Abstract
We consider a Lazer-Mckenna-type problem involving the fractional Laplacian and singular nonlinearity. We investigate existence, regularity and uniqueness of solutions in light of the interplay between the nonlinearities and the summability of the datum.
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The authors are thankful to the anonymous referees for their critical reviews and constructive suggestions that improved the quality of the manuscript.
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Appendix
Appendix
We start by proving the following lemma which we have used in the proof of Lemma 4.4.
Lemma 5.3
Let \(F(x)=x^r\), \(0<r<1\), for every \(x>0\). Then for every function \(v :\mathbb {R}^N\rightarrow ]0,+\infty [\) that satisfies
we have
Proof
Following [20, Lemma 2.3.], we can use Taylor’s formula obtaining for every \((x,y)\in \mathbb {R}^{N}\times \mathbb {R}^{N}\)
where
On other hand, since the function \(F^{\prime \prime }\) is increasing we have
Hence, it follows
Then, from (5.2) we obtain
Dividing both sides of this inequality by \(|x-y|^{N+2s}\) and then integrating with respect to the variable y we arrive at
which proves (5.1). \(\square \)
In the following result we extend the space of admissible test functions in (2.4).
Lemma 5.4
Let \(u\in X_{0}^{s}(\Omega )\) be a solution of the problem (1.1) taken in the sense of Definition 2.1 with \(f\in L^1(\Omega )\). Then for every \(\phi \in X_{0}^{s}(\Omega )\) we get \(\frac{f\phi }{u^\gamma }\in L^1(\Omega )\) and
Proof
Take an arbitrary \(\phi \in X_{0}^{s}(\Omega )\). By [29, Theorem 6] there exists a sequence \(\{\varphi _n\}_n\subset \mathcal {C}_{0}^{\infty }(\Omega )\) such that \(\varphi _n\rightarrow \phi \) in norm in \(H^{s}(\mathbb {R}^N)\). Writing (2.4) with \(\varphi _n\in \mathcal {C}_{0}^{\infty }(\Omega )\) we obtain
in which we shall pass to the limit as n tends to \(+\infty \). Starting with the left-hand side of (5.4), we consider the following two functions
Notice that the convergence \(\varphi _n\rightarrow \phi \) in norm in \(H^{s}(\mathbb {R}^N)\) implies that the sequence \(\{F_n(x,y)\}_n\) converges to F(x, y) in \(L^{2}(\mathbb {R}^{2N})\) and, up to a subsequence if necessary, we can assume that \(\{F_n(x,y)\}_n\) converges almost everywhere in \(\mathbb {R}^{2N}\).
As \(u\in X_{0}^{s}(\Omega )\) we have \(\frac{(u(x)-u(y))}{|x-y|^{\frac{N+2s}{2}}}\in L^{2}(\mathbb {R}^{2N})\) implying
For the term in the right-hand side of (5.4), we first note that thanks to [38, Proposition 3.] the two functions \((\varphi _n-\varphi _k)^+\) and \((\varphi _n-\varphi _k)^-\) are both admissible test functions in (2.4). Taking them so we obtain
and
Then, summing up both the two equalities we have
and then the Hölder inequality implies
Thus, we deduce that \(\Big \{\frac{f\varphi _n}{u^{\gamma }}\Big \}_n\) is a Cauchy sequence in \(L^1(\Omega )\). Since \(\varphi _n\) converges to \(\varphi \) a.e. in \(\Omega \), the sequence \(\Big \{\frac{f\varphi _n}{u^{\gamma }}\Big \}_n\) converges to \(\frac{f\phi }{u^{\gamma }}\in L^1(\Omega )\) in norm in \(L^1(\Omega )\). So that the passage to the limit as n tends to infinity in (5.4) yields
for every \(\phi \in X_{0}^{s}(\Omega )\). \(\square \)
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Youssfi, A., Mahmoud, G.O.M. Nonlocal semilinear elliptic problems with singular nonlinearity. Calc. Var. 60, 153 (2021). https://doi.org/10.1007/s00526-021-02034-1
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DOI: https://doi.org/10.1007/s00526-021-02034-1