Invariance of critical points under Kelvin transform and multiple solutions in exterior domains of $\mathbb {R}^2$

Abstract

Let $\Omega ={\mathbb {R}}^2{\setminus }\overline{B(0,1)}$ be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions as well as sign-changing solutions of the following boundary value problem

$$\begin{aligned} -\Delta u=a(x) f(u)\ \text{ in } \Omega ,\quad u=0\ \text{ on } \partial \Omega =\partial B(0,1), \end{aligned}$$

where the nonnegative coefficient a satisfies a certain integrability condition. We are looking for solutions in the space $D^{1,2}_0(\Omega )$ which is the completion of $C^\infty _c(\Omega )$ with respect to the $\Vert \nabla \cdot \Vert _{2,\Omega }$-norm. Unlike in the situation of ${\mathbb {R}}^N$ with $N\ge 3$, the behavior of solutions in the borderline case $N=2$ considered here is qualitatively significantly different, such as for example, constant-sign solutions in the borderline case are not decaying to zero at infinity, and instead are bounded away from zero. Our main tool in studying the above problem will be the Kelvin transform. We will first show that the Kelvin transform provides an isometric isomorphism between $D^{1,2}_0(\Omega )$ and the Sobolev space $H^1_0(B(0,1))$, which is order-preserving. This allows us to establish a one-to-one mapping between solutions of the problem above and solutions of an associated problem in the bounded domain B(0, 1) of the form:

$$\begin{aligned} -\Delta u=b(x) f(u)\ \text{ in } B(0,1),\quad u=0\ \text{ on } \partial B(0,1), \end{aligned}$$

where b satisfies an integrability condition in terms of the coefficient a. This duality approach given via the Kelvin transform allows us to handle nonlinearities under sub, super or asymptotically linear hypotheses.