Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 21, Article ID 12646, 7 pages doi:11155/21/12646 Research Article Existence and Localization Results for -Laplacian via Topological Methods B Cekic and R A Mashiyev Department of Mathematics, Dicle University, 2128 Diyarbakir, Turkey Correspondence should be addressed to B Cekic, bilalcekic@gmailcom Received 23 February 21; Revised 16 April 21; Accepted 2 June 21 Academic Editor: J Mawhin Copyright q 21 B Cekic and R A Mashiyev This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We show the existence of a week solution in W 1, to a Dirichlet problem for Δ u f x, u in, and its localization This approach is based on the nonlinear alternative of Leray-Schauder 1 Introduction In this work, we consider the boundary value problem Δ u f x, u in, u on, P where R N,N 2, is a nonempty bounded open set with smooth boundary, Δ u div u 2 u is the so-called -Laplacian operator, and CAR : f : R R is a Caratheodory function which satisfies the growth condition f x, s a x C s q x /q x for ae x and all s R, 11 with C const >, 1/q x 1/q x 1 for ae x, anda L q x, a x for ae x We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces L, W 1,, andw 1, In that context, we refer to 1, 2 for the fundamental properties of these spaces
2 Fixed Point Theory and Applications Set L p : p L, ess inf > 1 12 x For p L, let p 1 : ess inf x p 2 : ess sup x <, for ae x Let us define by U the set of all measurable real functions defined on For any p L, we define the variable exponent Lebesgue space by L u U : ρ u u x dx < 13 We define a norm, the so-called Luxemburg norm, on this space by the formula ) u u inf δ>:ρ 1, 14 δ and L, becomes a Banach space The variable exponent Sobolev space W 1, is W 1, u L u : L, i 1,,N x i 15 and we define on this space the norm u u u 16 for all u W 1, The space W 1, is the closure of C in W 1, Proposition 11 see 1, 2 If p L, then the spaces L, W 1,, and W 1, are separable and reflexive Banach spaces Proposition 12 see 1, 2 If u L and p 2 <, then we have i u < 1 1; > 1 ρ u < 1 1; > 1, ii u > 1 u p 1 ρ u u p 2, iii u < 1 u p 2 ρ u u p 1, iv u a> ρ u/a 1
Fixed Point Theory and Applications 3 Proposition 13 see 3 Assume that is bounded and smooth Denote by C h C : h 1 > 1 i Let p, q C If N if <N, q x <p x N if N, 17 then W 1,, is compactly imbedded in L q x ii Poincaré inequality, see [1, Theorem 27]) If p C, then there is a constant C> such that u C u, u W 1, 18 Consequently, u 1, u and u are equivalent norms on W 1, In what follows, W 1,, withp C, will be considered as endowed with the norm u 1, Lemma 14 Assume that r L and p C If u r x L, then we have min u r 1, u r 2 u r x max u r 1, u r 2 19 Proof By Proposition 12 iv, we have 1 u r x u r x dx u u u u max u u r x dx u r 1, u r 2 u r x dx 11 By the mean value theorem, there exists ξ such that 1 max u r 1p ξ, u r 2p ξ u r x p ξ u u dx 111
4 Fixed Point Theory and Applications and we have u r x max u r 1, u r 2 112 Similarly 1 min u r 1p ξ, u r 2p ξ u r x p ξ dx, u r x min u r 1, u r 2 113 Remark 15 If r x r const, then u r u r r 114 For simplicity of notation, we write, X W 1,, X W 1, ) Y L q x, Y L q x, X 1,, Y q x 115 In 4, a topological method, based on the fundamental properties of the Leray- Schauder degree, is used in proving the existence of a week solution in X to the Dirichlet problem P that is an adaptation of that used by Dinca et al for Dirichlet problems with classical p-laplacian 5 In this work, we use the nonlinear alternative of Leray-Schauder and give the existence of a solution and its localization This method is used for finding solutions in Hölder spaces, while in 6, solutions are found in Sobolev spaces Let us recall some results borrowed from Dinca 4 about -Laplacian and Nemytskii operator N f Firstly, since q x < < p x for all x, X is compactly embedded in Y Denote by i the compact injection of X in Yand by i : Y X, i υ υ i for all υ Y, its adjoint Since the Caratheodory function f satisfies CAR, the Nemytskii operator N f generated by f, N f u x f x, u x, is well defined from Y into Y, continuous, and bounded 3, Proposition22 In order to prove that problem P has a weak solution in X it is sufficient to prove that the equation Δ u i N f i ) u 116 has a solution in X
Fixed Point Theory and Applications 5 Indeed, if u X satisfies 116 then, for all υ X, one has Δ u, υ i N X,X f i ) u, υ N X,X f iu,iυ Y,Y 117 which rewrites as u u υdx fυdx 118 and tells us that u is a weak solution in X to problem P Since Δ is a homeomorphism of X onto X, 116 may be equivalently written as u Δ ) 1 i N f i ) u 119 Thus, proving that problem P has a weak solution in X reduces to proving that the compact operator K Δ ) 1 i N f i ) : X X 12 has a fixed point Theorem 16 Alternative of Leray-Schauder, 7 Let B,R denote the closed ball in a Banach space E, u E : u R, and let K : B,R E be a compact operator Then either i the equation λku u has a solution in B,R for λ 1 or ii there exists an element u E with u R satisfying λku u for some λ, 1 2 Main Results In this work, we present new existence and localization results for X-solutions to problem P, under CAR condition on f Our approach is based on regularity results for the solutions of Dirichlet problems and again on the nonlinear alternative of Leray-Schauder We start with an existence and localization principle for problem P Theorem 21 Assume that there is a constant R>, independent of λ>, with u X / R for any solution u X to Δ u λf x, u in, u on P λ and for each λ, 1 Then the Dirichlet problem P has at least one solution u X with u X R Proof By 3, Theorem 31, Δ is a homeomorphism of X onto X We will apply Theorem 21 to E X and to operator K : X X, Ku Δ ) 1 i N f i ) u, 21
6 Fixed Point Theory and Applications where i N f i : X X is given by N f u x f x, u x Notice that, according to a wellknown regularity result 4, the operator Δ 1 from X to X is well defined, continuous, and order preserving Consequently, K is a compact operator On the other hand, it is clear that the fixed points of K are the solutions of problem P Now the conclusion follows from Theorem 16 since condition ii is excluded by hypothesis Theorem 22 immediately yields the following existence and localization result Theorem 22 Let R N,N 2, be a smooth bounded domain and let p, q C be such that q x < for all x Assume that f : R R is a Caratheodory function which satisfies the growth condition CAR) Suppose, in addition, that C i Y X max i q 1 1 < 1, 22 where C is the constant appearing in condition CAR) Let R 1 be a constant such that i R 1 C i Y X max i q 1 1 23 Then the Dirichlet problem P has at least a solution in X with u X R Proof Let u X be a solution of problem P λ with u X R 1, corresponding to some λ, 1 Then by Propositions 12, 13, andlemma 14,weobtain u p 1 X u dx λ i N f i ) u, u λ N X,X f iu,iu Y,Y λ i Y X N f iu Y u X ) λ i Y X u X a Y C max iu q 1 1 Y, iu q 2 1 Y ) λ i Y X u X a Y C u q 2 1 X max i q 1 1, i q 2 1 ) λ i Y X u X a Y C u p 1 1 X max i q 1 1, i q 2 1 24 Therefore, we have u p 1 1 X λ i Y X a Y 1 λc i Y X max i q 1 1 25
Fixed Point Theory and Applications 7 Substituting u X R in the above inequality, we obtain λ i R 1 λc i Y X max i q 1 1, 26 which, taking into account 23 and λ, 1, gives R λ 1/p 1 1 i 1 Cλ i Y X max i q 1 1 λ 1/p 1 1 i 1 C i Y X max i q 1 1 27 λ 1/p 1 1 R<R, a contradiction Theorem 21 applies Acknowledgment The authors would like to thank the referees for their valuable and useful comments References 1 X Fan and D Zhao, On the spaces L and W m,, Journal of Mathematical Analysis and Applications, vol 263, no 2, pp 424 446, 21 2 O Kováčik and J Rákosník, On spaces L and W k,, Czechoslovak Mathematical Journal, vol 41 116, no 4, pp 592 618, 1991 3 X-L Fan and Q-H Zhang, Existence of solutions for -Laplacian Dirichlet problem, Nonlinear Analysis: Theory, Methods & Applications, vol 52, no 8, pp 1843 1852, 23 4 G Dinca, A fixed point method for the -Laplacian, Comptes Rendus Mathématique, vol 347, no 13-14, pp 757 762, 29 5 G Dinca, P Jebelean, and J Mawhin, Variational and topological methods for Dirichlet problems with p-laplacian, Portugaliae Mathematica, vol 58, no 3, pp 339 378, 21 6 D O Regan and R Precup, Theorems of Leray-Schauder Type and Applications, vol 3 of Series in Mathematical Analysis and Applications, Gordon and Breach, Amsterdam, The Netherlands, 21 7 J Dugundji and A Granas, Fixed Point Theory I, vol 61 of Monografie Matematyczne, PWN-Polish Scientific, Warsaw, Poland, 1982