Two solutions for an elliptic problem with two singular terms

Two solutions for an elliptic problem with two singular terms We study the Dirichlet boundary value problem with 0-boundary data for the semilinear elliptic equation $$-\Delta u= (\lambda u^{s-1}-u^{r-1})\chi _{\{u>0\}}$$ - Δ u = ( λ u s - 1 - u r - 1 ) χ { u > 0 } in a bounded domain $$\Omega $$ Ω , where $$0<r<s<1$$ 0 < r < s < 1 and $$\lambda \in (0,\infty )$$ λ ∈ ( 0 , ∞ ) . In particular, for $$\lambda $$ λ large enough, we prove the existence of at least two nonnegative solutions, one of which is positive, satisfies the Hopf’s boundary condition and corresponds to a local minimum of the energy functional. This paper is motivated by a recent result of the authors where the same conclusion was obtained for the case $$0<r\le 1< s<2$$ 0 < r ≤ 1 < s < 2 . Calculus of Variations and Partial Differential Equations Springer Journals

Two solutions for an elliptic problem with two singular terms

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Springer Berlin Heidelberg
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
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