NONHOMOGENEOUS DIRICHLET PROBLEMS WITHOUT THE AMBROSETTI RABINOWITZ CONDITION. Gang Li Vicenţiu D. Rădulescu

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1 TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS Vol. 5, No. March 208 NONHOMOGENEOUS DIRICHLET PROBLEMS WITHOUT THE AMBROSETTI RABINOWITZ CONDITION Gang Li Vicenţiu D. Rădulescu Dušan D. Repovš Qihu Zhang Topol. Methods Nonlinear Anal. 5 (208), DOI: /TMNA Published by the Juliusz Schauder Center TORUŃ, 208 ISSN ISBN

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3 Topological Methods in Nonlinear Analysis Volume 5, No., 208, DOI: /TMNA c 208 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University NONHOMOGENEOUS DIRICHLET PROBLEMS WITHOUT THE AMBROSETTI RABINOWITZ CONDITION Gang Li Vicenţiu D. Rădulescu Dušan D. Repovš Qihu Zhang Abstract. We consider the existence of solutions of the following p(x)- Laplacian Dirichlet problem without the Ambrosetti Rabinowitz condition: { div( u p(x) 2 u) = f(x, u) in, u = 0 on. We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a p(x)-laplacian Dirichlet problem without the Ambrosetti Rabinowitz condition, Computers and Mathematics with Applications, 205) and we establish the existence of solutions under weaker hypotheses on the nonlinear term. 200 Mathematics Subject Classification. Primary: 35J60; Secondary: 35J20, 35J25, 58E05. Key words and phrases. Nonhomogeneous differential operator; Ambrosetti Rabinowitz condition; Cerami compactness condition; Sobolev space with variable exponent. V. Rădulescu and D. Repovš were supported in part by the Slovenian Research Agency grants P-0292, J-7025, and J-83. V. Rădulescu acknowledges the support through a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4- ID-PCE Q. Zhang has been partially supported by the key projects of Science and Technology Research of the Henan Education Department (4A00) and the National Natural Science Foundation of China (3266 and ). 55

4 56 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang. Introduction In recent years, the study of differential equations and variational problems with variable exponent growth conditions has been a topic of great interest. This type of problems has very strong background, for instance in image processing, nonlinear electrorheological fluids and elastic mechanics. Some of these phenomena are related to the Winslow effect, which describes the behavior of certain fluids that become solids or quasi-solids when subjected to an electric field. The result was named after American engineer Willis M. Winslow. There are many papers dealing with problems with variable exponents, see [] [8], [0] [25], [28], [33], [34], [37], [38], [40] [46], [48] [49]. On results concerning the existence of solutions of these kinds of problems, we refer to [8], [4], [5], [8], [2], [33], [36], [45]. We also refer to the recent monograph [35] which treats variational methods in the framework of nonlinear problems with variable exponent. In this paper, we consider the existence of solutions of the following class of Dirichlet problems: p(x) u := div( u p(x) 2 u) = f(x, u) in, (P) u = 0 on, where R N is a bounded domain with C,α smooth boundary, and p( ) > is of class C (). Since the elliptic operator with variable exponent is not homogeneous, new methods and techniques are needed to study these types of problems. We point out that commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. For instance, the eigenvalues of the p(x)-laplacian Dirichlet problem were studied in [6]. In this case, if R N is a smooth bounded domain, then the Rayleigh quotient (.) λ p( ) = inf u W,p( ) 0 ()\{0} p(x) u p(x) dx p(x) u p(x) dx is in general zero, and λ p( ) > 0 holds only under some special conditions. In [4], the author generalized the Picone identities for half-linear elliptic operators with p(x)-laplacian. In the same paper some applications to Sturmian comparison theory are also presented, but the formula is different from the constant exponent case. In a related setting, we point out that the formula u(x) p dx = p t p {x ; u(x) > t} dt has no variable exponent analogue. 0

5 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 57 In [23] and [46], the authors deal with the local boundedness and the Harnack inequality for the p(x)-laplace equation. However, it was shown in [23] that even in the case of a very nice exponent, for example, 3 for 0 < x p(x) := ( 2, 3 2 x ) for 2 2 < x <, the constant in the Harnack inequality depends on the minimizer, that is, the inequality sup u c inf u does not hold for any absolute constant c. The standard norm in variable exponent Sobolev spaces is the so-called Luxemburg norm u p( ) (see Section 2) and the integral u(x) p(x) dx does not satisfy the constant power relation. On several occasions, it is difficult to judge whether or not results about p-laplacian can be generalized to p(x)-laplacian, and even if this can be done, it is still difficult to figure out the form in which the results should be. Our main goal is to obtain a couple of existence results for problem (P) without the Ambrosetti Rabinowitz condition via critical point theory. For this purpose, we use a new method for checking the Cerami compactness condition under a new growth condition. Our results can be regarded as extensions of the corresponding results for the p-laplacian problems, but the growth condition and the methods for checking the Cerami compactness condition are different with respect to quasilinear equations with constant exponent. Next, we give a review of some results related to our work. Since the Ambrosetti Rabinowitz type condition is quite restrictive and excludes many cases of nonlinearity, there are many papers dealing with the problem without the Ambrosetti Rabinowitz type growth condition. For the constant exponent case p( ) p, we refer to [26], [27], [3], [39]. In [26], the authors considered problem (P) for p( ) p, and proved the existence of weak solutions under the following assumptions: F (x, t) lim t + t p = +, where F (x, t) = t 0 f(x, s) ds; and there exists a constant C > 0 such that H(x, t) H(x, s) + C for each x, 0 < t < s or s < t < 0, where H(x, t) = tf(x, t) pf (x, t). In [27], the author studied problem (P) for p( ) p. Under the assumption that f(x, s)/ s p 2 s is increasing when s s 0 and decreasing when s s 0, for all x, the existence of weak solutions was obtained. In [3], the authors studied problem (P) for p( ) 2, which becomes a Laplacian problem. The main result in [3] establishes the existence of weak solutions by assuming that f(x, s)/s is increasing when s s 0 and decreasing when s s 0, for all x.

6 58 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang In [39], the author also studied problem (P) for p( ) 2 and proved the existence of weak solutions under the assumption sf(x, u) C 0 s µ, where µ > 2 and C 0 > 0. If p( ) is a general function, results on variable exponent problem without the Ambrosetti Rabinowitz type growth condition are rare due to the complexity of p(x)-laplacian (see [3], [5], [20], [9], [42]). However, their assumptions imply G p +(x, t) = f(x, t)t p + F (x, t) 0 and F (x, t) > 0 as t +, so we can see that F (x, t) Ct p+ as t +. This is too strong and unnatural for the p(x)-laplacian problems. In [45], the author considered problem (P) under the following growth condition: there exist constants M, C, C 2 > 0, a > p on such that, for all x and all t M, (.2) C t p(x) [ln(e + t )] a(x) tf(x, t) C 2 tf(x, t) p(x)f (x, t). ln(e + t ) A typical example is f(x, t) = t p(x) 2 t [ln( + t )] a(x). This function satisfies the above condition (.2), but does not satisfy the Ambrosetti Rabinowitz condition. Our paper was motivated by [45]. We further weaken condition (.2). To begin, we point out that the assumption a > p on is unnecessary in the present paper. Before stating our main results, we make the following assumptions: (f 0 ) f : R R satisfies the Carathéodory condition and f(x, t) C( + t α(x) ), for all (x, t) R, where α C() and p(x) < α(x) < p (x) on. (f ) There exist constants M, C > 0, such that (.3) C (.4) and tf(x, t) K(t) tf(x, t) p(x)f (x, t), for all t M and all x, tf(x, t) + uniformly as t + for x, t p(x) [K(t)] p(x) where K satisfies the following hypotheses: (K) K( ) C ([0, + ), [, + )) is increasing and [ln(e + t)] 2 K(t) + as t +, which satisfies tk (t)/k(t) σ 0 (0, ), where σ 0 is a constant. (f 2 ) f(x, t) = o( t p(x) ) uniformly for x as t 0. (f 3 ) f(x, t) = f(x, t), for all x, for all t R.

7 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 59 (f 4 ) F satisfies F (x, t) + uniformly as t + for x. t p(x) [ln(e + t )] p(x) (p ) There is a vector l R N \ {0} such that for any x, ρ(t) = p(x + tl) is monotone for t I x (l) = {t x + tl }. (p 2 ) p has a local maximum point, that is, there exist x 0 and δ > 0 such that B(x 0, 3δ) and min p(x) > max p(x). x x 0 δ 2δ x x 0 3δ (p 3 ) p has a sequence of local maximum points, that is, there exist a sequence of points x n and δ n > 0 such that B(x 0, 3δ n ) are mutually disjoint and min p(x) > max p(x). x x n δ n 2δ n x x n 3δ n We state our main results in what follows. Theorem.. Assume that hypotheses (f 0 ) (f 2 ), (p ) and (f 4 ) or (p 2 ) are fulfilled. Then problem (P) has a nontrivial solution. Theorem.2. Assume that hypotheses (f 0 ), (f ), (f 3 ) and (f 4 ) or (p 3 ) are fulfilled. Then problem (P) has infinitely many pairs of solutions. Remark.3. (a) The following functions satisfy hypothesis (K): K (t) = ln(e + t ), K 2 (t) = ln(e + ln(e + t )), K 3 (t) = [ln(e + ln(e + t ))] ln(e + t ). Let K = K, and f(x, t) = t p(x) 2 t[ln( + t )] p(x) ρ( t ), where ρ( t ) [ln(e + tt )] 2, ρ 0 and ρ( t ) + as t +, for example ρ( t ) = ln(e + ln(e + t )). Then f satisfies conditions (f 0 ) (f 4 ), but it does not satisfy the Ambrosetti Rabinowitz condition, and does not satisfy (.2). (b) We do not need any monotonicity assumption on f(x, ). This paper is organized as follows. In Section 2, we do some preparatory work including some basic properties of the variable exponent Sobolev spaces, which can be regarded as a special class of generalized Orlicz Sobolev spaces. In Section 3, we give proofs of the results stated above. 2. Preliminary results Throughout this paper, we use letters c, c i, C, C i, i =, 2,..., to denote generic positive constants which may vary from line to line, and we will specify them whenever necessary.

8 60 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang One of the reasons for the huge development of the theory of classical Lebesgue and Sobolev spaces L p and W,p (where p ) is its usefulness for the description of many phenomena arising in applied sciences. For instance, many materials can be modeled with sufficient accuracy by using the function spaces L p and W,p, where p is a fixed constant. For some materials with nonhomogeneities, for instance electrorheological fluids (sometimes referred to as smart fluids ), this approach is not adequate, but rather the exponent p should be allowed to vary. This leads us to the study of variable exponent Lebesgue and Sobolev spaces, L p( ) and W,p( ), where p is a real-valued function. In order to discuss problem (P), we need some results about the space W,p( ) 0 (), which we call the variable exponent Sobolev space. We first state some basic properties of W,p( ) 0 () (for details, see [2], [7], [5], [25], [35] and [38]). Denote C + () = {h h C(), h(x) > for x }, h + = max h(x), h = min h(x), for any h C(), { L p( ) () = u u is a measurable real-valued function, We introduce the norm on L p( ) () by { u p( ) = inf λ > 0 u(x) λ p(x) } dx. } u(x) p(x) dx <. Then (L p( ) (), p( ) ) becomes a Banach space and it is called the variable exponent Lebesgue space. Proposition 2. (see [2], [35]). (a) The space (L p( ) (), p( ) ) is a separable, uniform convex Banach space, and its conjugate space is L q( ) (), where /q( ) + /p( ). For any u L p( ) () and v L q( ) (), we have ( uv dx p + ) q u p( ) v q( ). (b) If p, p 2 C + (), p (x) p 2 (x) for any x, then L p2( ) () L p( ) (), and this imbedding is continuous. Proposition 2.2 (see [5], [35]). If f : R R is a Carathéodory function and satisfies f(x, s) a(x) + b s p(x)/p2(x) for any x, s R, where p, p 2 C + (), a L p2( ) (), a(x) 0, b 0, then the Nemytskiĭ operator from L p( ) () to L p2( ) () defined by (N f u)(x) = f(x, u(x)), is a continuous and bounded operator.

9 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 6 Proposition 2.3 (see [5], [35]). If we denote ρ(u) = u p(x) dx, for all u L p( ) (), then there exists ξ such that u p(ξ) p( ) = u p(x) dx and (a) u p( ) < (= ; > ) if and only if ρ(u) < (= ; > ); (b) if u p( ) > then u p p( ) ρ(u) u p+ p( ) ; if u p( ) < then u p p( ) ρ(u) u p+ p( ) ; (c) u p( ) 0 if and only if ρ(u) 0; u p( ) if and only if ρ(u). Proposition 2.4 (see [5], [35]). If u, u n L p( ) (), n =, 2,..., then the following statements are equivalent: (a) lim u k u p( ) = 0; k (b) lim ρ(u k u) = 0; k (c) u k u in measure in and lim ρ(u k) = ρ(u). k The space W,p( ) () is defined by W,p( ) () = {u L p( ) () u (L p( ) ()) N }, and it can be equipped with the norm u = u p( ) + u p( ), for all u W,p( ) (). We denote by W,p( ) 0 () the closure of C0 () in W,p( ) () and set Np(x) if p(x) < N, p (x) = N p(x) if p(x) N. Then we have the following properties. Proposition 2.5 (see [2], [5], [35]). (a) W,p( ) () and W,p( ) 0 () are separable reflexive Banach spaces; (b) if q C + () and q(x) < p (x) for any x, then the imbedding from W,p( ) () to L q( ) () is compact; (c) there is a constant C > 0 such that u p( ) C u p( ), for all u W,p( ) 0 (). It follows from (a) of Proposition 2.5 that u p( ) and u are equivalent norms on W,p( ) 0 (). From now on, we will use u p( ) instead of u as the norm on W,p( ) 0 (). The Lebesgue and Sobolev spaces with variable exponents coincide with the usual Lebesgue and Sobolev spaces provided that p is constant. These function

10 62 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang spaces L p(x) and W,p(x) have some unusual properties, see [35, p. 8 9]. Some of these properties are the following: (i) Assuming that < p p + < and p: [, ) is a smooth function, the following co-area formula u(x) p dx = p t p {x ; u(x) > t} dt 0 has no analogue in the framework of variable exponents. (ii) Spaces L p(x) do not satisfy the mean continuity property. More exactly, if p is nonconstant and continuous in an open ball B, then there is some u L p(x) (B) such that u(x + h) L p(x) (B) for every h R N with arbitrary small norm. (iii) Function spaces with variable exponent are never invariant with respect to translations. The convolution is also limited. For instance, the classical Young inequality holds if and only if p is constant. f g p(x) C f p(x) g L Proposition 2.6 (see [6]). If the assumption (p ) is satisfied, then λ p( ) defined in (.) is positive. Next, we prove some results related to the p(x)-laplace operator p(x) as defined at the beginning of Section. Consider the following functional: J(u) = p(x) u p(x) dx, u X := W,p( ) 0 (). Then (see [9]) J C (X, R) and the p(x)-laplace operator is the derivative operator of J in the weak sense. We denote L = J : X X, then (L(u), v) = u p(x) 2 u v dx, for all v, u X. Theorem 2.7 (see [5], [2]). (a) L: X X is a continuous, bounded and strictly monotone operator; (b) L is a mapping of type (S + ), that is, if u n u in X and lim (L(u n) n + L(u), u n u) 0, then u n u in X; (c) L: X X is a homeomorphism. Denote B(x 0, ε, δ, θ) = { x R N δ x x 0 ε, where θ (0, π/2). Then we obtain the following. x x 0 x x 0 } p(x 0 ) p(x 0 ) cos θ,

11 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 63 Lemma 2.8. If p C (), x 0 satisfy p(x 0 ) 0, then there exists a small enough ε > 0 such that (2.) (2.2) (x x 0 ) p(x) > 0, for all x B(x 0, ε, δ, θ), max{p(x) x B(x 0, ε)} = max{p(x) x B(x 0, ε, δ, θ), x x 0 = ε}. Proof. A proof of this lemma can be found in [45]. For readers convenience, we include it here. Since p C (), for any x B(x 0, ε, δ, θ), when ε > 0 is small enough, we have p(x) (x x 0 ) = ( p(x 0 ) + o()) (x x 0 ) = p(x 0 ) (x x 0 ) + o( x x 0 ) p(x 0 ) x x 0 cos θ + o( x x 0 ) > 0, where o() R N is a function and o() 0 uniformly as x x 0 0. When ε is small enough, condition (2.) is valid. Since p C (), there exists a small enough positive ε such that p(x) p(x 0 ) = p(y) (x x 0 ) = ( p(x 0 ) + o()) (x x 0 ), where y = x 0 + τ(x x 0 ) and τ (0, ), o() R N is a function and o() 0 uniformly as x x 0 0. Suppose that x B(x 0, ε) \ B(x 0, ε, δ, θ). Let x = x 0 + ε p(x 0 )/ p(x 0 ). Suppose that x x 0 x x 0 p(x 0 ) < cos θ. p(x 0 ) When ε is small enough, we have p(x) p(x 0 ) = ( p(x 0 ) + o()) (x x 0 ) < p(x 0 ) x x 0 cos θ + ε o() ( p(x 0 ) + o()) ε p(x 0 )/ p(x 0 ) = p(x ) p(x 0 ), where o() R N is a function and o() 0 as ε 0. Suppose that x x 0 < δ. When ε is small enough, we have p(x) p(x 0 ) = ( p(x 0 ) + o()) (x x 0 ) p(x 0 ) x x 0 + ε o() < ( p(x 0 ) + o()) ε p(x 0 )/ p(x 0 ) = p(x ) p(x 0 ), where o() R N is a function and o() 0 as ε 0. Thus (2.3) max{p(x) x B(x 0, ε)} = max{p(x) x B(x 0, ε, δ, θ)}. It follows from (2.) and (2.3) that relation (2.2) holds. Lemma 2.9. Suppose that F (x, u) satisfies (f 4 ). Let 0 if x x 0 > ε, h(x) = ε x x 0 if x x 0 ε,

12 64 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang where ε is defined as in Lemma 2.8. Then th p(x) dx F (x, th) dx Proof. Obviously, p(x) th p(x) dx C 2 B(x 0,ε,δ,θ) as t +. th p(x) dx. We make a spherical coordinate transformation. Denote r = x x 0. Since p C (), it follows from (2.) that there exist positive constants c and c 2 such that p(ε, ω) c 2 (ε r) p(r, ω) p(ε, ω) c (ε r), for all (r, ω) B(x 0, ε, δ, θ). Therefore (2.4) Denote Then B(x 0,ε,δ,θ) th p(x) dx = G(x, u) = B(x 0,ε,δ,θ) B(x 0,ε,δ,θ) ε N ε N t p(r,ω) r N dr dω t p(ε,ω) c(ε r) r N dr dω B(x 0,ε,δ,θ) B(x 0,,,θ) F (x, u) u p(x) [ln(e + u )] p(x). t p(ε,ω) c(ε r) dr dω t p(ε,ω) c ln t dω. (2.5) G(x, u) + uniformly as u + for x. Thus there exists a positive constant M such that G(x, u), for all u M and for all x. Denote E = {x B(x 0, ε) th M} = {x B(x 0, ε) x x 0 ε M/t}, E 2 = B(x 0, ε) \ E. Then we have F (x, th) dx = B(x 0,ε) F (x, th) dx = F (x, th) dx + E F (x, th) dx E 2 F (x, th) dx C. E

13 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 65 When t is large enough, we have F (x, th) dx = th p(x) [ln(e + th )] p(x) G(x, th) dx E E = C th p(x) [ln(e + th )] p(x) G(x, th) dx = B(x 0,ε M/t,δ,θ) B(x 0,ε M/t,δ,θ) C t(ε r) p(r,ω) r N [ln(e + t(ε r) )] p(r,ω) G(r, ω, t(ε r)) dr dω C δ N t p(ε,ω) c2(ε r) ε r p(ε,ω) c(ε r) B(x 0,ε M/t,δ,θ) [ln(e + t(ε r) )] p(r,ω) G(r, ω, t(ε r)) dr dω ε M/t = C δ N dω t p(ε,ω) c2(ε r) ε r p(ε,ω) c(ε r) B(x 0,,,θ) δ [ln(e + t(ε r) )] p(r,ω) G(r, ω, t(ε r)) dr ε /ln t C δ N dω t p(ε,ω) c2(ε r) ε r p(ε,ω) B(x 0,,,θ) [ln(e + t(ε r) )] p(r,ω) G(r, ω, t(ε r)) dr ( C 2 δ N G(r t, ω t, t(ε r t )) ln t ε /ln t δ t p(ε,ω) c2(ε r) dr dω C 3 δ N G(r t, ω t, t(ε r t )) C 4 δ N G(r t, ω t, t(ε r t )) where (r t, ω t ) E is such that δ B(x 0,,,θ) B(x 0,,,θ) B(x 0,,,θ) { G(r t, ω t, t(ε r t )) = min G(r, ω, t(ε r)) ) p(ε,ω) [ ln t p(ε,ω) c2/ln t c 2 ln t t p(ε,ω) c 2 ln t dω, Note that t(ε r t ) t/ln t + as t +. Thus ( e + dω ( (r, ω) B x 0, ε t )] p(ε,ω) ln t )} ln t, δ, θ. (2.6) F (x, th) dx G(r t, ω t, t(ε r t ))C 5 B(x 0,,,θ) t p(ε,ω) ln t dω C as t +. It follows from (2.4), (2.5) and (2.6) that Ψ(th).

14 66 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang Lemma 2.0. The following K i (i =, 2, 3) satisfy hypothesis (K) K (t) = ln(e + t ); K 2 (t) = ln(e + ln(e + t )); K 3 (t) = [ln(e + ln(e + t ))] ln(e + t ). Proof. We only need to check that K 3 (t) satisfies hypothesis (K). The proofs for the other functions are similar. We observe that K( ) C ([0, + ), [, + )) is increasing and K(t) + as t +. So we only need to prove that tk (t)/k(t) σ (0, ), where σ is a constant. By computation we obtain We have tk K = t K = { [ln(e + t )]sgn t [ln(e + ln(e + t ))]sgn t + [e + ln(e + t )](e + t ) (e + t ) t [ln(e + ln(e + t ))][e + ln(e + t )](e + t ) t + [ln(e + t )](e + t ). t [ln(e + ln(e + t ))][e + ln(e + t )](e + t ), 3 t [ln(e + t )](e + t ) 2 } and we complete the proof by observing that tk /K 5/6, for all t R. 3. Proofs of main results In this section we give the proofs of our main results. Definition 3.. We say that u W,p( ) 0 () is a weak solution of (P) if u p(x) 2 u v dx = f(x, u)vd x, for all v X := W,p( ) 0 (). The corresponding functional of (P) is ϕ(u) = p(x) u p(x) dx where F (x, t) = t f(x, s) ds. 0 F (x, u) dx, u X, Definition 3.2. We say that ϕ satisfies the Cerami condition in X, if any sequence {u n } X such that {ϕ(u n )} is bounded and ϕ (u n ) ( + u n ) 0 as n + has a convergent subsequence. Lemma 3.3. If f satisfies (f 0 ) and (f ), then ϕ satisfies the Cerami condition.

15 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 67 Proof. Let {u n } X be a Cerami sequence, that is ϕ(u n ) c and ϕ (u n ) ( + u n ) 0. Therefore ϕ (u n ) = L(u n ) f(x, u n ) 0 in X, so we have L(u n ) = f(x, u n ) + o n (), where o n () 0 in X as n. Suppose that {u n } is bounded. Then {u n } has a weakly convergent subsequence in X. Without loss of generality, we may assume that u n u. Then by Propositions 2.2 and 2.5, we have f(x, u n ) f(x, u) in X. Thus L(u n ) = f(x, u n ) + o n () f(x, u) in X. Since L is a homeomorphism, we have u n L (f(x, u)) in X, and so ϕ satisfies the Cerami condition. Therefore u = L (f(x, u)), so L(u) = f(x, u), which means that u is a solution of (P). Thus we only need to prove the boundedness of the Cerami sequence {u n }. We argue by contradiction. Then there exist c R and {u n } X satisfying: ϕ(u n ) c, ϕ (u n ) ( + u n ) 0, u n +. Obviously, p(x) u n p( ) p u n p( ), p(x) u n p( ) p u n p( ) + C u n p( ). Thus u n /p(x) C u n. Therefore (ϕ (u n ), u n /p(x)) 0. We may assume that ( ) c+ ϕ(u n ) ϕ (u n ), p(x) u n { = p(x) u n p(x) dx F (x, u n ) dx p(x) u n p(x) dx } p(x) f(x, u n)u n dx p 2 (x) u n u n p(x) 2 u n p dx { } p 2 (x) u n u n p(x) 2 u n p dx + p(x) f(x, u n)u n F (x, u n ) dx. Hence (3.) { f(x, un )u n p(x) σ } ( F (x, u n ) dx C 0 u n p(x) K( u n ) dx + C + C(σ) u n u n p(x) dx + ) u n p(x) [K( u n )] p(x) dx, where σ is a small enough positive constant. Due to hypothesis (K), it is easy to check that u n /K( u n ) X, and u n /K( u n ) C 2 u n. Let u n /K( u n ) be a test function. We have u n f(x, u n ) K( u n ) dx = = u n p(x) K( u n ) dx u n p(x) 2 u n u n u n p(x) 2 u n u n dx + o() K u n ) dx + o(). K( u n )

16 68 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang By computation, we obtain u n u n p(x) 2 u n K( u n ) dx u n u n p(x) K( u n ) K 2 ( u n ) u n p(x) K( u n ) Note that u n K ( u n )/K( u n ) σ 0 (0, ). Thus u n p(x) (3.2) C 3 K( u n ) dx C f(x, u n )u n 4 dx C 5 K( u n ) u n K u n ) K( u n ) dx. dx u n p(x) K( u n ) dx + C 6. By (3.), (3.2) and conditions (f 0 ) and (f ), we have { } u (f) n f(x, un )u n f(x, u n ) dx C 7 F (x, u n ) dx + C 7 K( u n ) p(x) u n C 7 {σ p(x) } K( u n ) dx + C 8 + C(σ) u n p(x) [K( u n )] p(x) dx C 7 σ u n p(x) K( u n ) dx + C 7C(σ) (3.2) f(x, u n )u n dx + C 7 C(σ) 2 K( u n ) u n p(x) [K( u n )] p(x) dx + C 9 u n p(x) [K( u n )] p(x) dx + C 0. Thus, by condition (f ) and the above inequality, we can see that u n (3.3) f(x, u n ) K( u n ) dx C u n p(x) [K( u n )] p(x) dx + C 2. Note that tf(x, t)/( t p(x) [K(t)] p(x) ) + uniformly as t + for x. We claim that u n p(x) [K( u n )] p(x) dx is bounded. This means that u n f(x, u n ) K( u n ) dx is bounded. In fact, by (K), we observe that there exists M > 0 large enough such that (3.4) tf(x, t) K(t) > 2C t p(x) [K(t)] p(x), for all t M. Denote n = {x u n M}. We have u n (3.5) f(x, u n ) K( u n ) dx 2C u n p(x) [K( u n )] p(x) dx C 2. n Combining (3.3) (3.5), we obtain and hence n C u n p(x) [K( u n )] p(x) dx C 3, C u n p(x) [K( u n )] p(x) dx C 4.

17 Thus Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 69 f(x, u n ) This combined with (f 0 ) implies that { f(x, u n )u n (3.6) K( u n ) u n K( u n ) dx C 4, for any n =, 2,... } dx is bounded. Let ε > 0 satisfy ε < min{, p, /p +, (p /α) }. Since ϕ (u n ) u n 0, we get u n p(x) dx = f(x, u n )u n dx + o() ε [ ] ε f(x, f(x, u n )u n ε [K( u n )] ε un )u n dx + o(). K( u n ) By condition (f ), we have f(x, u n )u n u n p(x) for large enough vertu n, and [K( u n )] ε [ln(e + u n )] 2( ε) for large enough u n, so we have Therefore u n p(x) dx = f(x, u n )u n ε [K( u n )] ε C 5 ( f(x, u n )u n ε(+ε) + ). C 5 ( + u n ) +ε By Young s inequality, we have (3.7) u n p(x) dx f(x, u n )u n dx + o() C 5 ( + u n ) +ε [ f(x, un )u n +ε + ( + u n ) (+ε)/ε According to the definition of ε, we have and Therefore f(x, u n )u n +ε + dx ( + u n ) (+ε)/ε Thus, the sequence ] ε [ ] ε f(x, un )u n dx + o(). K( u n ) f(x, u n )u n +ε + + f(x, u n)u n dx + o(). ( + u n ) (+ε)/ε K( u n ) f(x, u n )u n +ε + C( u n p (x) + ) ( + u n ) (+ε)/ε ( + u n ) (+ε)(p ) +. { C( u n p (x) + ) dx ( + u n )(+ε)/ε C( u n (p ) + + ) ( + u n ) C #( u n (p ) + + ) (+ε)/ε ( + u n ). (+ε)/ε f(x, u n )u n +ε } + dx ( + u n ) (+ε)/ε

18 70 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang is bounded. This combined with (3.6) and (3.7) implies u n p(x) dx C 6 ( + u n ) +ε + C 7. Note that ε < p. This is a contradiction, hence {u n } is bounded in X, as claimed. Proof of Theorem.. We first establish the existence of a nontrivial weak solution. We show that ϕ satisfies conditions of the mountain pass lemma. By Lemma 3.3, ϕ satisfies the Cerami condition. Since p(x) < α(x) < p (x), the embedding X L α( ) () is compact. Hence there exists C 0 > 0 such that u p( ) C 0 u, for all u X. Let σ > 0 be small enough such that σ 4 λ p( ). By assumptions (f 0 ) and (f 2 ), we obtain F (x, t) σ p(x) t p(x) + C(σ) t α(x), for all (x, t) R. By (p ) and Lemma 2.6, we have λ p( ) > 0 and p(x) u p(x) dx σ p(x) u p(x) dx 3 4 p(x) u p(x). Since α C() and p(x) < α(x) < p (x), we can divide the domain into n 0 disjoint small subdomains i (i =,..., n 0 ) such that = n0 i and Let ε = that is { min i n 0 inf i sup i i p(x) p(x) < inf i α(x) sup p(x) i u u i α(x) sup i } p(x) i= α(x) < inf p (x). i and denote by u i the norm of u on i, dx + i p(x) u u i Then u i C u and there exist ξ i, η i i such that u α(ξi) α( ) = u α(x) dx, i u p(ηi) i = i p(x) ( p(x) u p(x) + p(x) u p(x) When u is small enough, we have n 0 C(σ) u α(x) dx = C(σ) u α(x) dx i i= n 0 = C(σ) i= dx =. ) dx. u α(ξi) α( ) (where ξ i i )

19 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 7 Thus ϕ(u) n 0 C u α(ξi) i (by Proposition 2.5) i= n 0 C u ε u p(ηi) i (where η i i ) i= n 0 = C u ε = C u ε 4 i= i ( p(x) u p(x) + ) p(x) u p(x) dx ( p(x) u p(x) + p(x) u p(x) p(x) u p(x) dx. p(x) u p(x) σ p(x) u p(x) dx C(σ) 2 u α(x) dx ) dx p(x) u p(x) when u is small enough. Therefore, there exist r > 0 and δ > 0 such that ϕ(u) δ > 0 for every u X and u = r. Suppose (p 2 ) is satisfied. Define h C 0 (B(x 0, 3δ)) as follows: 0 if x x 0 3δ, h(x) = 3δ x x 0 if 2δ x x 0 < 3δ, δ if x x 0 < 2δ. Note that min p(x) > max p(x). It is now easy to check that x x 0 δ 2δ x x 0 3δ ϕ(th) = p(x) th p(x) F (x, th) dx p(x) th p(x) C th p(x) dx + C 2 B(x 0,3δ)\(B(x 0,2δ)) (B(x 0,δ)) as t +. Since ϕ(0) = 0, the functional ϕ satisfies the conditions of the mountain pass lemma. So ϕ admits at least one nontrivial critical point, which implies that problem (P) has a nontrivial weak solution u. Suppose (f 4 ) is satisfied. We may assume that there exists x 0 such that p(x 0 ) 0. Define h C 0 (B(x 0, ε)) as follows: 0 if x x 0 ε, h(x) = ε x x 0 if x x 0 < ε. By (f 4 ) and Lemma 2.9, there exists ε > 0 small enough such that ϕ(th) = p(x) th p(x) F (x, th) dx as t +.

20 72 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang Since functional ϕ(0) = 0, ϕ satisfies the conditions of the mountain pass lemma. So ϕ admits at least one nontrivial critical point, which implies that problem (P) has a nontrivial weak solution u. In order to prove Theorem.2, we need to make some preparations. Note that X := W,p( ) 0 () is a reflexive and separable Banach space (see [47], Section 7, Theorems 2 and 3). Therefore there exist {e j } X and {e j } X such that and X = span{e j, j =, 2,...}, X = span W {e j, j =, 2,...}, For convenience, we write X j = span{e j }, Y k = e if i = j, j, e j = 0 if i j. k X j and Z k = X j. j= Lemma 3.4. Assume that α C + (), α(x) < p (x) for any x. If then lim k β k = 0. β k = sup{ u α( ) u =, u Z k }, Proof. Obviously, 0 < β k+ β k, so β k β 0. Let u k Z k satisfy u k =, 0 β k u k α( ) < k. Then there exists a subsequence of {u k } (which we still denote by u k ) such that u k u, and e j, u = lim k e j, u k = 0, for all e j. This implies that u = 0, and so u k 0. Since the embedding from W,p( ) 0 () into L α( ) () is compact, we can conclude that u k 0 in L α( ) (). Hence we get β k 0 as k. In order to prove Theorem.2, we need the following auxiliary result, see [50, Theorem 4.7]. If the Cerami condition is replaced by the PS condition, we can use the following property, see [9, Theorem 3.6]. Lemma 3.5. Suppose that ϕ C (X, R) is even and satisfies the Cerami condition. Let V +, V X be closed subspaces of X with codim V + + = dim V. Suppose that: () ϕ(0) = 0; (2) there exist τ > 0, γ > 0 such that, for all u V +, if u = γ then ϕ(u) τ; and (3) there exists ρ > 0 such that, for all u V, if u ρ then ϕ(u) 0. j=k

21 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 73 Consider the set: Then Γ = { g C 0 (X, X) g is odd, g(u) = u if u V and u ρ }. (a) for all δ > 0, g Γ, S + δ g(v ), and it satisfies S + δ = {u V + u = δ}; and (b) the number ϖ := inf sup ϕ(g(u)) τ > 0 is a critical value for ϕ. g Γ u V Proof of Theorem.2. We first establish the existence of infinitely many pairs of weak solutions. According to (f 0 ), (f ) and (f 3 ), the functional ϕ is an even functional and it satisfies the Cerami condition. Let V + k = Z k be a closed linear subspace of X and V + k Y k = X. Suppose that (f 4 ) is satisfied. We may assume that there exists x n such that p(x n ) 0. Define h n C 0 (B(x n, ε n )) by 0 if x x n ε n, h n (x) = ε n x x n if x x n < ε n. Without loss of generality, we may assume that supp h i supp h j =, for all i j. By Lemma 2.9, we can let ε n > 0 be small enough, so that ϕ(th n ) = p(x) th n p(x) F (x, th n ) dx as t +. Suppose that (p 3 ) is satisfied. Define h n C 0 (B(x n, ε n )) by 0 if x x n 3δ n, h n (x) = 3δ n x x n if 2δ n x x n < 3δ n, δ n if x x n < 2δ n. Note that min p(x) > max p(x). It follows that x x n δ n 2δ n x x n 3δ n ϕ(th n ) = p(x) th n p(x) F (x, th n ) dx 2δ n x x n 3δ n p(x) th n p(x) C th n p(x) dx + C 2 x x n δ n as t +. Set V k = span{h,..., h k }. We will prove that there exist infinitely many pairs of V + k and V k, such that ϕ satisfies the conditions of Lemma 3.5 and the corresponding critical value satisfies ϖ k := inf sup ϕ(g(u)) + g Γ u V k

22 74 G. Li V.D. Rădulescu D.D. Repovš Q. Zhang when k +. This shows that there are infinitely many pairs of solutions of problem (P). For any m =, 2,..., we will prove that there exist ρ m > γ m > 0 and large enough k m such that (A ) b km := inf {ϕ(u) u V + k m, u = γ m } + (m + ); and (A 2 ) a km := max{ϕ(u) u V k m, u = ρ m } 0. First, we prove (A ) as follows. By computation, for any u Z km with u = γ m = m, we have ϕ(u) = p(x) u p(x) dx F (x, u) dx p + u p(x) dx C u α(x) dx C u dx p + u p C u α(ξ) α( ) C 2 u α( ) (where ξ ) p + u p Cβk α m u α C 2 β km u if u α( ), p + u p Cβk α+ m u α+ C 2 β km u if u α( ) >, p + u p Cβk α m ( u α+ + ) C 2 β km u. Obviously, there exists a large enough k m such that, for all u Z km with u = γ m = m, p + u p Cβk α m ( u α+ + ) C 2 β km u 2p + u p. Therefore ϕ(u) u p /2p +, for all u Z km with u = γ m = m. Hence b km + as m. Next, we give a proof of (A 2 ). According to the above discussion, it is easy to see that Ψ(th km ) as t +. We conclude that, for all h V k m = span{h,..., h km } with h =, ϕ(th) as t +. References [] E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 56 (200), [2] E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 64 (2002), [3] C.O. Alves and S.B. Liu, On superlinear p(x)-laplacian equations in R N, Nonlinear Anal. 73 (200), [4] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006),

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25 Nonhomogeneous Problems without the Ambrosetti Rabinowitz Condition 77 [48] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 29 (987), [49] V.V. Zhikov, Existence theorem for some pairs of coupled elliptic equations, Dokl. Math. 77 (2008), [50] C.K. Zhong, X.L. Fan and W.Y. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 998. Manuscript received December 5, 206 accepted June 2, 207 Gang Li and Qihu Zhang College of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou, Henan , P.R. CHINA address: Vicenţiu D. Rădulescu Department of Mathematics Faculty of Sciences King Abdulaziz University P.O. Box Jeddah 2589, SAUDI ARABIA and Department of Mathematics University of Craiova Craiova, ROMANIA address: Dušan D. Repovš Faculty of Education and Faculty of Mathematics and Physics University of Ljubljana 000 Ljubljana, SLOVENIA address: TMNA : Volume N o