Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity

Lemma 2.1

Let

where q is as in condition (A2), r ≥ 1 , R > 0 , and ν > 0 with p − 1 + ν > 1 . Here,

Let a measurable function z in R N satisfy

Then the following properties hold.

1. Assume that r ≥ q j for some j ∈ { 0 , 1 , … , j ∗ − 2 } . Then

   G ∗ z ∈ g B C ( R N ) + L R q j + 1 ( R N ) .

2. Assume that r ≥ q j ∗ − 1 . Then G ∗ z ∈ g B C ( R N ) . Furthermore, there exist C ∗ > 0 and θ ∈ ( 0 , 1 ) such that

   ∣ [ G ∗ z ] ( x ) ∣ ≤ C ∗ g ( x ) , ∣ [ G ∗ z ] ( x ) − [ G ∗ z ] ( y ) ∣ ≤ C ∗ ∣ x − y ∣ θ ,

   for x , y ∈ R n . Here, C ∗ and θ depend only on N , F , q , ν , and r .

Proof

By (2.3) we find R ′ ∈ ( 0 , R ) such that f and h are continuous functions in R N \ B ( 0 , R ′ ) . Set

Then

On the other hand, by (2.3) and (2.4), we have

This together with (G3) and (2.2) implies that assertions (i) and (ii) hold with G ∗ z replaced by I 2 . It suffices to prove that assertions (i) and (ii) hold with G ∗ z replaced by I 1 .

We prove that assertion (i) holds with G ∗ z replaced by I 1 . Since F ′ ( f ) ∈ L q ∕ ( p − 1 ) ( B ( 0 , R ) ) and r ∗ < q ∕ ( p − 1 ) , it follows from (2.3) that

Let r ≥ q j , where j ∈ { 0 , 1 , … , j ∗ − 1 } . Then, by (2.5), we have

where 1 ∕ r j = 1 ∕ r ∗ + 1 ∕ q j . Here,

Since

by (G2), we see that I 1 ∈ L q j + 1 ( R N ) . Furthermore, for any R ″ ∈ ( R ′ , ∞ ) ,

for x , x ′ ∈ R N \ B ( 0 , R ″ ) . Then assertion (i) holds with G ∗ z replaced by I 1 . Thus, assertion (i) follows.

On the other hand, since j ≥ j ∗ − 1 , similar to (2.6), we see that q j + 1 < 0 . Then, by (G3) and (2.7), we see that assertion (ii) holds with G ∗ z replaced by I 1 . Thus, assertion (ii) follows. The proof of Lemma 2.1 is complete.□

Lemma 2.2

Let μ ∈ ℳ + . Assume conditions (A1) and (A2). Let κ > 0 and j = 0 , 1 , … . Then

1. U j κ ∈ g B C ( R N ) + L R q ( R N ) ;

2. V j κ ∈ g B C ( R N ) + L R q j ( R N ) if j < j ∗ and V j κ ∈ g B C ( R N ) if j ≥ j ∗ .

Lemma 2.3

Let μ ∈ ℳ + and p > 1 . Assume conditions (A1) and (A2). For any κ > 0 , the following statements are equivalent:

1. u = w + U j ∗ κ is a ( C 0 + L c q ) -solution to problem (P);

2. w ∈ g B C ( R N ) is positive in R N and w satisfies

   w = G ∗ [ F ( w + U j ∗ κ ) − F ( U j ∗ − 1 κ ) ] i n R N .

Lemma 2.4

Let μ ∈ ℳ + , 1 < p < p J L , and 0 < κ ≤ κ ∗ . Assume conditions (A1) and (A2). Let ϕ κ be the first eigenfunction to problem ( E κ ) such that ϕ κ > 0 in R N and

Then ϕ κ ∈ C 0 ( R N ) and

Furthermore,

1. C − 1 g ( x ) ≤ ϕ κ ( x ) ≤ C g ( x ) in R N for some C > 0 ,

2. 1 < λ κ ′ ≤ λ κ if 0 < κ ≤ κ ′ < κ ∗ and λ κ ∗ = 1 .

Lemma 2.5

Assume conditions (A1) and (A2). Let ν ∈ ( 0 , 1 ] with p − 1 + ν > 1 . Then the operator T κ defined by (2.9) is a bounded linear operator from g ν B C ( R N ) to g B C ( R N ) . Furthermore, if ν < 1 , then T κ is a compact operator on g ν B C ( R N ) .

Proof

Let f ∈ g ν B C ( R N ) be such that ∣ ∣ ∣ f ∣ ∣ ∣ ν ≤ 1 . By (2.8), we apply Lemma 2.1 (ii) to find C ∗ > 0 and θ ∈ ( 0 , 1 ) , which are independent of f , such that

for all x , y ∈ R N . These imply that T κ is a bounded linear operator on g ν B C .

Let { f n } be a bounded sequence of g ν B C with ν < 1 . By (2.10) and (2.11), we apply the Arzelà-Ascoli theorem and the diagonal argument to find a subsequence { T κ f n i } of { T κ f n } such that

for all compact sets K of R N . Furthermore, for any ε > 0 , we find L > 0 such that

for all x ∈ R N \ B ( 0 , L ) and n = 1 , 2 , … . By (2.12) and (2.13), we see that

Since ε is arbitrary, we deduce that