We study the asymptotic behavior of the least energy nodal radial solution $u_p$, as $p\rightarrow +\infty$. Assuming w. Read More

We study the nodal solutions of the Lane Emden Dirichlet problem $-\Delta u = |u|^{p-1}u with DBC on a smooth bounded domain $\Omega$ in $\IR^2$ and where $p>1$. We consider solutions $u_p$ satisfying $p \int_{\Omega}\abs{\nabla u_p}^2\to 16\pi e\quad\hbox{as}p\rightarrow+\infty\qquad (*)$ and we are interested in the shape and the asymptotic behavior as $p\rightarrow+\infty$. First we prove that (*) holds for least energy nodal solutions. Read More

We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns to be a constrained minimum of the associated energy functional. Read More

In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates which enable us to associate a "limit problem" to the initial one. Read More