Dual variational methods in critical point theory and applications
Consider the nonlinear elliptic partial differential equation \[ L(u) \equiv -\sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u = p(x,u),\quad x\in\Omega,\ u = 0,\ x \in\partial\Omega, \tag{*}\] where \(\Omega\subset\mathbb R^n\) is a smooth bounded domain. Formally, the critical points of the functional \[ I(u) = \int_\Omega \left[ \frac12 \sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u^2 - P(x,u(x))\right] \,dx, \] where \(P(x,u)\) is a primitive of \(p(x,u)\), are solutions of (*). The authors construct dual variational methods to enable them to prove the existence and estimate the number of critical points possessed by a real continuously differentiable functional on a real Banach space, and then apply their results to various existence problems for equations of type (*). They also apply them to problems with linear term added, i.e. \[ L(u) = a(x)u + p(x,u),\quad x\in\Omega;\ u=0,\ x \in\partial\Omega, \] as well as to nonlinear integral equations of the form \[ v(x) = \int_\Omega g(x,y)q(y,v(y))\,dy. \]
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