arXiv subjects

Sérgio Neves

Publications and source records attributed to Sérgio Neves.

3 recordsLinked to original sources

arXiv2015

Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane

In this paper we study the problem -Δu =\left(\frac{2+α}{2}\right)^2\abs{x}^αf(λ,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 $\a$, $ł$ 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 $ł$. The case $f(λ,u)=łe^u$ provides more detailed information.

math.AP↗

arXiv2013

Nonradial solutions for the Hénon equation in $R^N$

In this paper we consider the problem $$ {ll} -Δu=(N+\a)(N-2)|x|^{\a}u^\frac{N+2+2\a}{N-2} & in R^N u>0& in R^N u\in D^{1,2}(R^N). $$ 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 $α$ is an even integer.

math.AP↗

arXiv2012

Exact multiplicity results for a singularly perturbed Neumann problem

In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 Δu + u = u^p, &\text{in}Ωu > 0, &\text{in}Ω\frac{\partial u}{\partial ν} = 0,& \text{on}\partial Ω{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.

math.AP↗