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Existence of three solutions for the Dirichlet problem involving the p-Laplacian.

Abstract

In this paper, the existence of at least three weak solutions for Dirichlet problems involving the p-Laplacian is established. The approach is based on variational methods and critical points.

AMS Subject Classification: 35J20; 34A15

Keywords: Three solutions; Critical point; Multiplicity results; Dirichlet problem.

1. Introduction

In this work, we consider the boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

where [[DELTA].sub.p]u =div([|[nabla]u|.sup.p-2] [nabla]u) is the p-Laplacian operator, [OMEGA] [subset] [R.sup.N](N [greater than or equal to] 1) is nonempty bounded open set with smooth boundary [partial derivative] [OMEGA],p > N, [lambda] > 0 and f : [OMEGA] X R [right arrow] R is a positive [L.sup.1]-Caratheodory function. As usual, a weak solution of (1.1) is any u [member of] [W.sup.1,p.sub.0] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all v [member of] [W.sup.1,p.sub.0] ([OMEGA]).

In this paper, under novel assumptions, we are interested in ensuring the existence of at least three weak solutions for the problem (1.1). In recent years, many authors have studied multiple solutions from several points of view and with different approaches (see, for example, [1,2,4,5]); for instance, in [1], using variational methods, the authors ensure the existence of a sequence of arbitrarily small positive solutions for problem (1.1) when the function f has a suitable oscillating behaviour at zero. Also, in [2] authors studied problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.2)

by using a multiple fixed-point theorem to obtain three symmetric positive solutions under growth conditions on f.

In [4], the author proves multiplicity results for the problem (1.2) which for each [lambda] [member of] [0, + [infinity] [, admits at least three solutions in [W.sup.1,2.sub.0] ([0, 1]) when f is continuous function. Our approach is based on a three critical points Theorem proved in [7], recalled below for the reader's convenience (Theorem A), and on technical lemma that allow us to apply it.

Theorem 2.2 which is our main result, under novel assumptions ensures the existence of an open interval [LAMBDA] [[subset].bar] [0, [infinity][ and a positive real number q such that, for each [lambda] [member of] [LAMBDA], problem (1.1) admits at least three weak solutions whose norms in [W.sup.1,p.sub.0] ([OMEGA]) are less than q.

We here recall its equivalent formulation [3, Theorem 1.1 and Remark 1.1]:

Theorem A. Let X be a separable and reflexive real Banach space; [PHI] : X [right arrow] R a continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on [X.sup.*]; [psi] : X [right arrow] R a continuously Gateaux differentiable functional whose Gateaux derivative is compact.

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all [lambad] [member of] [0,+ [infinity][, and that there exists [rho] [member of] R such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then, there exists an open interval [LAMBDA] [[subset].bar] [0,+ [infinity][ and a positive real number q such that , for each [lambda] [member of] [LAMBDA], the equation

[PHI]'(u) + [lambda] [psi]'(u) = 0

has at least three solutions in X whose norms are less than q.

2. Main Results

Here and in the sequel, X will denote the Sobolev space [W.sup.1,p.sub.0 ([OMEGA]) with the norm

[parallel] u [parallel] = [([[integral].sub.[OMEGA]] [|[nabla]u(x)|.sup.p]dx).sup.1/p],

and put

g(x, t) = [[integral].sup.t.sub.0] f (x, [xi])d [xi]

for each (x, t) [member of] [OMEGA] x R.

Now, fix [x.sup.0] [member of] [OMEGA] and pick [r.sub.1], [r.sub.2] with o < [r.sub.1] < [r.sub.2] such that

S([x.sup.0], [r.sub.1]) [subset] S([x.sup.0], [r.sub.2]) [[subset].bar] [OMEGA].

Put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where [GAMMA] denotes the Gamma function, c = c(N, p) is a positive constant and |[OMEGA]| is the measure of the set [OMEGA].

Our main results fully depend on the following Lemma:

Lemma 2.1. Assume that there exist two positive constants [tau] and d with kd > [tau], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where k is given in (2.1).

Then, there exist r > 0 and w [member of] X such that [[parallel]w[parallel].sup.p] > pr and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. We put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and r = [[tau].sup.p]/[pc.sup.p][|[OMEGA]|.sup.p/N-1] . It is easy to see that w [member of] X and, in particular, one has

[[parallel]w[parallel].sup.p] = ([r.sup.N.sub.2] - [r.sup.N.sub.1]) [[pi].sup.N/2/[GAMMA](1 + N/2) [(d/[r.sup.2] - [r.sup.1]).sup.p]

Hence, taking into account that kd > [tau], one has

pr < [[parallel]w[parallel].sup.p].

Moreover, owing to our assumptions, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So, the Proof is complete.

Now, we state our main result:

Theorem 2.2. Assume that there exist three positive constants [tau], d, s with kd > [tau], s < p and a positive function a [member of [L.sup.1]([OMEGA]) such that

(i) |[OMEGA]|[(kd).sup.p] [max.sub.(x,t)[member of] [bar.[OMEGA]] x [-[tau], [tau]]] g(x, t) < [[tau].sup.p] [[integral].sub.S([x.sup.0, [r.sup.1])] g(x, d)dx,

(ii) g(x, t) [less than or equal to] a(x)(1 + [|t|.sup.s]) almost everywhere in [OMEGA] and for each t [member of] R, where k is given in (2.1).

Then, there exists an open interval [LAMBDA] [member of] [0, + [infinity][ and a positive real number q such that, for each [lambda] [member of] [LAMBDA], problem (1.1) admits at least three solutions in X whose norms are less than q.

Proof. For each u [member of]X, we put

[phi](u) = [[parallel]u[parallel].sup.p]/p, [psi](u) = - [[integral].sub.[omega]] g(x, u(x))dx and J (u) = [phi](u) + [lambda][psi](u).

In particular, for each u, v [member of] X one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is well known that the critical points of J are the weak solutions of (1.1), our goal is to prove that [phi] and [psi] satisfy the assumptions of Theorem A. Clearly, [phi] is a continuously Gateaux differentiable and sequentially weakly lower semi continuous functional whose Gateaux derivative admits a continuous inverse on [X.sup.*] and [psi] is a continuously Gateaux differentiable functional whose Gateaux derivative is compact.

Thanks to (ii), for each [lambda] > 0 one has that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We claim that there exist r > 0 and w [member of] X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, taking into account that for every u [member of] X, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for each u [member of] X, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thanks to Lemma 2.1, there exist r > 0 and w [member or] X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Fix [rho] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from Proposition 3.1 of [6], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, our conclusion follows from Theorem A. _

Let [h.sub.1] [member of] C([bar.[OMEGA]]) and [h.sub.2] [member of] C(R) be two positive functions. Put

f (x,u) = [h.sub.1](x)[h.sub.2](u)

for each (x, u) [member of] [OMEGA] x R,

H(t) = [[integral].sup.t.sub.0] [h.sub.2]([xi])d[xi]

for all t [member of] R, and

b(x) = a(x)/[h.sub.1](x).

Then, with use the Theorem 2.2, we have the following result:

Corollary 2.3. Assume that there exist three positive constants [tau], d, s with kd > [tau], s < p and a positive function b [member of] [L.sup.1]([OMEGA]) such that

(jj) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(ii) H(t) [less than or equal to] a(x)(1 + [|t|.sup.s]) for each t [member of] R, where k is given in (2.1).

Then, there exists an open interval [nabla] [[subset].bar] [0,+ [infinity][ and a positive real number q such that, for each [lambda] [member of] [nabla], problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

admits at least three solutions in X whose norms are less than q.

References

[1] Anello G., Cordaro G., 2002, Positive infinitely many arbitrarily small solutions for the Dirichlet problem involving the p-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 132, pp. 511-519.

[2] Avery R.I., Henderson J., 2000, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. lett., 13, pp. 1-7.

[3] Bonanno G., 2002,Aminimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl., 270, pp. 210-229.

[4] Bonanno G., 2000, Existence of three solutions for a two point boundary value problem, Appl. Math. Lett., 13, pp. 53-57.

[5] Korman P., Ouyang T., 1993, Exact multiplicity results for two classes of boundary value problem, Diff. Integral Equations, 6, pp. 1507-1517.

[6] Ricceri B., Existence of three solutions for a class of elliptic eigenvalue problem, Math. Comput. Modelling., 32, pp. 1485-1494.

[7] Ricceri B., 2000, On a three critical points theorem, Arch. Math. (Basel), 75, pp. 220-226.

G.A. Afrouzi and S. Heidarkhani

Department of Mathematics, Faculty of Basic Sciences,

Mazandaran University, Babolsar, Iran

E-mail: afrouzi@umz.ac.ir
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Author:Afrouzi, G.A.; Heidarkhani, S.
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:7IRAN
Date:Apr 1, 2006
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