Existence of an unbounded branch of the set of solutions for Neumann problems involving the $p(x)$-Laplacian

In recent years, there has been much interest in studying differential equations and variational problems involving $p(x)$-growth conditions since they can model physical phenomena which arise in the study of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [1–5] and references therein. In the case of $p(x)$ a constant, called the p-Laplacian, there are a lot of papers, for instance, [6–14] and references therein.

where Ω is a bounded domain in ${\mathbb{R}}^N$ with the Lipschitz boundary ∂ Ω, $\frac{\partial u}{\partial n}$ denotes the outer normal derivative of u with respect to ∂ Ω, the variable exponent $p:\overline{\mathrm{\Omega }}\to (1,\mathrm{\infty })$ is a continuous function, $g\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, w is a weighted function in Ω and $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ satisfies a Carathéodory condition.

Since the inceptive study of bifurcation theory by Krasnoselskii [15], Rabinowitz [16] claimed that the bifurcation occurring in the Krasnoselskii theorem is actually a global phenomenon. As regards the p-Laplacian and generalized operators, the nonlinear eigenvalue and bifurcation problems have been widely studied by many researchers in various approaches in the spirit of Rabinowitz [16]; see also [6–9, 13, 17].

which is based on the fact [19] that the first eigenvalue of the p-Laplacian is simple and isolated under suitable conditions on m.

While many researchers considered global branches bifurcating from the first eigenvalue of the p-Laplacian, Väth [20] came at it from another viewpoint to establish the existence of a global branch of solutions for the p-Laplacian with Dirichlet boundary condition by applying nonlinear spectral theory for homogeneous operators. From this point of view, for the case that $p(x)$ is a constant function, the existence of a global branch of solutions for the problem (B) was attained in [11] (for generalization to equations involving nonhomogeneous operators, see also [12]) when μ is not eigenvalue of (E).

Compared to the p-Laplacian equation, an analysis for the $p(x)$-Laplacian equation has to be carried out more carefully because it has complicated nonlinearities (it is nonhomogeneous) and includes a weighted function. As mentioned before, the fact that the principal eigenvalue for nonlinear eigenvalue problems related to the p-Laplacian under either Dirichlet boundary condition or Neumann boundary condition is isolated plays a key role in obtaining the bifurcation result from the principal eigenvalue of the p-Laplacian. However, unlike the p-Laplacian case, under some conditions on $p(x)$, the first eigenvalue for the $p(x)$-Laplacian Neumann problems is not isolated (see [21]), that is, the infimum of all eigenvalues of the problem might be zero (see [22] for Dirichlet boundary condition). Thus we cannot investigate the existence of global branches bifurcating from the principal eigenvalue of the $p(x)$-Laplacian. For this reason, the global behavior of solutions for nonlinear problems involving the $p(x)$-Laplacian had been considered in [23]. To the best of our knowledge, there are no papers concerned with the bifurcation theory for the $p(x)$-Laplacian Neumann problems with weighted functions.

This paper is organized as follows. We first state some basic results for the weighted variable exponent Lebesgue-Sobolev spaces which were given in [23]. Next we give some properties of the corresponding integral operators. Finally we show the existence of a global bifurcation for a Neumann problem involving the $p(x)$-Laplacian by using a bifurcation result in an abstract setting.

In this section, we state some elementary properties for the (weighted) variable exponent Lebesgue-Sobolev spaces which will be used in the next sections. The basic properties of the variable exponent Lebesgue-Sobolev spaces, that is, when $w(x)\equiv 1$ can be found from [24].

To make a self-contained paper, we recall some definitions and basic properties of the weighted variable exponent Lebesgue spaces $L^{p(x)}(w,\mathrm{\Omega })$ and the weighted variable exponent Lebesgue-Sobolev spaces $W^{1,p(x)}(w,\mathrm{\Omega })$.

for every $x,y\in \mathrm{\Omega }$ with $|x-y|\le 1/2$, then smooth functions are dense in variable exponent Sobolev spaces and there is no confusion in defining the Sobolev space with zero boundary values, $W_0^{1,p(x)}(\mathrm{\Omega })$, as the completion of $C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ with respect to the norm ${\parallel u\parallel }_{W^{1,p(x)}(\mathrm{\Omega })}$ (see [26]).

Lemma 2.1 ([24])

Lemma 2.2 ([23])

Lemma 2.3 ([27])

We assume that w is a measurable positive and a.e. finite function in Ω satisfying that

(w1) $w\in L_{\mathrm{loc}}^1(\mathrm{\Omega })$ and $w^{-1/(p(x)-1)}\in L_{\mathrm{loc}}^1(\mathrm{\Omega })$;

(w2) $w^{-s(x)}\in L^1(\mathrm{\Omega })$ with $s(x)\in (\frac{N}{p(x)},\mathrm{\infty })\cap [\frac{1}{p(x)-1},\mathrm{\infty })$.

The reasons that we assume (w1) and (w2) can be found in [23].

Lemma 2.4 ([23])

Let $p\in C_{+}(\overline{\mathrm{\Omega }})$ and (w1) hold. Then X is a reflexive and separable Banach space.

for almost all $x\in \mathrm{\Omega }$.

We shall frequently make use of the following (compact) imbedding theorem for the weighted variable exponent Lebesgue-Sobolev space in the next sections.

Lemma 2.5 ([23])

and the imbedding is compact if ${inf}_{x\in \mathrm{\Omega }}(p_s^{\ast }(x)-r(x))>0$.

In this section, we give the definitions and some properties of the integral operators corresponding to the problem (B), by applying the basic properties of the spaces $L^{p(x)}(w,\mathrm{\Omega })$ and X which are given in the previous section.

for any $\phi \in X$ where $〈\cdot ,\cdot 〉$ denotes the pairing of X and its dual $X^{\ast }$ and the Euclidean scalar product on ${\mathbb{R}}^N$, respectively.

The following estimate, which can be found in [17], plays a key role in obtaining the homeomorphism of the operator J.

From Lemma 3.1, we can obtain the following topological result, which will be needed in the main result. Compared to the case of $p(x)$ being constant (see [11]), the following result is hard to prove because it has complicated nonlinearities.

Theorem 3.2 Let (w1) and (w2) be satisfied. The operator $J:X\to X^{\ast }$ is homeomorphism onto $X^{\ast }$ with a bounded inverse.

and the integrands at the right-hand sides in (3.2) and (3.3) are dominated by some integrable functions. Since $u_{n_k}\to u$ in X as $k\to \mathrm{\infty }$, we can deduce that ${|u_{n_k}(x)|}^{p(x)-2}{u}_{n_k}(x)\to {|u(x)|}^{p(x)-2}u(x)$ and $w^{\frac{1}{p^{\prime }(x)}}(x){|\mathrm{\nabla }{u}_{n_k}(x)|}^{p(x)-2}\mathrm{\nabla }{u}_{n_k}(x)\to w^{\frac{1}{p^{\prime }(x)}}(x){|\mathrm{\nabla }u(x)|}^{p(x)-2}\mathrm{\nabla }u(x)$ as $k\to \mathrm{\infty }$ for almost all $x\in \mathrm{\Omega }$. Therefore, the Lebesgue dominated convergence theorem tells us that ${\mathrm{\Psi }}_1(u_{n_k})\to {\mathrm{\Psi }}_1(u)$ in $L^{p^{\prime }(x)}(\mathrm{\Omega })$ and ${\mathrm{\Psi }}_2(u_{n_k})\to {\mathrm{\Psi }}_2(u)$ in $L^{p^{\prime }(x)}(\mathrm{\Omega },{\mathbb{R}}^N)$ as $k\to \mathrm{\infty }$, that is, ${\mathrm{\Psi }}_1$, ${\mathrm{\Psi }}_2$ are continuous on X. Also it is easy to show that these operators are bounded on X.

and the right-hand side in (3.4) converges to zero as $n\to 0$. Therefore the operator J is continuous on X.

as ${\parallel u\parallel }_X\to \mathrm{\infty }$ and therefore the operator J is coercive on X.

By using Lemma 3.1 and (3.5), we find that J is strictly monotone on X. The Browder-Minty theorem hence implies that the inverse operator $J^{-1}:X^{\ast }\to X$ exists and is bounded; see Theorem 26.A in [28].

Since $\{u_n:n\in \mathbb{N}\}$ is bounded in X and $J(u_n)\to J(u)$ in $X^{\ast }$ as $n\to \mathrm{\infty }$, it follows that $(u_n)$ converges to u in X. Thus, $J^{-1}$ is continuous at each $h\in X^{\ast }$. This completes the proof. □