Abstract
AbstractThe present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$
R
+
N
:
=
{
x
∈
R
N
:
x
N
>
0
}
for some $$N \ge 1$$
N
≥
1
and $$p > 1$$
p
>
1
, $$c > 0$$
c
>
0
are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$
P
c
). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$
P
c
) depend in a striking way on the value of $$c > 0$$
c
>
0
and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$
c
p
∈
(
1
,
e
)
, depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$
N
≥
2
, we prove that, for $$0< c < {c_p}$$
0
<
c
<
c
p
, problem ($$P_c$$
P
c
) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$
c
>
c
p
, there are no bounded positive solutions to ($$P_c$$
P
c
).
Publisher
Springer Science and Business Media LLC
Reference34 articles.
1. Ao, W., Musso, M., Pacard, F., Wei, J.: Solutions without any symmetry for semilinear elliptic problems. J. Funct. Anal. 270(3), 884–956 (2016)
2. Bartsch, T., Willem, M.: Infinitely many nonradial solutions of a Euclidean scalar field equation. J. Funct. Anal. 117(2), 447–460 (1993)
3. Bartsch, T., Willem, M.: Infinitely many radial solutions of a semilinear elliptic problem on $${ R}^N$$. Arch. Ration. Mech. Anal. 124(3), 261–276 (1993)
4. Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 69–94 (1997)
5. Berestycki, H., Caffarelli, L.A., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipschitz domains. Commun. Pure Appl. Math. 50(11), 1089–1111 (1997)
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