Minimax Theory and its Applications 08 (2023), No. 2, 257--284
Copyright Heldermann Verlag 2023
Local Mountain Pass for a Class of Elliptic Systems without Homogeneity on the Nonlinearity
Giovany M. Figueiredo
Dep. de Matemática, Universidade de Brasília, Brazil
giovany@unb.br
Segundo Manuel A. Salirrosas
Dep. de Matemática, Universidade de Brasília, Brazil
semaarsa@gmail.com
[Abstract-pdf]
We consider the gradient elliptic system given by $$ \left\{ \begin{array}{l} -\varepsilon^2\mbox{div}(a(x) \nabla u) + u = Q_{u}(u,v)+\lambda K_u(u,v)~~\text{in}~~ \mathbb{R}^N, \\[2mm] -\varepsilon^2\mbox{div}(b(x) \nabla v) + v = Q_{v}(u,v)+\lambda K_v(u,v)~~\text{in}~~ \mathbb{R}^N, \end{array} \right. $$ where the potentials $a,b$ are continuous, the nonlinearity $Q+\lambda K$ is not homogeneous. We study the subcritical, critical and supercritical cases. For $\varepsilon >0$ small we show existence and concentration results using the penalization method.
Keywords: Gradient elliptic systems, Schroedinger equation.
MSC: 35J20, 35J50.
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