Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Definition 1. One says u ∈ E is a weak solution of problem (2) if

holds for all v ∈ E.

Lemma 2 (see [13].)If v is a conditional critical point of Φ(·), under the constraint ∥v∥_E = 1, then u = t(v)v is a critical point of J(·), where Φ(v) = J(t(v)v) and t(v) is a nonnegative solution of (13).

Lemma 3. Assume (H₂)–(H₄). Then the embedding E↪L^q(Ω; |f|) is compact.

Proof. Let u ∈ E. Since p/(p − q) ≥ δ > q₀, it follows that p ≤ qδ^′ = qδ/(δ − 1) < p^*. Let W^1,p(Ω) be the standard Banach space endowed with the norm $$. By assumptions (H₂)-(H₃), E ~ W^1,p(Ω). Similar to the proof of [10, Lemma 2] (see also the proof of [14, Theorem 7.9]), we can prove that $$ is compact and so is $$. Let S₁ be the best trace embedding constant; that is,

By Hölder′s inequality, we have This shows that the embedding E↪L^q(Ω; |f|) is continuous.

Assume {u_n} is a bounded sequence in E. Then by the compact embedding $$, there exist u ∈ E and a subsequence of {u_n} (not relabelled) such that u_n → u strongly in $$.

By Hölder′s inequality again, we infer

This completes the proof.

Lemma 4. Assume (H₂)-(H₃) and (H₅). Then the embedding E↪L^r(Γ; g) is compact.

Proof. Let u ∈ E. Since p/(p − r) ≥ σ > r₀, it follows that p ≤ rσ^′ = rσ/(σ − 1) < p_*. Hence the embedding $$ is compact (see [15, 16]). Let S₂ be the best trace embedding constant; that is,

By Hölder′s inequality, we have This shows that the embedding E↪L^r(Γ; g) is continuous.

Assume {u_n} is a bounded sequence in E. Then by the compact embedding $$, there exist u ∈ E and a subsequence of {u_n} (not relabelled) such that u_n → u strongly in $$.

By Hölder′s inequality again, we infer

This completes the proof.

Lemma 5. Let E be a real Banach space and J ∈ C¹(E, ℝ) with J(0) = 0. Suppose

•  

  A₁ there are ρ, α > 0 such that J(u) ≥ α for ∥u∥_E = ρ;

•  

  A₂ there is e ∈ E,∥e∥_E > ρ such that J(e) < 0.

Define

Then is finite and J(·) possess a (PS) _c sequence at level c. Furthermore, if J satisfies the (PS) condition, then c is a critical value of J.

Lemma 6 (fountain theorem [19]). Assume J ∈ C¹(X, ℝ¹) is an even functional that satisfies the (PS)_c condition. If for every k ∈ ℕ there exist ρ_k > r_k > 0 such that

•  

  B₁ $$ as k → ∞,

•  

  B₂ $$,

then J has a sequence of critical points {u_k} with J(u_k) → ∞.

Lemma 7 (see [20], [21].)Let J ∈ C¹(X, ℝ¹), where X is a Banach space. Assume that J satisfies the (PS)_c condition and is even and bounded from below, and J(0) = 0. If for any k ∈ ℕ there exists a k-dimensional subspace Y_k and ρ_k > 0 such that

then J has a sequence of critical values c_k < 0 satisfying c_k → 0 as k → ∞.

Theorem 8. Let (H₁)–(H₅) hold with either q < min {p, r} or q > max {p, r}. Then problem (2) admits a nonnegative nontrivial weak solution u ∈ E∖{0} which is also a ground state.

Proof. Suppose q < min {p, r}. Rewriting (13) as

we immediately see that for every v ∈ F₊ (where F₊ is defined by (9)) there exists a unique t(v) > 0 satisfying (25). Moreover, it can be easily checked that

Consider now the variational problem

Let {v_n} be a minimizing sequence in F₊ with $$. Then there exists v₀ ∈ E such that v_n⇀v₀ in E. By Lemmas 3 and 4, we have F(v_n) → F(v₀) ≥ 0 and G(v_n) → G(v₀) ≥ 0.

We first assert that v₀ ∈ F₊. Suppose the contrary; then F(v₀) = 0. In view of (25),

Letting n → ∞, it follows that t(v_n) → 0. Thus

which contradicts M < 0.

Next, we prove $$. If not, then $$. So, there exists μ > 1 such that $$. From (25), we have

This and (26) yield

On the other hand, it follows from (29) that {t(v_n)} is bounded and so there exists a subsequence (not relabelled) such that t(v_n) → t₀ > 0. Thus by (25), we have

Hence t₀ < t(v₀). Notice that

is strictly decreasing for all t > 0; we have which is a contradiction. So, $$ and v₀ is a critical point of Φ(·). By Lemma 2, u = t(v₀)v₀ is a nontrivial solution of problem (2). Since t = t(v) is a unique solution of (25), then (25) generates a bijection between E∖{0} and 𝒩 and so the obtained solution is actually a ground state.

The case q > max {p, r} can be treated in a similar way.

Remark 9. Afrouzi and Rasouli [12] consider the following problem:

where Ω is a bounded domain in ℝ^N and 1 < q < p < r < p_* = (N − 1)p/(N − p). The functions f and g are continuous functions which change sign in $$. Using the Nehari manifold and fibering map, they proved that problem (36) has at least two nontrivial nonnegative solutions if |λ| is sufficiently small. In fact, by slight modification, we can prove that the result they established is still true if Ω is a smooth exterior domain or the parameters satisfy 1 < r < p < q < p^* = Np/(N − p). But for g(x) ≥ 0, we can prove that problem (36) has at least one nontrivial nonnegative solution for |λ| is sufficiently small via the method used in [12]. Notice that our result (Theorem 8) does not need |λ| to be small.

Theorem 10. Let (H₁)–(H₅) hold with r < q < p. Then problem (2) has a nonnegative nontrivial weak solution.

Proof. From Lemma 3, we have

Thus So J(u) is coercive by p > q.

By Lemmas 3 and 4, it is easy to verify that J is weakly lower semicontinuous. So J has a minimum point u in E and u is a weak solution of (2).

In the following, we prove inf _u∈EJ(u) < 0. Let $$. Then

Thus J(tv) < 0 for t > 0 is sufficiently small. Notice that $$; we obtain Thus the minimum point of J is nontrivial.

Theorem 11. Let (H₁)–(H₅) hold with p < q < r. Then problem (2) has a nonnegative nontrivial weak solution in $$.

Proof. Let $$. Then

Thus J(tu) ≥ α > 0 for t small and J(tu) < 0 for t large.

Let $$ satisfy J(u_n) → c in E and J^′(u_n) → 0 in E^*. Then

This shows that $$ is bounded in E. Up to a subsequence, we obtain u_n⇀u in E. Thus It follows from Lemmas 3 and 4 that

Hence

Then (45) give that 〈J^′(u), u_n − u〉→0. Notice that J^′(u_n) → 0; we have where

Using the standard inequality in ℝ^N given by

we have from (46) that u_n → u in E. Thus J(·) satisfies (PS) condition. Then the assertion of this theorem follows from Lemma 5.

Theorem 12. Let (H₁)–(H₄) and ($$) hold with p < q < r. Suppose also

where ξ = (r − p) ^r−p/((q − p) ^q−p(r − q) ^r−q)(r/p) ^r−q. Then problem (2) has a nonnegative nontrivial weak solution in $$.

Proof. Define

Fix v₀ with $$. In view of assumption ($$), we have G(v₀) > 0. For all $$, there exist t > 0 and $$ with ∥v∥_E = 1 such that u = tv. Moreover, $$. Thus as t → ∞. Hence J(·) is coercive in $$. By Lemmas 3 and 4, it is easy to verify that J is weakly lower semicontinuous. So J has a minimum point u in $$ and u is a weak solution of (2).

In the following, we prove that $$. Notice that ψ(t, v) = 0 as t → 0, ψ(t, v) = −∞ as t → +∞, and $$; we infer that ψ(t, v) attain its maximum at t_M(v), where

If $$, then (13) has exactly two solutions t₁(v) and t₂(v) with 0 < t₁(v) < t_M(v) < t₂(v). Let t(v) = t₂(v). We have from (11) and (12) that

Let $$. It follows that

which ensures J(t(v)v) < 0. Thus $$. This implies that the weak solution of (2) is nontrivial.

Remark 13. Condition (49) may be viewed as grading the “strength” of interaction induced by F(u) and G(u). Hence, qualitatively speaking, one may rephrase Theorem 12 as saying that if p < q < r, then problem (2) admits a nontrivial weak solution provided that F(·) “prevails” over G(·).

Theorem 14. Let (H₁)–(H₃), ($$), and (H₅) hold with q > max {p, r}. Then problem (2) has a sequence of solutions u_k in E with J(u_k) → ∞ as k → ∞.

Proof. We will prove this theorem by fountain theorem. The proof is divided into three steps.

(1) Let Y_k and Z_k be defined by (23) and $$. Then it follows that β_k → 0 (see [22]). Therefore, we have

Choosing $$, we obtain that if u ∈ Z_k,∥u∥_E = r_k, then Thus (B₁) in Lemma 6 is proved.

(2) Since in the finite dimensional space Y_k all norms are equivalent, there exist C > 0 such that $$ hold for all u ∈ Y_k. Thus by (19),

Therefore (B₂) in Lemma 6 is satisfied for every ρ_k : = ∥u∥_E > 0 large enough.

(3) Let {u_n} be a (PS) _c sequence of J. Then we have

Therefore {u_n} is bounded in E. Similar to the proof of Theorem 11, we can verify that J(·) satisfies (PS) condition.

Obviously, J(·) is an even functional and J(0) = 0. Thus the assertion of Theorem 14 follows from Lemma 6.

Theorem 15. Let (H₁)–(H₃), ($$), and (H₅) hold with q < min {p, r}. Then problem (2) has a sequence of solutions u_k in E with J(u_k) < 0 and J(u_k) → 0 as k → ∞.

Proof. Since

and q < p,  J(·) is coercive and bounded from below. As before, we can verify that (PS) condition holds.

Let Y_k be defined by (23) and u ∈ Y_k. Since Y_k is a finite dimensional space and q < min {p, r}, we can choose ρ_k > 0 small enough such that

We obtain a sequence of solutions u_k by Lemma 7.