Multiplicity Results for the p(x)-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

Definition 1. u ∈ W^1,p(x)(Ω) is called a generalized solution of the equation

if for all $$

Theorem 2. Assume that (H0)–(H5) hold and a(x) ≤ p^∗(x) = Np(x)/(N − p(x)) in (6). Then there exists Λ > 0 with the following properties:

• (1)

  Problem (P_λ) has a solution u_λ for every λ ∈ (0, Λ).

• (2)

  Problem (P_λ) has a solution if λ = Λ.

• (3)

  Problem (P_λ) does not have any solution if λ > Λ.

Theorem 3. Assume that (H0)–(H5) hold and a(x) < p^∗(x) in (6). Then, problem (P_λ) has at least two distinct solutions u_λ(x) and v_λ(x) for every λ ∈ (0, Λ).

Lemma 4. Consider the following:

Theorem 5 (see [10], [11].)Let $$ be an open bounded domain with Lipschitz boundary and assume that $$ with p(x) > 1 for each $$. If $$ and p(x) ≤ r(x) ≤ p^∗(x) for all $$, then there exists a continuous embedding W^1,p(x)(Ω)↪L^r(x)(Ω). Also, the embedding is compact r(x) < p^∗(x) almost everywhere in $$, where

Theorem 6. Let $$ be an open bounded domain with Lipschitz boundary. If q ∈ C(∂Ω) such that

then W^1,p(x)↪L^q(x)(∂Ω).

Lemma 7 (see [8], Lemma 3.2.)Let u, v ∈ W^1,p(·)(Ω) be nonnegative functions satisfying

Then u ≥ v almost everywhere in Ω.

Theorem 8. Let $$, for some 0 < β < 1, satisfy 0≨u, 0≨v and

with u = v = 0 on ∂Ω, where g, h ∈ L^∞(Ω) are such that 0 ≤ g < h pointwise everywhere in Ω. If where n is the inward unit normal on ∂Ω, then, the following strong comparison principle holds:

Lemma 9. There exists λ₀ > 0 such that (P_λ) admits a solution for λ < λ₀.

Proof. Using (14) and the embeddings in Theorem 5, we estimate E_λ(u) for $$ as follows:

Hence, noting that 0 < δ⁺ < 1 and 1 − δ⁺ < p⁺ < a⁻, we can choose r₀ > 0 small enough, and there exists δ₀ > 0 such that

Moreover, since W^1,p(x)(Ω)↪L^1−δ(x)(Ω), we have, for λ > 0, small enough, Set Now, note that, for t → 0⁺ and u ∈ W^1,p(x)(Ω), we have

Since 1 − δ⁺ < p⁺ < θ, this implies that E_λ(tu) → −∞ as t → ∞. Thus, $$ Now, let $$ be a minimizing sequence for c₀. Then $$ for r₀ ∈ (0, r). Now by the Ekeland variational principle, there exists a sequence {v_j} such that $$ and

Moreover, using the Brézis-Lieb lemma in [14] combined with the p(x) version of Theorem 2.1 in Boccardo and Murat [15], it follows that ∇v_j(x)→∇u_λ(x) for almost every x ∈ Ω, Hence, from (28) and the compact boundary trace embedding in Theorem 5, we get E_λ(v_j) ≥ E_λ(u_λ) + o_j(1). Thus, it follows that E_λ(u_λ) = c₀ < 0. Hence u_λ≢0 and it is a local minimizer of E_λ in W^1,p(x)(Ω).

Lemma 10. Problem (P_λ) possesses a solution u_λ for 0 < λ < Λ.

Proof. Fix 0 < λ < λ₂ < Λ. λ₂ such that there exist solutions to (P_λ) for λ = λ₂, say $$. Note that $$ is a supersolution for (P_λ). It is clear that 0 is not a local minimizer of E_λ on $$ since E_λ(0) = 0 and E_λ(tv) < 0 for t → 0⁺. Now, we show the existence of a local minimizer of the functional energy. For this, we use the cut-off argument. Define

Also define the functional $$ by where $$, $$, and $$ From the dominated convergence and the compact boundary trace embedding in Theorem 5, it is easy to see that $$ is bounded below and is weakly lower semicontinuous in W^1,p(x)(Ω). Then, there exists $$ such that $$ achieves its global minimum in W^1,p(x)(Ω). Moreover, since $$ is $$ from Lemma A.4, $$ solves the equation

Now, using the strong maximum principle (see Theorem 8) $$ and $$ since 0 is not a local minimizer of E_λ on $$ By the definition of $$, and $$ we have

Again, by the strong comparison principle (see Lemma 7), we conclude that $$ in Ω and $$ on ∂Ω. Hence $$ is a solution to (P_λ). This completes the proof of Lemma 10.

Lemma 11. There exists at least one positive weak solution for λ = Λ to problem (P_λ).

Proof. Let $$, λ_k↑Λ as k → ∞, and $$ be a solution of $$ such that $$ for all x ∈ Ω. Now, taking $$ as test function in $$, we get

Moreover, as $$, we have It follows that Now, from (7) there exists C > 0 such that, for θ > p₊ and for all t > 0, Moreover, using Theorem 4.2 in [16], we get the existence of the constants C₁ > 0, C₂ > 0 such that where θ₁ = max_d(x)≤σ⁡(p(x)/(p(x) − 1 + δ(x))),   θ₂ = min_d(x)≤σ⁡(p(x)/(p(x) − 1 + δ(x))), and d(x) = dist⁡(x, ∂Ω). On the other hand, we recall the following inequality due to Lieberman [17]. There exists a constant K(Ω) > 0 such that

Inserting (36), (37), and (38) in (33), we get $$ It follows that $$ is bounded in W^1,p(x)(Ω) since $$ for all x ∈ Ω. Without loss of generality, $$ in W^1,p(x)(Ω) and then by the Sobolev imbedding $$ in L^q(x)(Ω) and $$ for a.e. x ∈ Ω. By the L^∞(Ω)-regularity results of [13], the boundedness of $$ implies the boundedness of $$ By the C^1,α(Ω)-regularity Theorem 16, the boundedness of $$ implies the boundedness of $$, where α ∈ (0,1) is a constant. Thus, we have $$ in $$ For every v ∈ W^1,p(x)(Ω), since $$ is a solution of problem $$, we have that, for each k,

Passing to the limit in (39) as k → ∞ yields which shows that u_Λ is a solution of (P_Λ). Obviously u_Λ ≥ 0 and u_Λ≢0. Hence u_Λ is a positive solution of (P_Λ) in W^1,p(x)(Ω). This completes the proof of Lemma 11.

Lemma 12. Consider the following:

Λ < ∞.

Proof. Let u_λ be a solution of (P_λ). Taking φ ≡ 1 as a test function in the weak formulation of (P_λ), we get

On the other hand, we have Using assumption (H2) we have $$, and using (6) we have $$ Therefore, from (41) and (42) we get Now, since p⁺ − 1 < γ and −δ⁺ < γ, by the embedding of L^γ(Ω) into $$ and by the embedding of L^γ(Ω) into $$ we obtain Substituting (44) in (43) we get Hence, $$ is bounded by a constant independent of λ. Now, taking $$ as a test function in the weak formulation of (P_λ) and using (H2) we get Now since it follows that Λ is finite. The proof of Lemma 12 is now completed.

Proof of Theorem 2. Theorem 2 follows from Lemmas 10, 11, and 12.

Lemma 13. Let u_λ be the weak solution of problem (P_λ) obtained in Lemma 10. Then, u_λ is a local minimum for $$.

Proof. Fix 0 < λ₁ < λ < λ₂ < Λ and let $$ be solutions to (P_λ) for λ = λ₁ and λ = λ₂, respectively, such that $$. By Lemma 7,  $$ on $$ Define the following cut-off functions:

Then the corresponding functional $$ given by where $$, $$, and $$(x ∈ ∂Ω). Firstly, from Lemma A.4  $$ is $$ Then, it is simple to see that $$ is coercive and bounded below. Let u_λ denote the global minimum of $$ on W^1,p(x)(Ω) which satisfies the equation Therefore, from the regularity results Theorem 16, we conclude that u_λ is in $$ for some 0 < α < 1. Now, using (H2) and since f(x, t) is nondecreasing, by the definition of $$, $$, and $$ we have Again, by the strong comparison principle (see Lemma 7), we conclude that $$ in Ω. Let $$ We claim that β > 0. If not, then there exists x ∈ ∂Ω such that $$ and $$ But this contradicts the boundary data as λ₁ ≠ λ₂. Therefore, if $$ then $$ in $$ since $$ on the set $$ Hence u_λ is a local minimum for $$ This completes the proof of Lemma 13.

Lemma 14. Suppose that conditions (H0)–(H5) are satisfied. Let $$ satisfying

be a local minimizer of E_λ in $$ topology. Then, u₀ is a local minimum of E_λ in W^1,p(x)(Ω) also.

Proposition 15. Let $$ be a family of solutions to (P_ϵ), where u₀ satisfies (52) and solves (P_λ). Let γ > 1 be such that

Then,

Theorem 16. Let u ∈ W^1,p(x)(Ω)∩L^∞(Ω) be a solution to problem (P_λ). Then, there exists $$ such that any weak solution to problem (P_λ) belongs to $$ for some α ∈ (0,1).

Proof of Lemma 14. Assume that the conclusion of Lemma 14 is not true. We define the following constraint for each ϵ > 0:

We consider the following constraint minimization problem: Firstly, clearly I_ϵ > −∞. Moreover, we note that $$ is a convex set. Using the trace embeddings we see that $$ is also a closed set in W^1,p(x)(Ω) which implies that $$ is weakly closed in W^1,p(x)(Ω); with the fact that E_λ is weakly low semicontinuous in W^1,p(x)(Ω), it follows that for ϵ   ∈ (0,1)  I_ϵ is achieved on some $$; that is Moreover, since $$ and $$, we may assume that u_ϵ ≥ 0.

We now consider the following two cases.

(1) Let ρ(u_ϵ) < ϵ. Then u_ϵ is also a local minimizer of E_λ in W^1,p(x)(Ω). We now show that E_λ admits Gâteaux derivatives on u_ϵ to derive that u_ϵ satisfies the Euler-Lagrange equation associated with E_λ. For this, according to Lemma A.2, in the Appendix, we need to prove that $$ such that

where d(x)≝dist⁡(x, ∂Ω). To prove (58), we argue by contradiction: ∀η > 0 let and suppose that Ω_η has a nonzero measure.

Let $$ and for 0 < t ≤ 1 set ξ(t) = E_λ(u_ϵ + tu_η). Then, there exists c(t) satisfying c(t) > ηt such that inf⁡((u_ϵ + tu_η)/d(x)) ≥ c(t)  for    t > 0. Then, from Lemma A.4  ξ is differentiable for 0 < t ≤ 1 and $$. Thus,

From (H1)–(H3), we see that for η > 0 small enough.

Now, since s^−δ(x) + f(x, s) + λs^q(x) is nonincreasing for 0 < s small enough uniformly to x ∈ Ω (by (H1)–(H3)) and from the monotonicity of the operator −Δ_p(x)u + |u|^p(x)−1u, we have that for 0 < η small enough 0 ≤ ξ^′(1) − ξ^′(t). Therefore, from Taylor’s expansion and since ρ(u_ϵ) ≤ ϵ, there exists 0 < γ < 1 such that

Letting t = γ we have ξ^′(γ) ≤ ξ^′(1) < 0. We obtain a contradiction with (62) and then u_ϵ ≥ ηd(x) for some η > 0 (which depends a priori on ϵ). Since u_ϵ is a local minimizer of E_λ and E_λ is Gâteaux differentiable in u_ϵ, we get that $$ is defined and $$ Recalling that $$ is the solution to the pure singular problem given by Theorem 4.2 in [16] and from the weak comparison principle, there exist C₁ > 0, C₂ > 0 such that where θ₁ = max_d(x)≤σ⁡(p(x)/(p(x) − 1 + δ(x))), for some η > 0 (independent of ϵ). Since $$ and from the fact that u_ϵ satisfies (P_λ), we get that $$ is uniformly bounded in W^1,p(x)(Ω). Now, using Proposition 15 and Theorem 16, we get and as ϵ → 0⁺ which contradicts the fact that u₀ is a local minimizer in $$.

Now, we deal with the second case.

(2) ρ(u_ϵ) = ϵ: we again show that u_ϵ ≥ ηd(x) in Ω for some η > 0. Taking $$,    ξ(t) = E_λ(u_ϵ + tu_η), we obtain as above that     ξ^′(t) ≤ ξ^′(1) < 0 for 0 < t < 1 and 0 < η small enough.

Then ξ(t) = E_λ(u_ϵ + tu_η) is decreasing. This implies that E_λ(u_ϵ) > E_λ(u_ϵ + tu_η) for t > 0 and using (52)

This yields a contradiction with the fact that u_ϵ is a global minimizer of E_λ on $$. In this case, using Lemma A.4 and from the Lagrange multiplier rule we have

We first show that μ_ϵ ≤ 0. We argue by contradiction. Suppose that μ_ϵ > 0; then there exists φ ∈ W^1,p(x)(Ω) such that

and then for t small we have This contradicts the fact that u_ϵ is a global minimizer of E_λ in $$.

We deal now with the two following cases.

Case 1 (inf_ϵ∈(0,1)⁡μ_ϵ≝l > −∞). In this case, we write (67) in its PDE form as

In this case, from (57), we have that $$ Hence, we can apply Proposition 15 to conclude that $$ for some constant K > 0 independent of ϵ. Therefore, using Theorem 16 we conclude that $$ for some constant C > 0 independent of ϵ and as ϵ → 0⁺

which contradicts the fact that u₀ is a local minimizer in $$.

Now, we deal with the second case.

Case 2 (inf_ϵ∈(0,1)⁡μ_ϵ = −∞). From above, we can assume that μ_ϵ ≤ −1 for 0 < ϵ small enough. Furthermore, we can find a number M > 0 independent of ϵ > 0 and $$, such that (1/s^δ(x) + f(x, s) + μ_ε|s|^α(x)−2s) and $$ are negative for all |s| ≥ M. Then, from the weak comparison principle (see Lemma 7 and using (u_ε − M) ⁺ as test function) we have that $$ for ϵ > 0 small enough. Now, since u₀ ∈ W^1,p(x)(Ω) is a C¹ local minimizer, u₀ is a weak solution to (P_λ); that is, satisfies ess inf_K⁡u₀ > 0 over every compact set K ⊂ Ω and

for all $$. Therefore, for every function w ∈ W^1,p(x)(Ω), u₀ satisfies Similarly, Now, substracting the above relations with w = (u_ϵ − u₀) | u_ϵ − u₀|^β−1, with β > 1, as a test function in (P_ϵ), integrate by parts and use the fact that u ↦ −Δ_p(x)u+|u|^p(x)−1u is a monotone operator to obtain Using the bounds of u_ϵ, u₀ we get where C does not depend on β and ϵ. Now, using Hölder’s inequality and the bounds of u_ϵ combined with Lemma 4 we obtain Therefore Thus for any β > 1 Passing to the limit in (79) β → +∞ we get Then, using (80) combined with Proposition 15, the uniform L^∞ bounds for $$ in Ω as well as ∂Ω, we get that the right-hand side terms in (P_ε) are uniformly bounded in L^∞(Ω) and in L^∞(∂Ω) from which as in the first case we obtain that u_ϵ  (0 < ϵ ≤ 1) is bounded in $$ independently of ϵ. Finally, using Ascoli-Arzela Theorem we find a sequence ϵ_n → 0⁺ such that It follows that, for ϵ > 0 sufficiently small, which contradicts the fact that u₀ is a local minimizer of E_λ for the $$ topology. The proof of Lemma 14 is now completed.

Lemma 17. The functional $$ satisfies the Palais-Smale condition.

Proof. Let {u_n} be a (PS) sequence; namely, $$ is bounded and $$ when n → ∞. Then,

Now, we estimate the boundary term from above as follows: Hence, taking (88) in (87) and using (H2) combined with (7) we get Now, using Lemma 4 and the fact that $$, we get Hence, $$ is bounded. Without loss of generality, we assume that there exists a subsequence of {u_n} such that u_n⇀u₀. Therefore, using Theorems 5 and 6 we get Observe that We already know that Using (91), we obtain This together with the convergence of u_n⇀u₀ in W^1,p(x)(Ω) implies that u_n → u₀ strongly in W^1,p(x)(Ω); that is, $$ satisfies the (PS) condition. The proof of Lemma 17 is now completed.

Proof of Theorem 3. Firstly, note that $$ for any solution u of (50). Hence, as in Section 3 we can conclude that u_λ is a local minimum for $$ in W^1,p(x)(Ω). By the strong comparison principle and Hopf lemma, we can conclude that any critical point of $$ is also a critical point of E_λ and hence u also solves (P_λ). It is easy to see that $$ and $$ are a subsolution and a supersolution to the problem associated with the functional energy $$ Therefore, using the approach as in Theorem 2, we prove that this problem has a solution $$ such that v_λ is a local minimizer of $$ in the C¹ topology. Now, by the comparison principle we can see that v_λ ≥ u_λ and also v_λ solves (P_λ). If v_λ≢u_λ the conclusion of Theorem 3 holds. That is, we can assume v_λ ≡ u_λ and u_λ is a strict local minimum of $$ in the W^1,p(x)(Ω) topology. Then, from Lemma A.4, the functional $$ and note that $$ as t → ∞. Thus, we can apply Lemma 17 combined with the Mountain Pass Theorem to conclude that problem (P_λ) has a solution v_λ such that v_λ ≠ u_λ. Therefore the proof of Theorem 3 is now completed.