Solutions for parametric double phase Robin problems

Abstract

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is $(p - 1)$ -superlinear and the solutions produced are asymptotically big as $λ \to 0^{+}$ . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as $λ \to 0^{+}$ .

Keywords

Unbalanced growth asymptotically big solutions asymptotically small solutions superlinear reaction

1. Introduction

Let $Ω \subseteq R^{N}$ be a bounded domain with a Lipschitz boundary $\partial Ω$ . In this paper we study the following parametric two phase Robin problem $\begin{matrix} (P_{λ}) & \{\begin{matrix} - div (a (z) | \nabla u |^{p - 2} \nabla u) - Δ_{q} u + ξ (z) | u |^{p - 2} u = λ f (z, u (z)) in Ω, \\ \frac{\partial u}{\partial n_{ϑ}} + β (z) | u |^{p - 2} u = 0 on \partial Ω, 1 < q < p < + \infty . \end{matrix} \end{matrix}$

In this problem $a \in L^{\infty} (Ω)$ with $a (z) > 0$ for a.a. $z \in Ω$ and $Δ_{q}$ denotes the q-Laplace differential operator defined by $\begin{matrix} Δ_{q} u = div (| \nabla u |^{q - 2} \nabla u) for all W^{1, q} (Ω) . \end{matrix}$

The differential operator in problem ( P λ ) is related to the two-phase integral functional $\begin{matrix} u \to \int_{Ω} [a (z) | \nabla u |^{p} + | \nabla u |^{q}] d z . \end{matrix}$

In the integral functional, the integrand is the function $\begin{matrix} ϑ (z, y) = a (z) | y |^{p} + | y |^{q} for all z \in Ω, all y \in R^{N} . \end{matrix}$

Since we do not assume that the coefficient $a (\cdot)$ is bounded away from zero, this integrand exhibits unbalanced growth, namely we have $\begin{matrix} | y |^{q} ⩽ ϑ (z, y) ⩽ c_{0} [1 + | y |^{p}] for some c_{0} > 0, all z \in Ω, all y \in R^{N} . \end{matrix}$

Such functionals were investigated first in the context of problems related to elasticity theory, by Marcellini [10] and Zhikov [20]. Recently the interest for such functional was revived with the remarkable works of Mingione and coworkers (see Baroni–Colombo–Mingione [1], Colombo–Mingione [3,4], De Filippis–Mingione [5]), who proved local regularity results for minimizers of such functionals. A global regularity theory is still elusive and so the tools and techniques used in the study of $(p, q)$ -equations (see, for example, Papageorgiou–Vetro–Vetro [15]) are not applicable in two-phase problems. Even the ambient space changes and it is no longer the Sobolev space $W^{1, p} (Ω)$ , but the Musielak–Orlicz–Sobolev space $W^{1, ϑ} (Ω)$ (see Section 2). In the left hand side of ( P λ ) we also have a potential term $x \to ξ (z) | x |^{p - 2} x$ with $ξ \in L^{\infty} (Ω)$ , $ξ (z) ⩾ 0$ for a.a. $z \in Ω$ . The reaction $λ f (z, x)$ is parametric, with $λ > 0$ being the parameter and $f (z, x)$ is a Carathéodory function (that is, for all $x \in R$ , $z \to f (z, x)$ is measurable and for a.a. $z \in Ω$ , $x \to f (z, x)$ is continuous). We prove two existence theorems and provide information about the asymptotic behavior of the solutions as $λ \to 0^{+}$ . In the first existence theorem we assume that $f (z, \cdot)$ exhibits $(p - 1)$ -superlinear growth near $\pm \infty$ . However, we do not employ the Ambrosetti–Rabinowitz condition (the AR-condition for short), which is common in the literature when dealing with superlinear problems. In this case we show that for the solution $u_{λ}$ , we have $‖ u_{λ} ‖ \to + \infty$ as $λ \to 0^{+}$ . In the second, the hypotheses on $f (z, \cdot)$ , aside from the “subcritical” growth condition, concern only its behavior near zero. In this case we show that $‖ u_{λ} ‖ \to 0^{+}$ as $λ \to 0^{+}$ . In the boundary condition $\frac{\partial u}{\partial n_{ϑ}}$ denotes the conormal derivative of u with respect to the modular function ϑ. We interpret this derivative using the nonlinear Green’s identity (see Papageorgiou–Rǎdulescu–Repovš [11], Corollary 1.5.16, p. 34). When $u \in C^{1} (\overline{Ω})$ , we have $\begin{matrix} \frac{\partial u}{\partial n_{ϑ}} = [a (z) | \nabla u |^{p - 2} + | \nabla u |^{q - 2}] \frac{\partial u}{\partial n}, \end{matrix}$ with $n (\cdot)$ being the outward unit normal on $\partial Ω$ .

We mention that recently existence and multiplicity results for two phase problems were proved by Gasiński–Papageorgiou [6], Ge–Lv–Lu [7], Liu–Dai [9], Papageorgiou–Rădulescu–Repovš [12–14], Papageorgiou–Vetro–Vetro [16]. In the framework of double-phase problems with variable growth we refer to Cencelj–Rădulescu–Repovš [2], Ragusa–Tachikawa [18] and Zhang–Rădulescu [19].

2. Mathematical background – Hypotheses

As we already mentioned in the Introduction, the right function space framework for the analysis of problem ( P λ ) is provided by the so-called Musielak–Orlicz–Sobolev spaces.

We consider the Carathéodory function $\begin{matrix} ϑ (z, x) = a (z) x^{p} + x^{q} for all z \in Ω, all x ⩾ 0 . \end{matrix}$

Then the Musielak–Orlicz space $L^{ϑ} (Ω)$ is defined by $\begin{matrix} L^{ϑ} (Ω) = {u : Ω \to R is measurable and ρ_{ϑ} (u) = \int_{Ω} ϑ (z, | u |) d z < + \infty} . \end{matrix}$

We furnish $L^{ϑ} (Ω)$ with the so-called “Luxemburg norm” defined by $\begin{matrix} ‖ u ‖_{ϑ} = inf [λ > 0 : ρ_{ϑ} (\frac{u}{λ}) ⩽ 1] . \end{matrix}$

Then $L^{ϑ} (Ω)$ becomes a separable, reflexive (in fact uniformly convex) Banach space. Also, we introduce the weighted Lebesgue space $\begin{matrix} L_{a}^{p} (Ω) = {u : Ω \to R is measurable and ‖ u ‖_{a, p} = {[\int_{Ω} a (z) | u |^{p} d z]}^{1 / p} < + \infty} . \end{matrix}$

We know that $\begin{matrix} L^{p} (Ω) ↪ L^{ϑ} (Ω) ↪ L^{q} (Ω) \cap L_{a}^{p} (Ω), \end{matrix}$ and $min {‖ u ‖_{ϑ}^{p}, ‖ u ‖_{ϑ}^{q}} ⩽ ‖ u ‖_{q}^{q} + ‖ u ‖_{a, p}^{p} ⩽ max {‖ u ‖_{ϑ}^{p}, ‖ u ‖_{ϑ}^{q}}$ for all $u \in L^{ϑ} (Ω)$ .

Then, we can define the corresponding Sobolev-type space $W^{1, ϑ} (Ω)$ by setting $\begin{matrix} W^{1, ϑ} (Ω) = {u \in L^{ϑ} (Ω) : | \nabla u | \in L^{ϑ} (Ω)} . \end{matrix}$

We furnish $W^{1, ϑ} (Ω)$ with the norm $\begin{matrix} ‖ u ‖ = ‖ u ‖_{ϑ} + ‖ \nabla u ‖_{ϑ} for all u \in W^{1, ϑ} (Ω) \end{matrix}$ (here $‖ \nabla u ‖_{ϑ} = ‖ | \nabla u | ‖_{ϑ}$ ). Normed this way, the space $W^{1, ϑ} (Ω)$ is separable and reflexive (in fact uniformly convex). We know that $\begin{matrix} W^{1, ϑ} (Ω) ↪ L^{r} (Ω) compactly \end{matrix}$ for every $r \in (1, q^{*})$ with $\begin{matrix} q^{*} = \{\begin{matrix} \frac{N q}{N - q} & if q < N, \\ + \infty & if N ⩽ q \end{matrix} \end{matrix}$ (the critical Sobolev exponent corresponding to q).

On $\partial Ω$ we consider the $(N - 1)$ -dimensional Hausdorff measure (surface measure) $σ (\cdot)$ . Using this measure, we can define in the usual way the boundary Lebesgue spaces $L^{s} (\partial Ω)$ ( $1 ⩽ s ⩽ + \infty$ ). We know that there exists a unique continuous linear map $γ_{0} : W^{1, q} (Ω) \to L^{q} (\partial Ω)$ , known as the “trace map”, such that $\begin{matrix} γ_{0} (u) = u |_{\partial Ω} for all u \in W^{1, q} (Ω) \cap C (\overline{Ω}) . \end{matrix}$

The trace map extends the notion of boundary values to all Sobolev functions. We know that $\begin{matrix} im γ_{0} = W^{\frac{1}{q^{'}}, q} (\partial Ω) (\frac{1}{q} + \frac{1}{q^{'}} = 1) and ker γ_{0} = W_{0}^{1, q} (Ω) . \end{matrix}$ Moreover, the trace map is compact into $L^{s} (\partial Ω)$ for all $s \in [1, \frac{(N - 1) q}{N - q})$ if $q < N$ and into $L^{s} (\partial Ω)$ for all $s ⩾ 1$ if $q ⩾ N$ . In the sequel, for the sake of notational simplicity, we drop the use of the trace map $γ_{0} (\cdot)$ . All restrictions of Sobolev functions on $\partial Ω$ are understood in the sense of traces.

If X is a Banach space and $φ \in C^{1} (X, R)$ , then we say that $φ (\cdot)$ satisfies the “C-condition”, if every sequence ${u_{n}}_{n ⩾ 1} \subseteq X$ such that ${φ (u_{n})}_{n ⩾ 1} \subseteq R$ is bounded and $(1 + ‖ u_{n} ‖_{X}) φ^{'} (u_{n}) \to 0$ in $X^{*}$ as $n \to + \infty$ , admits a strongly convergent subsequence. Also by $K_{φ}$ we denote the critical set of φ, that is, $K_{φ} = {u \in X : φ^{'} (u) = 0}$ .

Let $A : W^{1, ϑ} (Ω) \to W^{1, ϑ} {(Ω)}^{*}$ be the nonlinear map defined by $\begin{matrix} ⟨ A (u), h ⟩ = \int_{Ω} [a (z) | \nabla u |^{p - 2} + | \nabla u |^{q - 2}] {(\nabla u, \nabla h)}_{R^{N}} d z for all u, h \in W^{1, ϑ} (Ω) . \end{matrix}$

This map has the following properties (see Liu–Dai [9], Proposition 3.1).

Proposition 1.
If $a \in L^{\infty} (Ω)$ and $a (z) > 0$ for a.a. $z \in Ω$ , then $A (\cdot)$ is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type ${(S)}_{+}$ (that is, if $u_{n} \overset{w}{\to} u$ in $W^{1, ϑ} (Ω)$ and ${lim sup}_{n \to + \infty} ⟨ A (u_{n}), u_{n} - u ⟩ ⩽ 0$ , then $u_{n} \to u$ in $W^{1, ϑ} (Ω)$ ).

The hypotheses on the data of ( P λ ) are the following:

$a \in L^{\infty} (Ω)$ with $a (z) ⩾ 0$ for a.a. $z \in Ω$ , $ξ \in L^{\infty} (Ω)$ with $ξ (z) ⩾ 0$ for a.a. $z \in Ω$ , $β \in L^{\infty} (\partial Ω)$ with $β (z) ⩾ 0$ for σ-a.a. $z \in \partial Ω$ , $ξ ≢ 0$ or $β ≢ 0$ and $\frac{N p}{N + p - 1} < q$ .
Remark 1.
The last condition in hypotheses $H_{0}$ , which relates the two exponents p and q, implies that $W^{1, ϑ} (Ω) ↪ L^{p} (\partial Ω)$ compactly via the trace map $γ_{0} (\cdot)$ .

$f : Ω \times R \to R$ is a Carathéodory function such that $f (z, 0) = 0$ for a.a. $z \in Ω$ and

$| f (z, x) | ⩽ \hat{a} (z) [1 + | x |^{r - 1}]$ for a.a. $z \in Ω$ , all $x \in R$ , with $\hat{a} \in L^{\infty} (Ω)$ , $p < r < q^{}$ ;

if $F (z, x) = \int_{0}^{x} f (z, s) d s$ , then ${lim}_{x \to \pm \infty} \frac{F (z, x)}{| x |^{p}} = + \infty$ uniformly for a.a. $z \in Ω$ ;

there exists $τ \in ((r - q) max {1, \frac{N}{q}}, q^{})$ with $τ > q$ such that $\begin{matrix} 0 < \hat{η} ⩽ \underset{x \to \pm \infty}{lim inf} \frac{f (z, x) x - p F (z, x)}{| x |^{τ}} uniformly for a.a. z \in Ω; \end{matrix}$

there exist $1 < μ < q$ and $c_{1} > 0$ such that $\begin{matrix} - c_{1} ⩽ \underset{x \to 0}{lim inf} \frac{F (z, x)}{| x |^{μ}} ⩽ \underset{x \to 0}{lim sup} \frac{F (z, x)}{| x |^{μ}} ⩽ c_{1} uniformly for a.a. z \in Ω . \end{matrix}$
Remark 2.
From hypotheses $H_{1}$ (ii), (iii), we have that $\begin{matrix} lim_{x \to \pm \infty} \frac{f (z, x)}{| x |^{p - 2} x} = + \infty uniformly for a.a. z \in Ω . \end{matrix}$

So the reaction $f (z, \cdot)$ is $(p - 1)$ -superlinear. However, this superlinear growth of $f (z, \cdot)$ is not expressed using the AR-condition. Recall that the AR-condition says that there exist $η > p$ and $M > 0$ such that $\begin{array}{l} (1a) & 0 < η F (z, x) ⩽ f (z, x) x for a.a. z \in Ω, all | x | ⩾ M, \\ (1b) & 0 < \underset{Ω}{essinf} F (\cdot, \pm M) . \end{array}$

Integrating (1a) and using (1b), we obtain the following weaker condition $\begin{array}{l} c_{2} | x |^{η} ⩽ F (z, x) for a.a. z \in Ω, all | x | ⩾ M, some c_{2} > 0 \\ \Rightarrow c_{2} | x |^{η} ⩽ f (z, x) x for a.a. z \in Ω, all | x | ⩾ M . \end{array}$

In this paper instead of the AR-condition, we employ hypothesis $H_{1}$ (iii) which is less restrictive and incorporates in our framework superlinear nonlinearities which fail to satisfy the AR-condition. For example consider the following function (for the sake of simplicity we drop the z-dependence) $\begin{matrix} f (x) = \{\begin{matrix} | x |^{μ - 2} x & if | x | ⩽ 1, \\ | x |^{p - 2} x ln | x | + | x |^{s - 2} x & if 1 < | x |, \end{matrix} \end{matrix}$ with $1 < μ < q$ and $1 < s < p$ . The function satisfies hypothesis $H_{1}$ , but fails to satisfy the AR-condition.

Let ${\hat{γ}}_{p} : W^{1, ϑ} (Ω) \to R$ be the $C^{1}$ -functional defined by $\begin{matrix} {\hat{γ}}_{p} (u) = \int_{Ω} a (z) | \nabla u |^{p} d z + \int_{Ω} ξ (z) | u |^{p} d z + \int_{\partial Ω} β (z) | u |^{p} d σ for all u \in W^{1, ϑ} (Ω) . \end{matrix}$
Proposition 2.
If hypotheses $H_{0}$ hold, then $c_{3} ‖ u ‖^{p} ⩽ {\hat{γ}}_{p} (u)$ for some $c_{3} > 0$ , all $u \in W^{1, ϑ} (Ω)$ .
Proof.
We argue by contradiction. So, suppose that the result of the proposition is not true. Then on account of the p-homogeneity of ${\hat{γ}}_{p} (\cdot)$ , we can find ${u_{n}}_{n ⩾ 1} \subseteq W^{1, ϑ} (Ω)$ such that $\begin{matrix} (2) & ‖ u_{n} ‖ = 1 and {\hat{γ}}_{p} (u_{n}) < \frac{1}{n} for all n \in N . \end{matrix}$

We may assume that $\begin{matrix} (3) & u_{n} \overset{w}{\to} u in W^{1, ϑ} (Ω) and u_{n} \to u in L^{p} (Ω) and in L^{p} (\partial Ω) . \end{matrix}$

From (2) and (3) it follows that $\begin{array}{l} \int_{Ω} a (z) | \nabla u |^{p} d z = 0 \\ \Rightarrow | \nabla u (z) | = 0 for a.a. z \in Ω \\ \Rightarrow u \equiv c \in R . \end{array}$

Then from (2) in the limit as $n \to + \infty$ we have $\begin{array}{l} | c |^{p} [\int_{Ω} ξ (z) d z + \int_{\partial Ω} β (z) d σ] = 0 \\ \Rightarrow c = 0 (see hypotheses H_{0}) \\ \Rightarrow u_{n} \to 0 in W^{1, ϑ} (Ω), \end{array}$ which contradicts (2). □

For every $λ > 0$ , let $φ_{λ} : W^{1, ϑ} (Ω) \to R$ be the energy (Euler) functional for problem ( P λ ) defined by $\begin{matrix} φ_{λ} (u) = \frac{1}{p} {\hat{γ}}_{p} (u) + \frac{1}{q} ‖ \nabla u ‖_{q}^{q} - λ \int_{Ω} F (z, u) d z for all u \in W^{1, p} (Ω) . \end{matrix}$

Evidently $φ_{λ} \in C^{1} (W^{1, ϑ} (Ω), R)$ .
3. Asymptotically big solutions

In this section we show that for all $λ > 0$ small problem ( P λ ) has a solution $u_{λ} \in W^{1, ϑ} (Ω)$ such that $‖ u_{λ} ‖ \to + \infty$ as $λ \to 0^{+}$ .

Proposition 3.
If hypotheses $H_{0}$ , $H_{1}$ hold and $λ > 0$ , then the functional $φ_{λ} (\cdot)$ satisfies the C-condition.
Proof.
We consider a sequence ${u_{n}}_{n ⩾ 1} \subseteq W^{1, ϑ} (Ω)$ such that $\begin{array}{l} (4) & | φ_{λ} (u_{n}) | ⩽ c_{4} for some c_{4} > 0, all n \in N, \\ (5) & (1 + ‖ u_{n} ‖) φ_{λ}^{'} (u_{n}) \to 0 in W^{1, ϑ} {(Ω)}^{} as n \to + \infty . \end{array}$

From (5) we have $\begin{array}{l} | ⟨ A (u_{n}), h ⟩ + \int_{Ω} ξ (z) | u_{n} |^{p - 2} u_{n} h d z + \int_{\partial Ω} β (z) | u_{n} |^{p - 2} u_{n} h d σ - λ \int_{Ω} f (z, u_{n}) h d z | \\ (6) & ⩽ \frac{ε_{n} ‖ h ‖}{1 + ‖ u_{n} ‖} for all h \in W^{1, ϑ} (Ω), with ε_{n} \to 0^{+} . \end{array}$

In (6) we choose $h = u_{n} \in W^{1, ϑ} (Ω)$ and obtain $\begin{matrix} (7) & - {\hat{γ}}_{p} (u_{n}) - ‖ \nabla u_{n} ‖_{q}^{q} + λ \int_{Ω} f (z, u_{n}) u_{n} d z ⩽ ε_{n} for all n \in N . \end{matrix}$ Also from (4) we have $\begin{matrix} (8) & {\hat{γ}}_{p} (u_{n}) + \frac{p}{q} ‖ \nabla u_{n} ‖_{q}^{q} - λ \int_{Ω} p F (z, u_{n}) d z ⩽ p c_{4} for all n \in N . \end{matrix}$

We add (7) and (8) and recall that $q < p$ . Then $\begin{matrix} (9) & λ \int_{Ω} [f (z, u_{n}) u_{n} - p F (z, u_{n})] d z ⩽ c_{5} for some c_{5} > 0, all n \in N . \end{matrix}$

Hypotheses $H_{1}$ (i), (iii) imply that $\begin{matrix} (10) & c_{6} | x |^{τ} - c_{7} ⩽ f (z, x) x - p F (z, x) for a.a. z \in Ω, all x \in R, some c_{6}, c_{7} > 0 . \end{matrix}$

We use (10) in (9) and obtain $\begin{array}{l} ‖ u_{n} ‖_{τ}^{τ} ⩽ c_{8} for some c_{8} > 0, all n \in N \\ (11) & \Rightarrow {u_{n}}_{n ⩾ 1} \subseteq L^{τ} (Ω) is bounded . \end{array}$

First assume that $q < N$ . From hypothesis $H_{1}$ (iii) it is clear that we may assume that $τ < r < q^{}$ . Let $t \in (0, 1)$ be such that $\begin{matrix} (12) & \frac{1}{r} = \frac{1 - t}{τ} + \frac{t}{q^{}} . \end{matrix}$

Using the interpolation inequality (see Papageorgiou–Winkert [17], Proposition 2.3.17, p. 116), we have $\begin{array}{l} ‖ u_{n} ‖_{r} ⩽ ‖ u_{n} ‖_{τ}^{1 - t} ‖ u_{n} ‖_{q^{}}^{t} \\ \Rightarrow ‖ u_{n} ‖_{r}^{r} ⩽ c_{9} ‖ u_{n} ‖^{t r} for some c_{9} > 0, all n \in N \\ (13) & (see (11) and recall that W^{1, ϑ} (Ω) ↪ L^{q^{}} (Ω)) . \end{array}$

From (6) with $h = u_{n} \in W^{1, ϑ} (Ω)$ we obtain $\begin{array}{l} {\hat{γ}}_{p} (u_{n}) + ‖ \nabla u_{n} ‖_{q}^{q} - λ \int_{Ω} f (z, u_{n}) u_{n} d z ⩽ ε_{n} for all n \in N \\ \Rightarrow c_{3} ‖ u_{n} ‖^{p} ⩽ λ \int_{Ω} f (z, u_{n}) u_{n} d z + ε_{n} (see Proposition 2) \\ ⩽ λ c_{10} [1 + ‖ u_{n} ‖^{t r}] + ε_{n} for some c_{10} > 0, all n \in N \\ (14) & (see hypothesis H_{1} (i) and (13)) . \end{array}$

From (12) we have $\begin{array}{l} t = \frac{q^{} (r - τ)}{r (q^{} - τ)} \\ (15) & \Rightarrow t r = \frac{q^{} (r - τ)}{q^{} - τ} . \end{array}$

On account of hypothesis $H_{1}$ (iii) we have $\begin{array}{l} (r - q) \frac{N}{q} < τ (recall that we have assumed that q < N) \\ \Rightarrow N (r - q) < τ q \\ \Rightarrow N r - N τ < N q - N τ + τ q \\ \Rightarrow \frac{N q (r - τ)}{N q - N τ + τ q} < q \\ \Rightarrow \frac{q^{} (r - τ)}{q^{} - τ} < q \\ \Rightarrow t r < q (see (15)) . \end{array}$

Then from (14) and since $q < p$ , we infer that $\begin{matrix} (16) & {u_{n}}_{n ⩾ 1} \subseteq W^{1, ϑ} (Ω) is bounded . \end{matrix}$

Next suppose that $q ⩾ N$ . In this case we know that $q^{} = + \infty$ , while from the Sobolev embedding theorem, we have $\begin{matrix} W^{1, ϑ} (Ω) ↪ W^{1, q} (Ω) ↪ L^{s} (Ω) (for all 1 ⩽ s < + \infty) . \end{matrix}$

So, in the previous argument we need to replace $q^{}$ by $l > r$ .

Then again from (12) we have $\begin{matrix} t r = \frac{l (r - τ)}{l - τ} \to r - τ < q as l \to + \infty (see hypothesis H_{1} (iii)) . \end{matrix}$

So, by choosing $l > r$ big, we will have $\begin{matrix} t r < q < p, \end{matrix}$ hence (16) holds again.

From (16) it follows that we may assume that $\begin{matrix} (17) & u_{n} \overset{w}{\to} u in W^{1, ϑ} (Ω) and u_{n} \to u in L^{p} (Ω) and in L^{p} (\partial Ω) . \end{matrix}$

In (6) we choose $h = u_{n} - u \in W^{1, ϑ} (Ω)$ , pass to the limit as $n \to + \infty$ and use (17). Then $\begin{array}{l} lim_{n \to + \infty} ⟨ A (u_{n}), u_{n} - u ⟩ = 0 \\ \Rightarrow u_{n} \to u in W^{1, ϑ} (Ω) (see Proposition 1) . \end{array}$

We conclude that for every $λ > 0$ the functional $φ_{λ} (\cdot)$ satisfies the C-condition. □
Proposition 4.
If hypotheses* $H_{0}$ , $H_{1}$ hold, then we can find $λ^{} > 0$ such that* $0 < m_{λ} ⩽ φ_{λ} (u)$ for all $‖ u ‖ = ρ_{λ}$ , all $λ \in (0, λ^{})$ .
Proof.
On account of hypotheses $H_{1}$ (i), (iv), we have $\begin{matrix} (18) & | F (z, x) | ⩽ c_{11} [| x |^{μ} + | x |^{r}] for a.a. z \in Ω, all x \in R, some c_{11} > 0 . \end{matrix}$

Then for every $u \in W^{1, ϑ} (Ω)$ we have $\begin{array}{l} φ_{λ} (u) ⩾ \frac{c_{3}}{p} ‖ u ‖^{p} - λ c_{12} [‖ u ‖^{μ} + ‖ u ‖^{r}] for some c_{12} > 0 \\ (19) & (see Proposition 2 and (18)) . \end{array}$

Consider $u \in W^{1, ϑ} (Ω)$ with $‖ u ‖ = ρ_{λ} = λ^{- δ}$ where $0 < δ < \frac{1}{r - p}$ . Then from (19) we have $\begin{array}{l} φ_{λ} (u) & ⩾ \frac{c_{3}}{p} λ^{- δ p} - c_{12} [λ^{1 - δ μ} + λ^{1 - δ r}] \\ (20) & = [\frac{c_{3}}{p} - c_{12} (λ^{1 - δ (μ - p)} + λ^{1 - δ (r - p)})] λ^{- δ p} = m_{λ} . \end{array}$

Note that $\begin{matrix} 0 < 1 - δ (r - p) < 1 - δ (μ - p) . \end{matrix}$

Then we can find $λ^{} > 0$ such that $\begin{matrix} λ^{1 - δ (μ - p)} + λ^{1 - δ (r - p)} < \frac{c_{3}}{c_{12} p} for all λ \in (0, λ^{}) . \end{matrix}$

From (20) we infer that $\begin{matrix} φ_{λ} (u) ⩾ m_{λ} > 0 for all u \in W^{1, ϑ} (Ω) with ‖ u ‖ = ρ_{λ}, all 0 < λ < λ^{} . \end{matrix}$ □
Remark 3.
From the above proof we see that $m_{λ} \to + \infty$ as $λ \to 0^{+}$ (see (20)).

Now we can produce solutions of ( P λ ) which asymptotically as $λ \to 0^{+}$ become arbitrarily big in the $W^{1, ϑ} (Ω)$ -norm. Theorem 1.
If hypotheses $H_{0}$ , $H_{1}$ hold, then we can find $λ^{} > 0$ such that for all* $λ \in (0, λ^{})$ problem (* P λ ) has a nontrivial solution $u_{λ} \in W^{1, ϑ} (Ω)$ and $‖ u_{λ} ‖ \to + \infty$ as $λ \to 0^{+}$ .
Proof.
Let $u \in W^{1, ϑ} (Ω)$ with $u (z) > 0$ for a.a. $z \in Ω$ . Then on account of hypothesis $H_{1} (i i)$ we have $\begin{matrix} (21) & φ_{λ} (t u) \to - \infty as t \to + \infty . \end{matrix}$

Then (21) together with Propositions 3 and 4, permit the use of the mountain pass theorem. So, we can find $u_{λ} \in W^{1, ϑ} (Ω)$ such that $\begin{matrix} (22) & u_{λ} \in K_{φ_{λ}} and φ_{λ} (0) = 0 < m_{λ} ⩽ φ_{λ} (u_{λ}) . \end{matrix}$

So, $u_{λ}$ is a nontrivial solution of ( P λ ) ( $λ \in (0, λ^{*})$ ). Using (18), we have $\begin{array}{l} φ_{λ} (u_{λ}) ⩽ c_{13} [‖ u_{λ} ‖^{p} + ‖ u_{λ} ‖^{μ} + ‖ u_{λ} ‖^{r}] for some c_{13} > 0 \\ \Rightarrow m_{λ} ⩽ c_{14} [1 + ‖ u_{λ} ‖^{r}] for some c_{14} > 0 (see (22) and recall that 1 < μ < p < r) \\ \Rightarrow ‖ u_{λ} ‖ \to + \infty as λ \to 0^{+} (recall that m_{λ} \to + \infty as λ \to 0^{+}) . \end{array}$ □

4. Asymptotically small solutions

In this section, we provide conditions on $f (z, x)$ which guarantee that for all $λ > 0$ small problem ( P λ ) has a solution ${\hat{u}}_{λ} \in W^{1, ϑ} (Ω)$ such that $‖ {\hat{u}}_{λ} ‖ \to 0^{+}$ as $λ \to 0^{+}$ .

The new conditions on the function $f (z, x)$ in the reaction are the following:

$f : Ω \times R \to R$ is a Carathéodory function such that $f (z, 0) = 0$ for a.a. $z \in Ω$ and

$| f (z, x) | ⩽ \hat{a} (z) [1 + | x |^{r - 1}]$ for a.a. $z \in Ω$ , all $x \in R$ , with $\hat{a} \in L^{\infty} (Ω)$ , $p < r < q^{*}$ ;

there exists $τ \in (1, q)$ and δ, $\hat{c}$ , $\tilde{c}$ such that $\begin{array}{l} \hat{c} | x |^{τ} ⩽ F (z, x) for a.a. z \in Ω, all | x | ⩽ δ, \\ \underset{x \to 0}{lim sup} \frac{F (z, x)}{| x |^{τ}} ⩽ \tilde{c} uniformly for a.a. z \in Ω . \end{array}$

Remark 4.

The hypotheses on $f (z, \cdot)$ are minimal. We stress that no asymptotic condition as $x \to \pm \infty$ is imposed on $f (z, \cdot)$ . Only the subcritical growth condition $H_{2}$ (i), which guarantees that the energy functional of the problem is $C^{1}$ . It is an interesting open question whether we can drop hypothesis $H_{2}$ (i) and use cut-off techniques like those in Leonardi-Papageorgiou [8]. The lack of global regularity results for double phase problems, make such an approach problematic.

Theorem 2.

If hypotheses $H_{0}$ , $H_{2}$ hold, then we can find ${\hat{λ}}^{*} > 0$ such that for all $λ \in (0, {\hat{λ}}^{*})$ problem ( P λ ) has a nontrivial solution ${\hat{u}}_{λ} \in W^{1, ϑ} (Ω)$ and $‖ {\hat{u}}_{λ} ‖ \to 0^{+}$ as $λ \to 0^{+}$ .

Proof.

As before $φ_{λ} : W^{1, ϑ} (Ω) \to R$ is the energy functional for problem ( P λ ) defined by $\begin{matrix} φ_{λ} (u) = \frac{1}{p} {\hat{γ}}_{p} (u) + \frac{1}{q} ‖ \nabla u ‖_{q}^{q} - λ \int_{Ω} F (z, u) d z for all u \in W^{1, ϑ} (Ω) . \end{matrix}$

We know that $φ_{λ} \in C^{1} (W^{1, ϑ} (Ω), R)$ . Hypotheses $H_{2}$ imply that $\begin{matrix} (23) & | F (z, x) | ⩽ c_{15} [| x |^{τ} + | x |^{r}] for a.a. z \in Ω, all x \in R, some c_{15} > 0 . \end{matrix}$

Let $0 < δ < \frac{1}{p}$ . Then for $u \in W^{1, ϑ} (Ω)$ with $‖ u ‖ = λ^{δ}$ , we have $\begin{array}{l} φ_{λ} (u) & ⩾ \frac{c_{3}}{p} λ^{δ p} - c_{16} [λ^{δ τ} + λ^{δ r}] for some c_{15} > 0 (see Proposition 1 and (23)) \\ = [\frac{c_{3}}{p} λ^{δ p - 1} - c_{16} (λ^{δ τ} + λ^{δ r})] λ . \end{array}$

Note that $δ p - 1 < 0$ and so we see that we can find ${\hat{λ}}^{*} > 0$ such that for all $λ \in (0, {\hat{λ}}^{*})$ we have $\begin{matrix} (24) & φ_{λ} (u) > 0 for all u \in W^{1, ϑ} (Ω) with ‖ u ‖ = λ^{δ} . \end{matrix}$

Let $B_{λ} = {u \in W^{1, ϑ} (Ω) : ‖ u ‖ < λ^{δ}}$ . The reflexivity of $W^{1, ϑ} (Ω)$ and the Eberlein–Smulian theorem imply that ${\overline{B}}_{λ}$ is sequentially weakly compact. The functional $φ_{λ} (\cdot)$ is sequentially weakly lower semicontinuous (recall that $W^{1, ϑ} (Ω) ↪ L^{p} (Ω)$ compactly). So, by the Weierstrass–Tonelli theorem, we can find ${\hat{u}}_{λ} \in W^{1, ϑ} (Ω)$ such that $\begin{matrix} (25) & φ_{λ} ({\hat{u}}_{λ}) = min [φ_{λ} (u) : u \in {\overline{B}}_{λ}] . \end{matrix}$

Let $u \in C^{1} (\overline{Ω}) \subseteq W^{1, ϑ} (Ω)$ with $u (z) > 0$ for all $z \in \overline{Ω}$ . Then we can find $t \in (0, 1)$ small such that $0 < t u (z) ⩽ δ$ for all $z \in \overline{Ω}$ , where $δ > 0$ is as postulated by hypothesis $H_{2}$ (ii). We have $\begin{matrix} φ_{λ} (t u) ⩽ \frac{t^{p}}{p} {\hat{γ}}_{p} (u) + \frac{t^{q}}{q} ‖ \nabla u ‖_{q}^{q} - \hat{c} t^{τ} ‖ u ‖_{τ}^{τ} (see hypothesis H_{2} (ii)) . \end{matrix}$

Since $1 < τ < q < p$ , choosing $t \in (0, 1)$ even smaller if necessary, we have $\begin{array}{l} φ_{λ} (t u) < 0 \\ \Rightarrow φ_{λ} ({\hat{u}}_{λ}) < 0 = φ_{λ} (0) (see (25)) \\ (26) & \Rightarrow {\hat{u}}_{λ} \neq 0 . \end{array}$

Also from (24) and (26) it follows that $\begin{matrix} (27) & ‖ {\hat{u}}_{λ} ‖ < λ^{δ} . \end{matrix}$

Therefore ${\hat{u}}_{λ} \in B_{λ} ∖ {0}$ . On account of (25) we have $\begin{array}{l} {\hat{u}}_{λ} \in K_{φ_{λ}} \\ \Rightarrow {\hat{u}}_{λ} is a nontrivial solution of (P_{λ}), λ \in (0, {\hat{λ}}^{*}) . \end{array}$

From (27) we see that $‖ u_{λ} ‖ \to 0^{+}$ as $λ \to 0^{+}$ . □

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