Existence and multiplicity of solutions for a locally coercive elliptic equation

David Arcoya, Francisco Odair Paiva () and José M. Mendoza
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David Arcoya: Universidad de Granada
Francisco Odair Paiva: Universidade Federal de São Carlos
José M. Mendoza: Universidade Federal do Espírito Santo

Partial Differential Equations and Applications, 2024, vol. 5, issue 2, 1-15

Abstract: Abstract For a bounded domain $$\Omega $$ Ω , we establish existence and multiplicity of nontrivial solutions for the semilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x) f(u), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}$$ - Δ u = g ( u ) - h ( x ) f ( u ) , in Ω u = 0 , on ∂ Ω , where $$h\in L^\infty (\Omega )$$ h ∈ L ∞ ( Ω ) is nonnegative and nontrivial, g is asymptotically linear, f is superlinear and $${g(0)}=f(0)=0$$ g ( 0 ) = f ( 0 ) = 0 . We also study the existence of solutions for the problem $$\begin{aligned} \left\{ \begin{array}{rcll} -\Delta u &{} = &{} {g(u)} - h(x)f(u)+k(x), &{} \text{ in } \Omega \\ u &{} = &{} 0, &{} \text{ on } \partial \Omega ,\\ \end{array} \right. \end{aligned}$$ - Δ u = g ( u ) - h ( x ) f ( u ) + k ( x ) , in Ω u = 0 , on ∂ Ω , when $$k\in L^2(\Omega )$$ k ∈ L 2 ( Ω ) .

Keywords: Semilinear elliptic problem; Linking theorem; Critical groups; 35J20; 35J61; 58E05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42985-024-00275-1

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