PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 271–278 DOI: 10.2298/PIM1614271K

INFINITELY MANY WEAK SOLUTIONS FOR SOME ELLIPTIC PROBLEMS IN ℝ^{N}

Mehdi Khodabakhshi, Abdolmohammad Aminpour, and Mohamad Reza Heidari Tavani

Abstract. We investigate the existence of infinitely many weak solutions to some elliptic problems involving the p-Laplacian in ℝ^{N} by using variational method and critical point theory.

1. Introduction

We are going to establish infinitely many weak solutions to the following elliptic problem:

(1.1)  \begin{aligned}− & Δ_{p}u + |u|^{p−2} u = λf(x, u), in ℝ^{N} , \\ & u ∈ W^{1,p}(ℝ^{N}) \end{aligned}

where N > 1 , Δ_{p}u := div(|∇u|^{p−2} ∇u) denotes the p-Laplace operator with p > N, and f : ℝ^{N} ×ℝ → ℝ be an L^{1}-Carathéodory function, λ is a positive real parameter. Problems on unbounded domains have been intensively studied. For instance, we mention the works [1–3,9–12]. In [8] Condito and Molica Bisci considered problem (1.1) by taking f(x, u) := α(x)f(u), where f: ℝ → ℝ is a continuous function, α ∈ L^{1}(ℝ^{ℕ}) ∩ L^{∞} (ℝ^{ℕ}) is a nonnegative (not identically zero) map. It should be mentioned the different cases of problem (1.1) and even in general have been considered on the variety of bounded domains. We just refer in the large literature on the subject, the papers [5, 6, 13–15]. For instance, in [14] the authors have studied the following elliptic problem of Kirchhoff type:

− ( a + b ∫_{Ω} ∇u |^{p}dx) Δ_{p} u + α(x)|u|^{p−2} u = λh(x)f(x) in Ω,
u|_{∂Ω} = 0,

where a and b are two non-negative constant, α ∈ L^{∞} (Ω) with ess inf_{x∈Ω} α(x) ⩾ 0, λ is a positive parameter, f: ℝ → ℝ is a continuous function and finally, h ∈ L^{∞} (Ω) with ess inf x ∈ Ω h(x) > 0. Recall that a function f: ℝ^{N} × ℝ → ℝ is said to be an L^{1}-Carathéodory function, if (C_{1}) the function x ↦ f(x, t) is measurable for every t ∈ ℝ (C_{2}) the function t ↦ f(x, t) is continuous for almost every x ∈ ℝ^{ℕ} (C_{3}) for every ϱ > 0 there exists a function l_{ϱ}(x) ∈ L^{1}(ℝ^{ℕ}) such that

2010 Mathematics Subject Classification: 35J20.

Key words and phrases: p-Laplace operator; variational methods; critical point theory.

Communicated by Stevan Pilipović.

sup_{|t|⩽ϱ} |f(x, t)| ⩽ l_{ϱ}(x),

for a.e. x ∈ ℝ^{ℕ}.

2. Preliminaries

Let N > 1, denote by X the space W^{1,p}(ℝ^{N}) endowed with the norm

∥u∥ := ( ∫_{ℝ^{ℕ}} |∇u(x)|^{p} dx + ∫_{ℝ^{ℕ}} |u(x)|^{p}dx) ^{1/p}.

Let p > N, we recall continuous embedding W^{1,p}(ℝ^{ℕ}) ↪ L^{∞} (ℝ^{ℕ}), and one has

∥u∥_{∞} ⩽ \frac{2p}{p − N} ∥u∥,

for every u ∈ W^{1,p}(ℝ^{ℕ}) (see [7, Morrey’s theorem]). Let us define F (x, ξ) = ∫_{0}^{ξ} f(x, t) dt, for every (x, ξ) ∈ ℝ^{ℕ} × ℝ, and introduce the functional I_{λ} : X → ℝ associated with (1.1),

I_{λ}(u) := Φ(u) − λΨ(u), (∀u ∈ X)

where

Φ(u) = \frac{1}{p}∥u∥^{p}, Ψ(u) = ∫_{ℝ^{ℕ}} F (x, u(x)) dx

for every u ∈ X. Now, in [8], it is shown that Φ is a Gâteaux differentiable and sequentially weakly lower semicontinuous and coercive in X whose Gâteaux derivative is given by

Φ′(u)(v) = ∫_{ℝ^{ℕ}} |∇u(x)|^{p−2} ∇u(x).∇v(x) dx + ∫_{ℝ^{ℕ}} |u(x)|^{p−2} u(x)v(x) dx,

In this section, we summarize the for every v ∈ X. On the other hand, standard arguments show that Ψ is a well defined, Gâteaux differentiable and sequentially weakly upper semicontinuous functional whose Gâteaux derivative is given by

Ψ′(u)(v) = ∫_{ℝ^{ℕ}} f(x, u(x)) dx,

for every v ∈ X. Fixing the real parameterλ, a function u: ℝ^{ℕ} → ℝ is said to be a weak solution of (1.1) if u ∈ X and

∫_{ℝ^{ℕ}}|∇u(x)|^{p−2} ∇u(x)∇v(x) dx+∫_{ℝ^{ℕ}}|u(x)|^{p−2} u(x)v(x) dx−λ∫_{ℝ^{ℕ}}f(x, u(x))v(x) dx=0,

for every v ∈ X. Hence, the critical points of I_{λ} are exactly the weak solutions of (1.1). Our main tool to investigate the existence of infinitely many weak solutions to the problem (1.1) is the following critical points theorem.

Theorem 2.1. [4, Theorem 2.1] Let X be a reflexive real Banach space, and let Φ, Ψ: X → ℝ be two Gâteaux differentiable functionals such that Φ is strongly continuous, sequentially weakly lower semicontinuous, coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf_{X} Φ, put

φ(r) := inf_{φ(v)<r} \frac{(sup_{Φ(u)<r} Ψ(u)) − Ψ(v)}{r − Φ(v)}

and

γ := lim inf_{r→+∞} φ(r), δ := lim inf_{r→(inf_{X} Φ)^{+}} φ(r).

then, one has

(a) For every r > inf_{X} Φ and λ ∈ ]0, 1/Φ(r)[, the restriction of the functional I_{λ} = Φ − λΨ to Φ^{−1} ]−∞, r[ admits a global minimum, which is a critical point (local minimum) of I_{λ} in X.

(b) If γ < +∞, then for each λ ∈ ]0, \frac{1}{γ} [, the following alternative holds: either (b_{1}) I_{λ} possesses a global minimum, or (b_{2}) there is a sequence {u_{n}} of critical points (local minima) of I_{λ} such

that lim_{n→∞} Φ(u_{n}) = +∞.

(c) If δ < +∞, then for each λ ∈ ]0, \frac{1}{δ}[, the following alternative holds: either (c_{1}) there is a global minimum of Φ which is a local minimum of I_{λ}, or (c_{2}) there is a sequence {u_{n}} of pairwise distinct critical points (local min-

ima) of I_{λ} which weakly converges to a global minimum of Φ, with lim_{n→∞} Φ(u_{n}) = inf_{X} Φ.

3. Main results

Fix D > 0 such that

(3.1)  κ := \frac{1}{m(D)( \frac{2p}{p−N} )^{p}( \frac{σ(N,p)}{D^{p}} + g(p, N))} ,

where σ(N, p) := 2^{p−N} (2^{N} − 1), as well as

g(p, N) := \frac{1 + 2^{N+p} N ∫ 1_{\frac{1}{2}} t^{N−1} (1 − t)^{p}dt}{2^{N}} .

We also note that m(D) := D^{N} \frac{π^{\frac{N}{2}}}{Γ(1+ \frac{N}{2} )} , where Γ is the Gamma function defined by

Γ(t) := ∫_{0}^{+∞} z^{t−1} e^{−z} dz, (∀t > 0).

Theorem 3.1. Let f: ℝ^{N} × ℝ → ℝ be an L^{1}-Carathéodory function, and assume that

i) F (x, t) ⩾ 0 for every (x, t) ∈ ℝ^{ℕ} × ℝ^{+} ;
ii) A := lim inf_{ϱ→+∞} \frac{∥l_{ϱ} ∥_{1}}{ϱ^{p−1}} < κ B,

where B := lim sup_{ϱ→+∞} \frac{1}{ϱ^{p}} ∫_{B(0, \frac{D}{2} )} F (x, ϱ) dx and l_{ϱ} ∈ L^{1}(ℝ^{ℕ}) that satisfies (c_{3}) condition on f(x, t) for every ϱ > 0 and κ is given by (3.1). Then for every

λ ∈ ] \frac{m(D)}{B} ( \frac{σ(N, p)}{pD^{p}} + \frac{g(p, N)}{p}) , \frac{1}{p( \frac{2p}{p−N} )^{p}A}[

the problem (1.1) admits a sequence of many weak solutions which is unbounded in X.

Proof. Fix λ as in our conclusion. Our aim is to apply Theorem 2.1, part (b) with X = W^{1,p}(ℝ^{N}), and Φ, Ψ are the functionals introduced in Section 2. As seen before, the functionals Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.1. Now, we look on the existence of critical points of the functional I_{λ} in X. To this end, we take {ϱ_{n}} to be a real sequence such that lim_{n→∞} ϱ_{n} = +∞ and

lim_{n→∞} \frac{∥l_{ϱ_{n}} ∥_{1}}{ϱ_{n}^{p−1}} = A.

Set r_{n} := ϱ_{n}^{p}/p( \frac{2p}{p−N} )^{p}, for every n ∈ ℕ. By relation (1.1), one has ∥v∥_{∞} ⩽ ϱ_{n} for all v ∈ X such that ∥v∥^{p} < pr_{n}, we have the following inequalities:

φ(r_{n}) = inf_{∥u∥^{p} <pr_{n}}  \frac{sup_{∥v∥^{p} <pr_{n}} ∫_{ℝ^{ℕ}} F (x, v(x)) dx − ∫_{ℝ^{ℕ}} F (x, u(x)) dx}{r_{n} − \frac{∥u∥^{p}}{p}}

⩽ \frac{sup_{∥v∥^{p} <pr_{n}} ∫_{ℝ^{ℕ}} F (x, v(x)) dx}{r_{n}} ⩽ p( \frac{2p}{p − N} )^{p} \frac{∥l_{ϱ_{n}} ∥_{1}}{ϱ_{n}^{p−1}} ,

for every n ∈ ℕ. Hence, it follows that

γ ⩽ lim inf_{n→∞} Φ(r_{n}) ⩽ p( \frac{2p}{p − N} )^{p}A < +∞,

since condition ii) shows A < +∞. Now, we prove that the functional I_{λ} is unbounded from below. For our goal, let {d_{n}} be a real sequence such that lim_{n→∞} d_{n} = +∞ and

(3.2)  lim_{n→+∞} \frac{1}{d_{n}^{p}} ∫_{B(0, \frac{D}{2} )} F (x, d_{n}) dx = B.

Let {v_{n}} be a sequence in X which is defined by

v_{n}(x) := \{ \begin{matrix}0 & x ∈ ℝ^{ℕ} ∖ B(0, D) \\ d_{n} & x ∈ B(0, \frac{D}{2} ) \\ \frac{2d_{n}}{D} (D − |x|) & x ∈ B(0, D) ∖ B(0, \frac{D}{2} ), \end{matrix}

One can compute that

∥v_{n}∥^{p} = d_{n}^{p}m(D)( \frac{σ(N, p)}{D^{P}} + g(p, N)) ,

see for instance, the paper [8]. At this point, by using condition i), we infer

∫_{ℝ^{ℕ}} F (x, v_{n}(x)) dx ⩾ ∫_{B(0, \frac{D}{2} )} F (x, d_{n}) dx,

for every n ∈ ℕ. Then, we have

I_{λ}(v_{n}) ⩽ \frac{d_{n}^{p}m(D)}{P}( \frac{σ(N, p)}{D^{p}} + g(p, N)) − λ ∫_{B(0, \frac{D}{2} )} F (x, d_{n}) dx,

for every n ∈ ℕ. If B < +∞, let

ϵ ∈ ] \frac{m(D)}{λB} ( \frac{σ(N, p)}{pD^{P}} + \frac{g(p, N)}{p}) , 1[

By (3.2) there exists N_{ϵ} such that

∫_{B(0, \frac{D}{2} )} F (x, d_{n}) dx > ϵBd_{n}^{p}, (∀n > N_{ϵ}).

Consequently, one has

I_{λ}(v_{n}) ⩽ \frac{d_{n}^{p}m(D)}{p}( \frac{σ(N, p)}{D^{p}} + g(p, N)) − λϵBd_{n}^{p},

= d_{n}^{p}( m(D)( \frac{σ(N, p)}{pD^{p}} + \frac{g(p, N)}{p}) − λϵB) ,

for every n > N_{ϵ}. Thus, it follows that

lim_{n→∞} I_{λ}(v_{n}) = −∞.

If B = +∞, let us consider

M > \frac{m(D)}{λp} ( \frac{σ(N, p)}{D^{p}} + g(p, N)) .

By (3.2) there exists N(M) such that

∫_{B(0, \frac{D}{2} )} F (x, d_{n}) dx > Md_{n}^{p}, (∀n > N(M)).

Hence, we have

I_{λ}(v_{n}) ⩽ \frac{d_{n}^{p}m(D)}{p}( \frac{σ(N, p)}{D^{p}} − g(p, N)) − λMd_{n}^{p}

= d_{n}^{p}( m(D)( \frac{σ(N, p)}{pD^{p}} + \frac{g(p, N)}{p}) − λM) ,

for every n > N(M). Taking into account the choice of M, also in this case, one has

lim_{n→∞} I_{λ}(v_{n}) = −∞.

Thanks to Theorem 2.1, the functional I_{λ} admits an unbounded sequence {u_{n}} ⊂ X of critical points. In conclusion, problem (1.1) admits a sequence of many weak solutions which is unbounded in W^{1,p}(ℝ^{ℕ}). □

The following result is a consequence of Theorem 3.1.

Corollary 3.1. Let f : ℝ^{ℕ} ×ℝ → ℝ be an L^{1}-Carathéodory function. Assume that condition i) of Theorem 3.1 holds. Further, require that

ii_{1}) lim sup_{ϱ→+∞} \frac{1}{ϱ^{p}} ∫_{B(0, \frac{D}{2} )} F (x, ϱ) dx > \frac{m(D)}{p} ( \frac{σ(N,p)}{D^{p}} + g(p, N));

276 KHODABAKHSHI, AMINPOUR, AND HEIDARI TAVANI

ii_{2}) lim inf_{ϱ→+∞} \frac{∥l_{ϱ} ∥_{1}}{ϱ^{p−1}} < p ( \frac{2p}{p−N} )^{−p} .

Then, the problem

− Δ_{p}u + |u|^{p−2} u = f(x, u), in ℝ^{N} ,
u ∈ W^{1,p}(ℝ^{N})

possesses a sequence of many weak solutions which is unbounded in W^{1,p}(ℝ^{ℕ}). Finally, We point out the following result. Corollary 3.2. Let f: ℝ^{ℕ} ×ℝ → ℝ be an L^{1}-Carathéodory function. Assume

that condition i) of Theorem 3.1 holds. Further, require that

lim sup_{ϱ→+∞} \frac{1}{ϱ^{p}} ∫_{B(0, \frac{D}{2} )} F (x, ϱ) dx = ∞, lim inf_{ϱ→+∞} \frac{∥l_{ϱ}∥_{1}}{ϱ^{p−1}} = 0,

Then, for every λ > 0, the problem

− Δ_{p}u + |u|^{p−2} u = λf(x, u), in ℝ^{N} ,
u ∈ W^{1,p}(ℝ^{N})

possesses a sequence of many weak solutions which is unbounded in W^{1,p}(ℝ^{ℕ}).

Remark 3.1. We note that assumption ii) in Theorem 3.1 could be replaced by the following more general hypotheses:

ii′) There exists two sequences {a_{n}} and {b_{n}} such that

0 ⩽ a_{n} < \frac{(p − N)b_{n}}{2p( \frac{σ(N,p)}{D^{p}} + g(p, N)) ^{\frac{1}{p}} (m(D))^{\frac{1}{p}}}

for every n ∈ ℕ and lim_{n→∞} b_{n} = +∞ such that

Â < κ lim sup_{ϱ→+∞} \frac{1}{ϱ^{p}} ∫_{B(0, \frac{D}{2} )} F (x, ϱ) dx,

where κ is given by (3.1) and

Â := lim_{n→∞} \frac{(b_{n}∥l_{b_{n}} ∥_{1} − ∫_{B(0, \frac{D}{2} )} F (x, a_{n}) dx)p( \frac{2p}{p−N} )^{p}}{b_{n}^{p} − m(D)a_{n}^{p}( \frac{2p}{p−N} )^{p}( \frac{σ(N,p)}{D^{p}} + g(p, N))} .

For instance, in this setting by choosing

λ ∈ ] \frac{m(D)}{B} ( \frac{σ(N, p)}{pD^{p}} + \frac{g(p, N)}{p}) , \frac{1}{p( \frac{2p}{p−N} )^{p}Â}[,

problem (1.1) admits a sequence of many weak solutions which is unbounded in W^{1,p}(ℝ^{ℕ}). Indeed, if

λ ∈ ] \frac{m(D)}{B} ( \frac{σ(N, p)}{pD^{p}} + \frac{g(p, N)}{p}) , \frac{1}{p( \frac{2p}{p−N} )^{p}Â}[,

and r_{n} := \frac{b_{n}^{p}}{p( \frac{2p}{p−N} )^{p}} for every n ∈ ℕ, then one has

φ(r_{n}) = inf_{∥u∥^{p} <pr_{n}}  \frac{sup_{∥v∥^{p} <pr_{n}} ∫_{ℝ^{ℕ}} F (x, v(x)) dx − ∫_{ℝ^{ℕ}} F (x, u(x)) dx}{r_{n} − \frac{∥u∥^{p}}{p}}

⩽ \frac{sup_{∥v∥^{p} <pr_{n}} ∫_{ℝ^{ℕ}} F (x, v(x)) dx − ∫_{ℝ^{ℕ}} F (x, v_{n}(x)) dx}{r_{n} − \frac{∥v_{n} ∥^{p}}{p}}

⩽ \frac{(b_{n}∥l_{b_{n}} ∥_{1} − ∫_{B(0, \frac{D}{2} )} F (x, a_{n}) dx)p( \frac{2p}{p−N} )^{p}}{b_{n}^{p} − ( \frac{2p}{p−N} )^{p}a_{n}^{p}m(D)( \frac{σ(N,p)}{D^{p}} + g(p, N))} ,

by choosing

v_{n}(x) := \{ \begin{matrix}0 & x ∈ ℝ^{ℕ} ∖ B(0, D) \\ a_{n} & x ∈ B(0, \frac{D}{2} ) \\ \frac{2a_{n}}{D} (D − |x|) & x ∈ B(0, D) ∖ B(0, \frac{D}{2} ), \end{matrix}

for each n ∈ ℕ. Hence, by assumption ii′) one has Â < +∞, we obtain

γ ⩽ lim inf_{n→∞} φ(r_{n}) ⩽ p( \frac{2p}{p − N} )^{p}Â < +∞.

From now on, arguing as in the proof of Theorem 3.1, the conclusion follows.

Acknowledgement. The authors would like to thank the unknown referee for her/his valuable comments and suggestions.

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Department of Mathematics (Received 04 02 2015) Faculty of Mathematical and Computer Sciences (Revised 19 06 2016) Shahid Chamran University of Ahwaz Ahwaz Iran

m.khodabakhshi11@gmail.com
aminpour@scu.ac.ir

Department of Mathematics Ramhormoz Branch Islamic Azad University Ramhormoz Iran

m.reza.h56@gmail.com