Sesión Ecuaciones Diferenciales y AplicacionesMulti-Bump Solutions for Anisotropic Elliptic Problems
Carolina Camargo
Universidad Nacional de Cuyo-CONICET, Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In this talk we study the existence, multiplicity, and concentration of positive solutions for a class of quasilinear elliptic problems governed by the anisotropic \(p\)-Laplacian \[-\Delta_{\vec{p}} u + (\lambda V(x) + Z(x)) u^{p_0-1} = u^{q-1}, \quad u>0 \text{ in } \mathbb{R}^N,\] where, for a vector of exponents \(\vec{p}=(p_1,\dots,p_N)\), the anisotropic operator is defined as \[-\Delta_{\vec{p}} u := -\sum_{i=1}^{N} \partial_{x_i}\left(|\partial_{x_i} u|^{p_i-2}\partial_{x_i} u\right),\] \(\lambda>0\) is a parameter, and the potentials \(V, Z\) are continuous, nonnegative, satisfy \(\lambda V(x)+Z(x)\geq M_0>0\) for all \(x\in\mathbb{R}^N\) and all \(\lambda\geq 1\), and are such that the set where \(V\) vanishes has \(k\) disjoint connected components (the potential wells).
We extend to the anisotropic setting the variational techniques developed in [1] for the isotropic case (when all exponents \(p_1,\dots,p_N\) coincide). A truncated energy functional is constructed on the weighted anisotropic Sobolev space \(E_\lambda\), and it is shown that this functional satisfies the Palais-Smale condition for every \(\lambda\geq 1\). Using multidimensional minimax theory, we prove that for \(\lambda\) sufficiently large the problem admits at least \(2^k-1\) positive solutions, where \(k\) is the number of wells. Moreover, we establish that the resulting families of solutions concentrate, as \(\lambda\to\infty\), on the chosen components of the wells, converging to least energy solutions of the corresponding limit problems.
Trabajo en conjunto con: Pablo Ochoa (Universidad Nacional de Cuyo-CONICET, Argentina) y Analía Silva (Universidad Nacional de San Luis-CONICET, IMASL, Argentina).
Referencias
[1] C.O. Alves, Existence of multi-bump solutions for a class of quasilinear problems, Advanced Nonlinear Studies 6 (2006), 491-509.