Three solutions to a p(x)-laplacian problem in weighted-variable-exponent Sobolev space

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1 DOI: /uom An. Şt. Univ. Ovidius Constnţ Vol. 2(2),203, Three solutions to p(x)-lplcin problem in weighted-vrible-exponent Sobolev spce Wen-Wu Pn, Ghsem Alizdeh Afrouzi nd Lin Li Abstrct In this pper, we verify tht generl p(x)-lplcin Neumnn problem hs t lest three wek solutions, which generlizes the corresponding result of the reference [R. A. Mshiyev, Three Solutions to Neumnn Problem for Elliptic Equtions with Vrible Exponent, Arb. J. Sci. Eng. 36 (20) ]. Introduction Recently, elliptic equtions with vrible exponents hve been extensively investigted nd hve received much ttention. They hve been the subject of recent developments in nonliner elsticity theory nd electrorheologicl fluids dynmics [6]. In tht context, let us mention tht there ppered series of ppers on problems which led to spces with vrible exponent, we refer the reder to Fn et l. [8, 9], Ruzick [6] nd the references therein. Let us point out tht when p(x) = p = constnt, there is lrge literture which del with problems involving the p-lplcin with Dirichlet boundry conditions both in bounded or unbounded domins, which we do not need to cite here since the reder my esily find such ppers. Note tht mny ppers del with problems relted to the p-lplcin with Neumnn conditions in the sclr cse. We cn cite, mong others, the rticles [, 4] nd refer to the references therein for detils. The cse of p(x)-lplcin Key Words: p(x)-lplcin problems, Neumnn problems, Ricceri s vritionl principle 200 Mthemtics Subject Clssifiction: Primry 34B5; Secondry 35A5, 35G99. Received: Jnury 202 Accepted: My

2 96 W.-W. Pn, G. A. Afrouzi nd L. Li with Neumnn conditions hs been studied by Di [6], Mihilescu [3] nd Liu []. In this pper, we will consider the Neumnn problems involving the p(x)- Lplcin opertor { div ( u p(x) 2 u ) + (x) u p(x) 2 u = λf(x, u) + µg(x, u), in, u ν = 0, on, (P) where R N (N 3) is bounded domin with smooth boundry, λ, µ > 0 re rel numbers, p(x) is continuous function on with inf x p(x) > N nd L () with essinf x (x) = 0 > 0. We denote by ν the outwrd unit norml to. The min interest in studying such problems rises from the presence of the p(x)-lplcin opertor div ( u p(x) 2 u ), which is generliztion of the clssicl p-lplcin opertor div ( u p 2 u ) obtined in the cse when p is positive constnt. When µ = 0, in [2], R. A. Mshiyev studied the prticulr cse f(t) = b t q 2 t d t s 2 t where b nd d re positive constnts, 2 < s < q < inf x p(x) nd N < inf x p(x); nd where 2 < inf x f(x, t) = t q(x) 2 t t s(x) 2 t s(x) sup s(x) < inf x x q(x) sup q(x) < inf p(x) x x nd N < inf x p(x) for ll x. He estblished the existence of t lest three wek solutions by using the Ricceri s vritionl principle. In this pper, we ssume f(x, u) nd g(x, u) stisfies the following generl conditions: (f) f, g : R R re Crthéodory functions nd stisfies f(x, t) c + c 2 t α(x), (x, t) R, g(x, t) c + c 2 t β(x), (x, t) R, where α(x), β(x) C(), α(x), β(x) > nd < α + = mx x α(x) < p = min x p(x), < β + = mx x β(x) < p = min x p(x) nd c, c 2, c, c 2 re positive constnts.

3 THREE SOLUTIONS TO A P (X)-LAPLACIAN PROBLEM 97 (f2) There exist constnt t 0 nd following conditions stisfies f(x, t) < 0 when t (0, t 0 ) f(x, t) > M > 0 where M is positive constnt. when t (t 0, + ), Following long the sme lines s in [2], we will prove tht there lso exist three wek solutions for such generl problem for λ sufficiently lrge nd requiring µ smll enough. 2 Preliminry results nd lemm In this prt, we introduce some theories of Lebesgue Sobolev spce with vrible exponent. The detiled description cn be found in [0, 7, 8, 9]. Denote by S() the set of ll mesurble rel functions on. Set For ny p C + (), denote C + () = { p : p C(), p(x) >, x. < p := inf p(x) p(x) p + := sup p(x) <. x x Let p C + (). Define the generlized Lebesgue spce by { L p(x) () = u u S(), u(x) p(x) dx <, then L p(x) () endowed with the norm { u p(x) = inf β > 0 : u(x) β p(x) dx becomes Bnch spce. Let S(), nd (x) > 0 for.e. x. Define the weighted vrible exponent Lebesgue spce L p(x) () by { L p(x) () = u u S(), (x) u(x) p(x) dx <, with the norm { u p(x) = inf β > 0 : (x) u(x) β p(x) dx,.

4 98 W.-W. Pn, G. A. Afrouzi nd L. Li From now on, we suppose tht L () nd essinf x (x) = 0 > 0. Then obviously L p(x) () is Bnch spce (see [5] for detils). The vrible exponent Sobolev spce W,p(x) () is defined by { W,p(x) () = u L p(x) () : u L p(x) (), with the norm u = u p(x) + u p(x). Next, the weighted-vrible-exponent Sobolev spce W,p(x) () is defined by with the norm { u = inf β > 0 : { W,p(x) () = ( u(x) β u L p(x) p(x) () : u L p(x) () + (x) u(x) p(x)) dx, u W,p(x) (). β Then the norms nd in W,p(x) () re equivlent. If < p p + <, then the spce W,p(x) () is seprble nd reflexive Bnch spce. We set ρ(u) = ( u p(x) + (x) u p(x)) dx. Proposition ([7], Proposition 2.5). For ll u W,p(x) (), we hve (i) u u p+ ρ(u) u p, (ii) u u p ρ(u) u p+. Remrk. If N < p p(x) for ny x, by Theorem 2.2. in [9] nd the equivlence of the norms nd, we deduce tht W,p(x) () W,p (). Since N < p, it follows tht W,p(x) () W,p () C(). Defining the norm u = sup u(x), x then there exists constnt k > 0 such tht, u k u, u W,p(x) (). To prove the existence of t lest three wek solutions for ech of the given problem (P), we will use the following result proved in [5] tht, on the bsis of [2], cn be equivlently stted s follows

5 THREE SOLUTIONS TO A P (X)-LAPLACIAN PROBLEM 99 Theorem. Let X be seprble nd reflexive rel Bnch spce; Φ : X R continuously Gâteux differentible nd sequentilly wekly lower semicontinuous functionl whose Gâteux derivtive dmits continuous inverse on X, Ψ : X R continuously Gâteux differentible functionl whose Gâteux derivtive is compct. Assume tht (i) lim u Φ(u) + λψ(u) = for ll λ > 0; nd there re r R nd u 0, u X such tht (ii) Φ(u 0 ) < r < Φ(u ); (iii) inf u Φ ([,r]) Ψ(u) > (Φ(u) r)ψ(u0)+(r Φ(u0))Ψ(u) Φ(u ) Φ(u 0) Then there exist n open intervl Λ (0, ) nd positive rel number q such tht for ech λ Λ nd every continuously Gâteux differentible functionl J : X R with compct derivtive, there exists σ > 0 such tht for ech µ [0, σ], the eqution Φ (u) + λψ (u) + µj (u) = 0 hs t lest three solutions in X whose norms re less thn q. 3 The min result nd proof of the theorem In this prt, we will prove tht for problem (P) there lso exist three wek solutions for the generl cse. Definition. We sy u W,p(x) ( ) u p(x) 2 u v + (x) u p(x) 2 u dx λ for ny v W,p(x) is wek solution of problem (P) if f(x, u)v dx µ g(x, u)v dx = 0 Theorem 2. Assume tht p > N nd f(x, u) stisfies (f), (f2). Then there exist n open intervl Λ (0, ) nd positive rel number q > 0 such tht ech λ Λ nd every function g : R R which stisfying (f), there exists δ > 0 such tht for ech µ [0, δ] problem (P) hs t lest three solutions whose norms re less thn q.

6 200 W.-W. Pn, G. A. Afrouzi nd L. Li Proof. Let X denote the weighted vrible exponent Lebesgue spce (). Define W,p(x) F (x, t) = t 0 f(x, s) ds nd G(x, t) = t 0 g(x, s) ds. In order to use Theorem, we define the functions Φ, Ψ, J : X R by Φ(u) = p(x) ( u p(x) + (x)u p(x) ) dx Ψ(u) = F (x, u) dx J(u) = G(x, u) dx Arguments similr to those used in the proof of Proposition 3. in [4], we know Φ, Ψ, J C (X, R) with the derivtives given by Φ (u), v = ( u p(x) 2 u v + (x)u p(x) 2 uv) dx Ψ (u), v = f(x, u)v dx J (u), v = g(x, u)v dx for ny u, v X. Thus, there exists λ, µ > 0 such tht u is criticl point of the opertor Φ(u) + λψ(u) + µj(u), tht is Φ (u) + λψ (u) + µj (u) = 0. For proving our result, it is enough to verify tht Φ, Ψ nd J stisfy the hypotheses of Theorem. It is obvious tht (Φ ) : X X exists nd continuous, becuse Φ : X X is homeomorphism by Lemm 2.2 in [2]. Moreover, Ψ, J : X X re completely continuous becuse of the ssumption (f) nd [0], which imply Ψ nd J re compct. Next, we will verify tht condition(i) of Theorem is fulfilled. In fct, by Proposition, we hve Φ(u) p + ( u p(x) +(x) u p(x) ) dx = p + ρ(u) p + u p, u X, u >. On the other hnd, due to the ssumption (f), we hve Ψ(u) = F (x, u) dx = F (x, u) dx

7 THREE SOLUTIONS TO A P (X)-LAPLACIAN PROBLEM 20 nd Therefore, Ψ(u) c F (x, t) c t + c 2 α(x) t α(x). u dx c 2 c 3 u c 2 α + α(x) u α(x) dx ( u α+ + u α ) dx = c 3 u c 4 ( u α+ α + + u α α ) Using Remrk, we know tht X is continuously embedded in L α+ nd L α. Furthermore, we cn find two positive constnts d, d 2 > 0 such tht Moreover It follows tht Φ(u) + λψ(u) u α + d u nd u α d 2 u u X. Ψ(u) c 3 u c 4 d u α+ c 4 d 2 u α. ( ) p + λc 3 u p λc 4 (d u α+ + d 2 u α ), u X. Since < α + < p, then lim u Φ(u) + λψ(u) = nd (i) is verified. In the following, we will verify the conditions (ii) nd (iii) in Theorem. By F t(x, t) = f(x, t) nd ssumption (f2), it follows tht F (x, t) is incresing for t (t 0, ) nd decresing for t (0, t 0 ), uniformly with respect to x. Obviously, F (x, 0) = 0. F (x, t) when t, becuse of ssumption (f2). Then there exists rel number δ > t 0 such tht F (x, t) 0 = F (x, 0) F (x, τ), x X, t > δ, τ (0, t 0 ). Let, b be two rel numbers such tht 0 < < min{t 0, k with k given in Remrk nd b > δ stisfies nd b p L () > b p+ L () >. Let b >. When t [0, ], we hve F (x, t) F (x, 0), it follows tht sup F (x, t) dx F (x, 0) dx = 0 0 t

8 202 W.-W. Pn, G. A. Afrouzi nd L. Li Furthermore, we cn get F (x, b) dx > 0 becuse of b > δ. Moreover, p + F (x, b) dx > 0. k p+ b p The bove two inequlities imply sup 0 t F (x, t) dx 0 < p + k p+ b p F (x, b) dx. Consider u 0, u X with u 0 (x) = 0 nd u (x) = b for ny x. We define r = p + ( k ) p +. Clerly, r (0, ). A simple computtion implies nd Φ(u 0 ) = Ψ(u 0 ) = 0 Φ(u ) = p(x) (x)bp(x) dx Ψ(u ) = F (x, u (x)) dx = p + bp L () > p + > p + ( k F (x, b) dx. Similrly for b <, by help of Proposition, we get the desired result. Thus, we obtin Φ(u 0 ) < r < Φ(u ) nd (ii) in Theorem is verified. On the other hnd, we hve (Φ(u ) r)ψ(u 0 ) + (r Φ(u 0 ))Ψ(u ) Φ(u ) Φ(u 0 ) = r Ψ(u ) Φ(u ) = r ) p + F (x, b) dx p(x) (x)bp(x) dx > 0. Next, we consider the cse u X with Φ(u) r <. Since p(x) ρ(u) Φ(u) r, we obtin ρ(u) p + r = ( ) p + k <, it follows tht u <. Furthermore, it is cler tht + p + u p ρ(u) Φ(u) r. p + Thus, using Remrk, we hve u(x) k u k(p + r) p + = x, u X, Φ(u) r. The bove inequlity shows tht inf u Φ ([,r]) Ψ(u) = sup Ψ(u) u Φ ([,r]) sup 0 t F (x, t) dx 0.

9 THREE SOLUTIONS TO A P (X)-LAPLACIAN PROBLEM 203 It follows tht Tht is inf Ψ(u) < r u Φ ([,r]) F (x, b) dx p(x) (x)bp(x) dx. inf Ψ(u) > (Φ(u ) r)ψ(u 0 ) + (r Φ(u 0 ))Ψ(u ) u Φ ([,r]) Φ(u ) Φ(u 0 ) which mens tht condition (iii) in Theorem is verified. Then the proof of Theorem 2 is chieved. Remrk 2. Applying ([3], Theorem2.) in the proof of Theorem 2, n upper bound of the intervl of prmeters λ for which (P) hs t lest three wek solutions is obtined when µ = 0. To be precise, in the conclusion of Theorem 2 one hs ] [ p(x) Λ 0, h (x)bp(x) dx F (x, b) dx for ech h > nd b s in the proof of Theorem 2. Acknowledgments The uthor would like to thnk reviewers for cler vluble comments nd suggestions. The first nd the third uthor ws supported by the Fundmentl Reserch Funds for the Centrl Universities (No. XDJK203D007), Scientific Reserch Fund of SUSE (No. 20KY03) nd Scientific Reserch Fund of SiChun Provincil Eduction Deprtment (No. 2ZB08). References [] G. Anello nd G. Cordro. Existence of solutions of the Neumnn problem for clss of equtions involving the p-lplcin vi vritionl principle of Ricceri. Arch. Mth. (Bsel), 79(4): , [2] G. Bonnno. A minimx inequlity nd its pplictions to ordinry differentil equtions. J. Mth. Anl. Appl., 270():20 229, [3] G. Bonnno. Some remrks on three criticl points theorem. Nonliner Anl., 54(4):65 665, [4] G. Bonnno nd P. Cndito. Three solutions to Neumnn problem for elliptic equtions involving the p-lplcin. Arch. Mth. (Bsel), 80(4): , 2003.

10 204 W.-W. Pn, G. A. Afrouzi nd L. Li [5] D. Cruz-Uribe, L. Diening, nd P. Hästö. The mximl opertor on weighted vrible lebesgue spces. Frc. Clc. Appl. Anl., 4:36 374, 20. [6] G. Di. Three solutions for Neumnn-type differentil inclusion problem involving the p(x)-lplcin. Nonliner Anl., 70(0): , [7] X. Fn. Solutions for p(x)-lplcin Dirichlet problems with singulr coefficients. J. Mth. Anl. Appl., 32(2): , [8] X. Fn, J. Shen, nd D. Zho. Sobolev embedding theorems for spces W k,p(x) (). J. Mth. Anl. Appl., 262(2): , 200. [9] X. Fn nd D. Zho. On the spces L p(x) () nd W m,p(x) (). J. Mth. Anl. Appl., 263(2): , 200. [0] O. Kováčik nd J. Rákosník. On spces L p(x) nd W k,p(x). Czechoslovk Mth. J., 4(6)(4):592 68, 99. [] Q. Liu. Existence of three solutions for p(x)-lplcin equtions. Nonliner Anl., 68(7):29 227, [2] R. Mshiyev. Three Solutions to Neumnn Problem for Elliptic Equtions with Vrible Exponent. Arb. J. Sci. Eng., 36: , 20. [3] M. Mihăilescu. Existence nd multiplicity of solutions for Neumnn problem involving the p(x)-lplce opertor. Nonliner Anl., 67(5):49 425, [4] M. Mihăilescu nd V. Rădulescu. A multiplicity result for nonliner degenerte problem rising in the theory of electrorheologicl fluids. Proc. R. Soc. Lond. Ser. A Mth. Phys. Eng. Sci., 462(2073): , [5] B. Ricceri. A three criticl points theorem revisited. Nonliner Anl., 70(9): , [6] M. Růžičk. Electrorheologicl fluids: modeling nd mthemticl theory, volume 748 of Lecture Notes in Mthemtics. Springer-Verlg, Berlin, [7] S. Smko. Denseness of C 0 (R N ) in the generlized Sobolev spces W M,P (X) (R N ). In Direct nd inverse problems of mthemticl physics (Newrk, DE, 997), volume 5 of Int. Soc. Anl. Appl. Comput., pges Kluwer Acd. Publ., Dordrecht, 2000.

11 THREE SOLUTIONS TO A P (X)-LAPLACIAN PROBLEM 205 Wen-Wu PAN, Deprtment of Science, Sichun University of Science nd Engineering, Zigong , P. R. Chin. Emil: @qq.com Ghsem Alizdeh AFROUZI, Deprtment of Mthemtics, Fculty of Mthemticl sciences, University of Mzndrn, Bbolsr, Irn. Emil: frouzi@umz.c.ir Lin LI, School of Mthemtics nd Sttistics, Southwest University, Chongqing 40075, P. R. Chin. Emil: lilin420@gmil.com