, Volume 12, Issue 1, pp 271-282

Semilinear equations in ℝ N without condition at infinity

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Abstract

In this paper we establish that some nonlinear elliptic (and parabolic) problems are well posed in all of ℝ N without prescribing the behavior at infinity. A typical example is the following: Let 1<p<∞. For everyf ∈ L loc 1 (ℝ N ) there is a uniqueu ∈ L loc p (ℝ N ) satisfying $$ - \Delta u + |u|^{p - 1} u = f(x) on \mathbb{R}^N $$ .

Communicated by D. Kinderlehrer