On the existence of positive solutions to a certain class of semilinear elliptic equations

Norihisa Ikoma ()
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Norihisa Ikoma: Keio University

Partial Differential Equations and Applications, 2021, vol. 2, issue 2, 1-27

Abstract: Abstract In this paper, we study the following semilinear elliptic equation $$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$ Δ u = φ V ( x ) u - f ( x , u ( x ) ) in R N , u ∈ H 1 ( R N ) where $$N \ge 1$$ N ≥ 1 and $$\varphi (s)$$ φ ( s ) , V(x), f(x, s) are given functions. Under some conditions on $$\varphi (s), V(x), f(x,s)$$ φ ( s ) , V ( x ) , f ( x , s ) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

Keywords: Positive solution; Mountain pass theorem; Concentration-compactness lemma; 35J20; 35J61; 35B09 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00079-7

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