Some existence results for a new class of elliptic Kirchhoff equation with logarithmic source terms

Abstract

In this paper, some new existence results of elliptic equation of Kirchhoff-type with changing sign data and logarithmic source terms are proved, by using three different methods: direct variational method, Galerkin approach and sub-super solutions method. Our study is natural extensions from the previous recent works in [2–19 , 38], where the authors have already studied the existence of positive solutions for some classes of Laplacian elliptic problems by using one classical method for a certain class of elliptic equations.

Keywords

Kirchhoff existence sub-supersolution galerkin approach elliptic equation

1 Introduction

Complex nonlinearity is known to exist in some physical systems and processes. Recently, there has been considerable interest regarding solutions and their stability (see for example [5 , 28]). In the appendix to the last paper [6], the authors provided details discuss the physical meaning of the Kirchhoff fracture problems and their applications. In fact, they propose in [6] the problem of the constant variance of Kirchhoff, which, as a particularly important case, represents the non-local aspect of tension arising from non-local measurements of the fractional length of the chain. In this case, Kirchhoff term measures the tension change on the chain caused by changing its length during vibration. For this, it means that the term Kirrchoff equals zero means that the fundamental tension of the chain is zero, which is a very realistic model.

Many recent research focuses on theoretical aspects and applications of non-local fractional models. Always in [6], the following critical fractional problem at the border was first studied. The authors demonstrated a non-negative and unclear solution to a certain class of Kirchhoff elliptic system in a heterogeneous environment, combining the truncation argument with the principle of concentration pressure. The state of degradation is studied in [2], by introducing a new technical approach based on the comparative characteristic of the critical mountain path level. Moreover, a solution for multiple critical fracture problems Kirchhoff is provided in the entire R^N region in [5 , 38] adapting an advanced contrast technique in [2]. As for the results of pluralism, we refer to [32], where they consider the Schrodinger-Kirchhoff equation to be heterogeneous. By combining the mountain pass theory with Iceland’s diverse principle, [32] it proves the existence of two solutions in a non-renewable position. With a similar approach to [32, 33], there are two solutions to the retracted Kirchhoff equation in $ℝ^{N}$ with a convex concave linear. In [33], according to Fountain’s theory and Fountain’s dual theory, the authors obtained the existence of many infinite solutions to the problem of Kirchhoff without a large analogue analogue, with heterogeneous assumptions suitable for Kirchhoff term. In [32, 35] using the genus Krasnoselsky theory, under degraded frameworks. Moreover, for many unlimited solutions, the Krasnoselsky genus theory is used in [9] on 17 pages 4 of 18 Y. Song and S. Shi MJOM. A critical problem similar to our study and the new Kajikiya version of the symmetric lemma in the mountain in [33] are applied to a critical problem similar to the present problem, but only in the case of negegenerate. Finally, in [36], the Kirchhoff subcritical system was studied using the symmetrical mountain pass theory presented by Ambrosetti and Rabinovich in [25]. Apart from the above-mentioned interest in nonlinear polynomial terms, the linear algorithm received considerable attention from both physicists and mathematicians. This type of nonlinearity is presented in irrelevant wave equations that describe spinning molecules moving in an external electromagnetic field and also in the relative wave equation of rotational particles [28]. Furthermore, logarithmic nonlinearity appears in many branches of physics such as inflationary cosmology [37], nuclear physics [25, 39] and geophysics [29]. He was martyred. With all this basic meaning defined in physics, the timely global interest in solving the problem of equation of evolution with this nonlinear logarithmic type attracts a lot of attention. In this paper, by using three different methods: direct variational method, Galerkin approach and sub-super solutions method, I can give some existence results of positive solutions for a new class of elliptic Kirchhoff equation with logarithmic source terms.

2 Preliminaries, assumptions and statement of the problem

This paper deals with the existence of positive solutions of the following boundary value problem ${\begin{array}{l} - M ({‖ u ‖}^{2}) Δ u = u {| u |}^{p - 1} In {| u |}^{a} + λ f (x) in Ω, \\ u = 0 in \partial Ω, \end{array}$ (1) where $Ω \subset ℝ^{N}$ is a bounded domain with sufficiently smooth boundary ∂Ω and $f \in C^{1} (\bar{Ω})$ changes sign, M is continuous positive function in $ℝ^{+},$ p, q and λ are positive parameters.

The problem (1) is related to the stationary version of the Kirchhoff equation: ${\begin{matrix} u_{tt} - M (‖ \nabla u (x, t) ‖^{2}) Δ u = g (x, t) in Ω \\ u = 0 in \partial Ω \end{matrix}$ (2) which is introduced earlier in the mathematical description of vibration of an elastic stretched string. The problem (6) was presented by Kirchhoff in 1883 [34] for M (r) = a + br is a C¹-function for r ≥ 0, with a, b > 0. This equation is an extension of the classical d’Alembert’s wave equation by considering the effect of the changes in the length of the string during the vibrations. The studied model takes into account the length changes of the string produced by the transverse vibrations. Such problems are often called nonlocal since the equation is no longer a pointwise identity because of the presence of the term M (||u||²). This type of problems received much attention after the work by Lions [32] where a functional analysis framework was proposed. By using variational methods, Alves et al. [1] gave conditions on M and g for which, the stationary problem (6) that corresponds to possessing positive solutions in the subcritical case. Correa and Menezes [20] improved the existence result via Galerkin method when g (x, u) = g (x) . They also noticed that the problem admits a positive solution when the function M is bounded and g (x, u) = (u⁺) ^α + λφ (x) with 0 < α < 1 and φ > 0 .

In [32] Azzouz and Bensedik (Theorem 2) investigated the existence of a positive solution for the nonlocal problem of the form

${\begin{array}{l} - M ({\int_{Ω} | \nabla u |}^{2} d x) Δ u = {| u |}^{p - 1} u + λ f (x) in Ω, \\ u = 0 on \partial Ω, \end{array}$ (3)

By using the sub and supersolution methods combining a comparison principle introduced in [1], the authors established the existence of a positive solution for (3), where the parameter λ > 0 is small enough.

Recently in [17], we have studied the existence of positive solutions for the following Kirrchoff elliptic systems in bounded domains with multiple parameters: ${\begin{matrix} - A (\int_{Ω} {| \nabla u |}^{2} dx) ▵ u \\ = λ_{1} α (x) f (v) + μ_{1} β (x) h (u) in Ω, \\ - B (\int_{Ω} {| \nabla v |}^{2} dx) ▵ v \\ = λ_{2} γ (x) g (u) + μ_{2} η (x) τ (v) in Ω, \\ u = v = 0 on \partial Ω, \end{matrix}$ (4) where $Ω \subset ℝ^{N}$ (N ≥ 3) is a bounded smooth domain with C² boundary ∂Ω, and A, B: $ℝ^{+} \to ℝ^{+}$ are continuous functions, $α, β \in C (\bar{Ω}),$ λ₁ and λ₂ are nonnegative parameters. Then in [10], we could extend the study of the existence of positive solutions for the same problem with right hand side defined as a multiplication of two separate functions.

The previous works in [1 , 20] have motivated us to prove some existence results for an elliptic equation of Kirchhoff-type with changing sign data with a logarithmic nonlinearity which defined in (1) by direct variational method, Galerkin approach and sub and super solutions method.

Apart the aforesaid attention given to polynomial nonlinear terms, logarithmic nonlinearity has also received a great deal of interest from both physicists and mathematicians. This type of nonlinearity was introduced in the nonrelativistic wave equations describing spinning particles moving in an external electromagnetic field and also in the relativistic wave equation for spinless particles [28]. Moreover, the logarithmic nonlinearity appears in several branches of physics such as inflationary cosmology [37], nuclear physics [39], optics [25] and geophysics [29]. With all those specific underlying meaning in physics, the global-in-time well-posedness of solution to the problem of evolution equation with such logarithmic type nonlinearity captures lots of attention. Birula and Mycielski [13, 14] studied the following problem: ${\begin{array}{l} u_{t t} - u_{x x} + u - ε u In {| u |}^{2} = 0 in [a, b] \times (0, T), \\ u (a, t) = u (b, t) = 0, (0, T), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x) in [a, b], \end{array}$ (5) which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit p goes to 1 for the p-adic string equation [22, 38]. In [19], Cazenave and Haraux considered $u_{t t} - Δ u = u in {| u |}^{k} in ℝ^{3}$ (6) and established the existence and uniqueness of the solution for the Cauchy problem. Gorka [24] used some compactness arguments and obtained the global existence of weak solutions, for all $(u_{0}, u_{1}) \in H_{0}^{1} (Ω) \times L^{2} ([a, b])$ to the initial-boundary value problem (5) in the one-dimensional case. Bartkowski and Gorka, [12] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for Equation (6). Hiramatsu et al. [26] introduced the following equation $u_{t t} - Δ u + u + u_{t} + {| u |}^{2} u = u in | u |$ (7) to study the dynamics of Q-ball in theoretical physics and presented a numerical study. However, there was no theoretical analysis for the problem.

Before the statement of the main results, let’s define some useful tools. Let $m_{0} = inf_{s \geq 0} M (s) > 0,$ for m₀ > 0 and u₁ be the unique positive solution of the linear Dirichlet problem ${\begin{matrix} - Δ u_{1} = 1, in Ω \\ u = 0, in \partial Ω \end{matrix}$

Lemma 1. Let p ≥ 1 and h the continuous function on $ℝ^{+}$ defined by $h (A) = m_{0} A - {qA}^{p} | u_{1} |_{\infty}^{p} ln (A | u_{1} |_{\infty})$ and the positive number A ₀ such that $h (A_{0}) = max_{A > 0} h (A) .$ Then we have $h (A_{0}) \geq \frac{m_{0}}{| u_{1} |_{\infty}}$ and $m_{0} A_{0} \geq \frac{m_{0}}{| u_{1} |_{\infty}} - \frac{q}{e p} .$

Proof. It is easy to see that h is continuous and satisfies $h (0) = 0 and lim_{A \to + \infty} h (A) = - \infty .$ So A₀ exists and satisfies the following $h (A_{0}) \geq h (\frac{1}{| u_{1} |_{\infty}}) = \frac{m_{0}}{| u_{1} |_{\infty}}$ and $\begin{matrix} m_{0} A_{0} & \geq {qA}_{0}^{p} | u_{1} |_{\infty}^{p} ln (A_{0} | u_{1} |_{\infty}) + \frac{m_{0}}{| u_{1} |_{\infty}} \\ \geq \frac{m_{0}}{| u_{1} |_{\infty}} - \frac{q}{e p} \end{matrix}$

The last inequality is justified by the fact that $x^{p} ln (x) \geq - \frac{1}{e p} for all x > 0 .$ □

Lemma 2. We have the following $s^{p} ln (s) \leq \frac{1}{e a} s^{p + a}, for all p, s, a > 0$

Proof. An obvious study shows that the function $s \to s^{p} (\frac{1}{e a} s^{a} - ln (s))$ is positive on $ℝ^{+}$ , with p and a > 0.□

Now, we consider the following hypothesis:

(f₁): f ∈ C¹ (Ω)

(f₂): ∃k > 0 such that, the problem ${\begin{matrix} - Δ u = f (x) - k in Ω \\ u = 0 in \partial Ω \end{matrix}$ (8) has a nonnegative solution.

Let k₀ be the largest possible k that verifying this condition, and u_f its corresponding solution.

(f₃): f (x) changes sign on $\bar{Ω}$ .

(M₁): M is continuous and positive.

(M₂): m₀ is large enough such that: $λ_{1} \geq \frac{q}{e {pk}_{0}} and λ_{2} \geq \frac{q}{e {pk}_{0}},$

where

$\begin{matrix} λ_{1} = λ_{1} (m_{0}) = sup {λ > 0, such that : \\ q e^{- 1} {(\frac{λ}{m_{0}} | u_{f} |_{\infty})}^{p + 1} + λ | f |_{\infty} \leq m_{0} A_{0}} \end{matrix}$ and $λ_{2} = λ_{2} (m_{0}) = \frac{h (A_{0})}{| f |_{\infty}} .$ Observe that condition (M₂) is consistent since when m₀ gets large enough, then m₀A₀ and h (A₀) also get large enough according to Lemma 1.

For the existence of the function f that verifying (f₂), we refer to [21]. In [21], Các et al. treated more general problem. We give here a version of their Theorem 4 for the problem (8).

Let $G : Ω \times Ω \to ℝ$ be the Green’s function for -Δ with homogeneous boundary condition, and set for ϵ > 0 $w_{ϵ} (x) = \int_{Ω} G (x, y) [f^{+} (y) - (1 + ϵ) f^{-} (y)] d y,$ where $f^{\pm} (x) = max (\pm (f (x) - k), 0) .$ Remark 1. [21] Suppose that (f₁) and (f (x) - k) changes their sign. Then if there exists ϵ > 0 such that w_ϵ (x) > 0 for all x ∈ Ω, problem (1) has at least one positive solution.

3 Existence results

In this section, we give three theorems which are the main results of the paper, we prove Theorem 1 by using the sub-super solutions method. Then, we prove Theorem 2 via Galerkin method. Finally, we treat the case M as nondecreasing for proving Theorem 3.

Theorem 1. Assume that (f₁) and (f₂) are being hold, p > 1 and M is a nonincreasing function that satisfying (M₁) and (M₂). Suppose further that function H (t) = tM (t²) is increasing on $ℝ$ .

Then there exist min(λ₁, λ₂) ≤ λ_∗ < λ^∗ such that problem (1) admits at least one positive solution for $λ \in (\frac{q}{e {pk}_{0}}, λ_{*})$ , and no positive solution for λ > λ^∗.

Lemma 3. ([3]) Under the hypotheses on the monotony of M and H of Theorem 1, if v and w are nonnegative functions verifying ${\begin{matrix} - M (‖ v ‖^{2}) Δ v \leq - M (‖ w ‖^{2}) Δ w in Ω \\ v = w = 0 in \partial Ω . \end{matrix}$ Then $v \leq w .$

Theorem 2. Assume that (f₁) is satisfied and m₀ is large enough, then

If 0 < p < 1 our problem admits a nontrivial solution for all λ > 0 .

If $1 < p < \frac{N + 2}{N - 2}$ then there exists λ₀ > 0 such that our problem admits a nontrivial solution for 0 ≤ λ ≤ λ₀.

Theorem 3. Assume that (f₁) and (f₃) are being hold, 0 < p < 1 and M is a nondecreasing continuous function on $ℝ^{+}$ , then problem (1) admits a nontrivial solution v with energy $I (v) \leq \frac{q}{e {(p + 1)}^{2}}$ .

3.1 Proof of Theorem 1

Now, we prove Theorem 1 by using the sub-super solution method.

Proof. Let $\frac{q}{e {pk}_{0}} \leq λ \leq min (λ_{1}, λ_{2})$ .

Since (f₂) is satisfied by f, we consider u_f the solution of (8) and so we define T as $T (γ) : = \frac{1}{‖ u_{f} ‖} H (γ ‖ u_{f} ‖)) = γ M (γ^{2} ‖ u_{f} ‖^{2}) .$ T is a continuous positive increasing function such that: $T (0) = 0, and lim_{γ \to + \infty} T (γ) = + \infty .$ Then, the equation $T (γ) = λ$ admits a unique solution γ₀ and we have

$γ_{0} = \frac{λ}{M (γ_{0}^{2} ‖ u_{f} ‖^{2})} .$ We give a subsolution of (1) by $\underline{u} = γ_{0} u_{f} .$ Indeed $\underline{u}$ satisfies

$\begin{matrix} - M (‖ \underline{u} ‖^{2}) Δ \underline{u} & = M (γ_{0}^{2} ‖ u_{f} ‖^{2}) γ_{0} f (x) \\ - γ_{0} k_{0} M (γ_{0}^{2} ‖ u_{f} ‖^{2}) \\ = λ f (x) - λ k_{0} \\ \leq λ f (x) - \frac{q}{e p} \\ \leq λ f (x) + {\underline{u}}^{p} ln | \underline{u} |^{q} . \end{matrix}$ Now, let us show that $\bar{u}$ can be defined as $\bar{u} = A_{0} u_{1},$ which is a super solution of the problem (1), we have

$\begin{matrix} - M (‖ \bar{u} ‖^{2}) Δ \bar{u} = A_{0} M (‖ \bar{u} ‖^{2}) \geq A_{0} m_{0} \\ \geq λ_{2} | f |_{\infty} + {qA}_{0}^{p} | u_{1} |_{\infty}^{p} ln (A_{0} | u_{1} |_{\infty}) \\ \geq λ f (x) + {\bar{u}}^{p} ln | \bar{u} |^{q} . \end{matrix}$ Then, $\bar{u}$ furnishes a super solution of (1).

By using Lemma 3, we have $\begin{matrix} - M (‖ \bar{u} ‖^{2}) Δ \bar{u} = A_{0} M (‖ \bar{u} ‖^{2}) \geq A_{0} m_{0} \\ \geq q e^{- 1} {(\frac{λ_{1}}{m_{0}} | u_{f} |_{\infty})}^{p + 1} + λ_{1} | f |_{\infty} \\ \geq q e^{- 1} {(\frac{λ}{m_{0}} u_{f})}^{p + 1} + λ f \\ \geq q e^{- 1} {(γ_{0} u_{f})}^{p + 1} + λ f \\ \geq {\underline{u}}^{p} ln | \underline{u} |^{q} + λ f \\ \geq - M (‖ \underline{u} ‖^{2}) Δ \underline{u} . \end{matrix}$ Thus $\bar{u} \geq \underline{u},$ which completes the proof of the existence result. Now we show the nonexistence result.

Let λ₁ be the first eigenvalue of $(- Δ, H_{0}^{1} (Ω))$ , and φ₁ is the corresponding eigenfunction, if u_λ is a positive solution of problem (1), then $\begin{matrix} - M (‖ u_{λ} ‖^{2}) \int_{Ω} φ_{1} Δ u_{λ} d x & = \int_{Ω} u_{λ}^{p} ln | u_{λ} |^{q} φ_{1} d x \\ + λ \int_{Ω} f (x) φ_{1} d x . \end{matrix}$

Thus $\begin{matrix} λ \int_{Ω} f (x) φ_{1} d x = \\ \int_{Ω} [λ_{1} M (‖ u_{λ} ‖^{2}) u_{λ} - {qu}_{λ}^{p} ln (u_{λ})] φ_{1} d x . \end{matrix}$ Since M is non increasing, then $\begin{matrix} λ \int_{Ω} f (x) φ_{1} d x \\ \leq \int_{Ω} [λ_{1} M (0) u_{λ} - {qu}_{λ}^{p} ln (u_{λ})] φ_{1} d x \end{matrix}$ and since the function g : x → λ₁M (0) x - qx^p ln(x) is continuous on $ℝ^{+}$ with $g (0) = 0 and lim_{x \to + \infty} g (x) = - \infty .$

Then, g will attain its maximum value on a certain $x_{0} \in ℝ^{+}$ such that $g (x_{0}) > g (1) = λ_{1} M (0) > 0 .$

Consequently $λ \int_{Ω} f (x) φ_{1} d x \leq g (x_{0}) \int_{Ω} φ_{1} d x .$ Taking into consideration the condition (f₃) where $u_{f} \in H_{0}^{1} (Ω)$ is nonnegative such that $\begin{matrix} \int_{Ω} f (x) φ_{1} d x & = & \int_{Ω} (k_{0} - Δ u_{f}) φ_{1} d x \\ = & \int_{Ω} (λ_{1} u_{f} + k_{0}) φ_{1} d x > 0 . \end{matrix}$ We deduce that $λ \leq \frac{g (x_{0}) \int_{Ω} φ_{1} d x}{\int_{Ω} f (x) φ_{1} d x} = λ^{*} .$ This implies that our problem has no positive solution for λ > λ^∗ .□

3.2 Proof of Theorem 2

We prove the existence result of Theorem 2 via Galerkin method.

We mean by solution of (1) a function $u \in H_{0}^{1} (Ω)$ such that $\begin{matrix} M (‖ u ‖^{2}) \int_{Ω} \nabla u . \nabla φ d x \\ - \int_{Ω} (| u |^{p - 1} u ln | u |^{q} + λ f (x)) φ d x = 0 \end{matrix}$ for all $φ \in H_{0}^{1} (Ω) .$ This proof is based on the following proposition.

Proposition 1. ([34]) Let $F : ℝ^{N} \to ℝ^{N}$ be a continuous mapping and B_r (O) is an open ball in $ℝ^{n}$ centered at the origin with radius r. If 〈F (x) , x〉>0 for all x ∈ ∂B_r (O), then there exists x₀ ∈ B_r (O) such that F (x₀) = 0.

Now we start to prove Theorem

Proof. Let (e_k) be a complete orthonormal system for $H_{0}^{1} (Ω)$ . For each $n \in ℕ^{*}$ , we consider the finite dimensional space $V_{n} = span {e_{1}, . . ., e_{n}} .$ Then (V, || . ||) is isometric to $(ℝ^{n}, ‖ . ‖)$ and for each $v = \sum_{i = 1}^{n} ξ_{i} e_{i}$ we have $‖ v ‖^{2} = ‖ ξ ‖_{n}^{2} = \sum_{i = 1}^{n} ξ_{i}^{2}$ where ξ = (ξ₁, . . . , ξ_n). We can make the identification $V_{n} ∋ v = (ξ_{1}, . . ., ξ_{n}) \in ℝ^{n} .$ A function u_n ∈ V_n is said to be an approximate solution of the problem (1) if $\begin{matrix} M (‖ u_{n} ‖^{2}) \int_{Ω} \nabla u_{n} . \nabla e_{k} d x \\ - \int_{Ω} (| u_{n} |^{p - 1} u_{n} ln | u_{n} |^{q} + λ f (x)) e_{k} d x = 0, \\ k = 1, . . ., n, \end{matrix}$ (9)

Let us define the operator $F : ℝ^{N} \to ℝ^{N}$ $F (u_{n}) = (F_{1} (u_{n}), F_{2} (u_{n}), . . . ., F_{n} (u_{n}))$ with $\begin{matrix} F_{k} (u_{n}) = M (‖ u_{n} ‖^{2}) \int_{Ω} \nabla u_{n} . \nabla e_{k} d x \\ - \int_{Ω} (| u_{n} |^{p - 1} u_{n} ln | u_{n} |^{q} + λ f (x)) e_{k} d x . \end{matrix}$ Using the above identification, with $u_{n} = \sum_{i = 1}^{n} ξ_{i} e_{i}$ , we write $\begin{matrix} F_{k} (u_{n}) = M (‖ u_{n} ‖^{2}) ξ_{k} \\ - \int_{Ω} | u_{n} |^{p - 1} u_{n} ln | u_{n} |^{q} e_{k} d x \\ - λ \int_{Ω} f (x) e_{k} d x \end{matrix}$ and then $\begin{matrix} 〈 F (u_{n}), u_{n} 〉 = M (‖ u_{n} ‖^{2}) ‖ u_{n} ‖^{2} \\ - λ \int_{Ω} f (x) u_{n} d x - q \int_{Ω} | u_{n} |^{p + 1} ln | u_{n} | d x \\ \geq M (‖ u_{n} ‖^{2}) ‖ u_{n} ‖^{2} - λ \int_{Ω} f (x) u_{n} d x \\ - \frac{q}{e a} \int_{Ω} | u_{n} |^{p + 1 + a} d x, \end{matrix}$ where a > 0 which can be taken small enough, then using (f₁) and Sobolev imbedding, we get $\begin{matrix} 〈 F (u_{n}), u_{n} 〉 \geq m_{0} ‖ u_{n} ‖^{2} - C ‖ u_{n} ‖^{p + 1 + a} \\ - λ C | f |_{2} ‖ u_{n} ‖ \\ \geq (m_{0} ‖ u_{n} ‖ - C ‖ u_{n} ‖^{p + a} - λ C | f |_{2}) ‖ u_{n} ‖ . \end{matrix}$ By comparing the curves C₁ and C₂ defined by $y = m_{0}^{- 1} {Ct}^{p + a}$ and

$y = t - m_{0}^{- 1} λ C | f |_{2}$ respectively we deduce that

If 0 < p < 1, we take a such that 0 < p + a < 1, then for large values of t = ||u_n||, the straight line C₂ is above C₁ for all λ > 0.

If $1 < p < \frac{N + 2}{N - 2}$ , we take a such that $1 < p + a < \frac{N + 2}{N - 2},$ then for small values of ||u_n|| and large values of m₀, C₂ is above C₁ until a limit value λ₀ > 0.

So in both cases there exists r = r (λ) > 0 such that 〈F (u_n) , u_n〉>0 when ||u_n|| = r. Then in view of the proposition 1, there exists u_n ∈ V_n, which is approximate solution of the problem (1) with ||u_n||<r.

The obtained sequence (u_n) is bounded in $H_{0}^{1} (Ω)$ and also in L² (Ω) by the compact embedding of $H_{0}^{1} (Ω)$ into L² (Ω) so, passing to the subsequence if necessary, there exists γ > 0 and $u \in H_{0}^{1} (Ω)$ such that

$\begin{matrix} ‖ u_{n} ‖ \to γ in ℝ \\ u_{n} ⇀ u weakly in H_{0}^{1} (Ω) \\ u_{n} \to u strongly in L^{2} (Ω) \\ u_{n} \to u a . e ., in Ω . \end{matrix}$

Fixing k in (9) and letting n→ + ∞ we get $\begin{matrix} M (γ^{2}) \int_{Ω} \nabla u . \nabla e_{k} d x \\ - \int_{Ω} (| u |^{p - 1} u ln | u |^{q} + λ f (x)) e_{k} d x = 0 \end{matrix}$ and since (e_k) is a basis for $H_{0}^{1} (Ω)$ $\begin{matrix} M (γ^{2}) \int_{Ω} \nabla u . \nabla φ d x \\ - \int_{Ω} (| u |^{p - 1} u ln | u |^{q} + λ f (x)) φ d x = 0 \end{matrix}$ $\forall φ \in H_{0}^{1} (Ω) .$

In particular for φ = u we obtain

$\begin{matrix} M (γ^{2}) ‖ u ‖^{2} - \int_{Ω} | u |^{p + 1} ln | u |^{q} \\ + λ f (x) u d x = 0 . \end{matrix}$ (10) Similarly we have from (9) $\begin{matrix} M (‖ u_{n} ‖^{2}) \int_{Ω} \nabla u_{n} . \nabla φ d x \\ - \int_{Ω} (| u_{n} |^{p - 1} u_{n} ln | u_{n} |^{q} + λ f (x)) φ d x = 0 \\ for all φ \in V_{n} . \end{matrix}$ For φ = u_n, we obtain $\begin{matrix} M (‖ u_{n} ‖^{2}) ‖ u_{n} ‖^{2} \\ - \int_{Ω} | u_{n} |^{p + 1} ln | u_{n} |^{q} + λ f (x) u_{n} d x = 0 \end{matrix} .$ Letting n→ + ∞ we claim that we get $M (γ^{2}) γ^{2} - \int_{Ω} | u |^{p + 1} ln | u |^{q} + λ f (x) u d x = 0 .$ (11) Indeed we have only to justify the limit $\int_{Ω} | u_{n} |^{p + 1} ln | u_{n} |^{q} \to \int_{Ω} | u |^{p + 1} ln | u |^{q} .$ (12)

Since u_n → u a.e.,in Ω and u → u^p+1 ln(u) is continuous we get $u_{n}^{p + 1} ln (u_{n}) \to u^{p + 1} ln (u) a . e ., in Ω .$ (13) Furthermore $u^{p + 1} ln (u) \leq \frac{1}{e a} u^{p + 1 + a},$ where a is a positive number small enough to ensure the compact embedding $H_{0}^{1} (Ω) \subset L^{p + 1 + a} (Ω),$ then for n large enough, we have $u_{n}^{p + 1} ln (u_{n}) \leq \frac{1}{e a} u^{p + 1 + a} + 1 \in L^{1} (Ω) .$ (14) By using (13), (14) and dominating convergence theorem, we justify (12). Thus (11) hold. Also by using (10) and (11) we get ||u|| = γ and so $u_{n} \to u strongly in H_{0}^{1} (Ω) .$ □

Remark 2. We see from the proof that it is sufficient to take f ∈ H^-1 (Ω) instead of C¹ (Ω).

3.3 Proof of Theorem 3

Proof. We suppose that the function M is increasing, continuous and 0 < p < 1.

We look for solutions which are critical points of the energy functional I defined on $H_{0}^{1} (Ω)$ by $\begin{matrix} I (u) = \frac{1}{2} \tilde{M} (‖ u ‖^{2}) \\ - \frac{q}{p + 1} \int_{Ω} | u |^{p + 1} (ln (u) - \frac{1}{p + 1}) d x \\ - λ \int_{Ω} f (x) u d x, \end{matrix}$ where $\tilde{M} (t) = \int_{0}^{t} M (s) d t .$ By the continuity of M, Holder inequality and Sobolev embedding we get $\begin{matrix} I (u) & \geq \frac{1}{2} m_{0} ‖ u ‖^{2} - λ C ‖ u ‖ | f |_{2} \\ - \frac{q}{p + 1} \int_{Ω} | u |^{p + 1} ln (u) d x \\ \geq \frac{1}{2} m_{0} ‖ u ‖^{2} - λ C ‖ u ‖ | f |_{2} \\ - \frac{q}{e a (p + 1)} ‖ u ‖^{p + 1 + a}, \end{matrix}$ where a is taken such that $1 < p + 1 + a < 2 .$ The functional I is coercive. Let $α = inf_{u \in H_{0}^{1} (Ω)} I (u)$ and (u_n) be a minimizing sequence that is, satisfying I (u_n) → α. By coerciveness (u_n) is bounded in $H_{0}^{1} (Ω)$ , and so we may assume as above in (12) that by passing to the subsequence if necessary there exists $v \in H_{0}^{1} (Ω)$ such that $u_{n} ⇀ v weakly in H_{0}^{1} (Ω)$ and $\int_{Ω} | u_{n} |^{p + 1} ln | u_{n} | \to \int_{Ω} | v |^{p + 1} ln | v |$ (15) The function M is positive, then $\tilde{M}$ is increasing and since $‖ v ‖ \leq \underset{n \to + \infty}{lim inf} ‖ u_{n} ‖$ $\tilde{M} (‖ v ‖^{2}) \leq \tilde{M} ({(\underset{n \to + \infty}{lim inf} ‖ u_{n} ‖)}^{2})$ Because $\tilde{M}$ is continuous and ||u_n||≥0 we obtain $\begin{matrix} \tilde{M} ({(\underset{n \to + \infty}{lim inf} ‖ u_{n} ‖)}^{2}) = \tilde{M} (\underset{n \to + \infty}{lim inf} ‖ u_{n} ‖^{2}) \\ = \underset{n \to + \infty}{lim inf} \tilde{M} (‖ u_{n} ‖^{2}) \end{matrix}$ Hence $\tilde{M} (‖ v ‖^{2}) \leq \underset{n \to + \infty}{lim inf} \tilde{M} (‖ u_{n} ‖^{2})$ (16) Further, we have from (15) and the compact Sobolev embedding $H_{0}^{1} (Ω) \subset L^{p + 1} (Ω)$ $\begin{matrix} \frac{q}{p + 1} \int_{Ω} | u_{n} |^{p + 1} (ln | u_{n} | - \frac{1}{p + 1}) d x \\ + λ \int_{Ω} f (x) u_{n} d x \\ ⟶ \frac{q}{p + 1} \int_{Ω} | v |^{p + 1} (ln | v | - \frac{1}{p + 1}) d x \\ + λ \int_{Ω} f (x) v d x \end{matrix}$ (17) From (16) and (17), we deduce that I is weakly lower semi continuous and in consequently $α \leq I (v) \leq \underset{n \to + \infty}{lim inf} I (u_{n}) = α .$ So α = I (v) which proves that I attains its infimum at the limit v.

Let us prove that $I (v) \leq \frac{q}{e {(p + 1)}^{2}} .$

Denote $\begin{matrix} Ω^{+} & = & {x \in Ω; f (x) > 0}, \\ Ω^{-} & = & {x \in Ω; f (x) < 0} \end{matrix}$ and choose a positive function $φ \in H_{0}^{1} (Ω)$ such that support { φ } ⊂ Ω⁺. For t ∈ (0, 1) we have $\begin{matrix} I (t φ) = \frac{1}{2} \tilde{M} (‖ t φ ‖^{2}) \\ - \frac{q}{p + 1} \int_{Ω} | t φ |^{p + 1} (ln (t φ) - \frac{1}{p + 1}) d x \\ - λ \int_{Ω^{+}} f (x) t φ d x \leq \frac{1}{2} M (t^{2} ‖ φ ‖^{2}) t^{2} ‖ φ ‖^{2} \\ - \frac{q}{p + 1} \int_{Ω} | t φ |^{p + 1} ln (t φ) d x \\ + \frac{q}{{(p + 1)}^{2}} \int_{Ω} | t φ |^{p + 1} d x \\ \leq \frac{1}{2} M (‖ φ ‖^{2}) t^{2} ‖ φ ‖^{2} \\ + \frac{q}{e {(p + 1)}^{2}} + \frac{{qt}^{p + 1}}{{(p + 1)}^{2}} \int_{Ω} φ^{p + 1} d x . \end{matrix}$ Then $I (v) = inf_{u \in H_{0}^{1} (Ω)} I (u) \leq inf_{0 < t < 1} I (t φ) = \frac{q}{e {(p + 1)}^{2}} .$ □

4 Conclusion

Our results are natural extensions from the previous recent ones [2–19 , 38], where the authors have already studied the existence of positive solutions for some classes of Laplacian elliptic problems by using one classical method which is sub-supersolution. Here, in addition to the latter method, we use two other known methods, a direct variational method and Galerkin approach for proving some existence results to a logarithmic nonlinear elliptic equation of Kirchhoff-type with changing sign data. The logarithmic nonlinearity is of much interest in physics, since it appears naturally in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics [11 , 24] This type of problems has applications in many branches of physics such as nuclear physics, optics, and geophysics [15 , 33]. In the future work, we try to extend this study for evolutionary case of the presented problem but by using the semigroup theory.

Footnotes

Acknowledgment

The author would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions which helped them to improve the paper.

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