In the following, we suppose that (E = W_{0}^{1, q}(mathcal{M})setminus lbrace 0rbrace ) is endowed with (|u|_{E} = ( int _{mathcal{M}} | nabla u |^{q} ,dv _{mathfrak{g}} (z) )^{frac{1}{q}}).

Definition 4

We say that a function (uin E) is a weak solution to problem (1), if

$$begin{aligned}& int _{mathcal{M}} bigl( vert nabla u vert ^{p-2} nabla u + mu (z) vert nabla u vert ^{q-2}nabla u bigr)nabla varphi ,dv _{ mathfrak{g}} (z) + int _{mathcal{M}}V(z) vert u vert ^{p-2}u varphi ,dv _{mathfrak{g}} (z) \& quad = lambda int _{mathcal{M}} a(z) vert u vert ^{r-2}ulog bigl( vert u vert bigr)varphi ,dv _{mathfrak{g}} (z), end{aligned}$$

for all (varphi in D( mathcal{M})).

Consider the functional (J_{lambda}: Erightarrow mathbb{R}) defined by

$$begin{aligned} J_{lambda}(u)={}&frac{1}{p} int _{mathcal{M}} vert nabla u vert ^{p},dv _{ mathfrak{g}}(z)+ frac{1}{q} int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}}(z)+ frac{1}{p} int _{ mathcal{M}}V(z) vert u vert ^{p} ,dv _{mathfrak{g}}(z) \ &{} -frac{lambda}{r} int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}}(z)+frac{lambda}{r^{2}} int _{ mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) end{aligned}$$

for all (uin E).

Then (J_{lambda}) is well defined and belongs to (C^{1}(E)). Furthermore, we have

$$begin{aligned} bigllangle J’_{lambda}(u), varphi bigrrangle ={}& int _{mathcal{M}} bigl( vert nabla u vert ^{p-2} nabla u+mu (z) vert nabla u vert ^{q-2} nabla u bigr) nabla varphi ,dv _{mathfrak{g}} (z) \ &{} + int _{mathcal{M}}V(z) vert u vert ^{p-2} u varphi ,dv _{ mathfrak{g}} (z) -lambda int _{mathcal{M}} a(z) vert u vert ^{r-2} u log bigl( vert u vert bigr) varphi ,dv _{mathfrak{g}} (z) end{aligned}$$

for all (u, varphi in E).

Consider the Nehari set defined by

$$ mathcal{N}= bigl{ uin E: bigllangle J’_{lambda}(u), u bigrrangle =0 bigr} .$$

We can deduce that the critical points of (J_{lambda}) lie on (mathcal{N}) and further that (uin mathcal{N}) if and only if u is a weak solution to problem (1). Let us define the maps (psi _{u}: mathbb{R}^{+}rightarrow mathbb{R}) by (psi _{u}(t)=J_{lambda}(tu)) and analyze (mathcal{N}) in terms of the stationary points of fibering maps (psi _{u}).

We have

$$begin{aligned} psi ‘_{u}(t) = {}& t^{p-1} int _{mathcal{M}} vert nabla u vert ^{p} ,dv _{mathfrak{g}}(z) + t^{q-1} int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z)+ t^{p-1} int _{mathcal{M}}V(z) vert u vert ^{p} ,dv _{mathfrak{g}} (z) \ &{} -lambda t^{r-1} int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) -lambda t^{r-1} log (t) int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{ mathfrak{g}} (z) end{aligned}$$

and

$$begin{aligned} psi ”_{u}(t) ={} & (p-1) t^{p-2} int _{mathcal{M}} vert nabla u vert ^{p} ,dv _{mathfrak{g}} (z) + (q-1) t^{q-2} int _{ mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) \ &{} + (p-1) t^{p-2} int _{mathcal{M}}V(z) vert u vert ^{p} ,dv _{ mathfrak{g}} (z) -lambda (r-1) t^{r-2} int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) \ &{} -lambda (r-1) t^{r-2} log (t) int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) -lambda t^{r-2} int _{ mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z). end{aligned}$$

It is easy to verify that (tuin mathcal{N} Longleftrightarrow psi ^{prime }_{u}(t)=0) for any (uin E) and (t>0).

We shall split (mathcal{N} ) into three subsets which correspond to local minima, local maxima, and points of inflection of fibering maps, that is,

$$begin{aligned}& begin{aligned} mathcal{N}^{+}&=bigllbrace uin mathcal{N}:psi ”_{u}(1)>0 bigrrbrace \ &= biggllbrace uin E: (q-p) int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) +lambda (p-r) int _{ mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{ mathfrak{g}} (z) \ &quad {}> lambda int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) biggrrbrace , end{aligned} \& begin{aligned}mathcal{N}^{0}&=bigllbrace uin mathcal{N}:psi ”_{u}(1) =0 bigrrbrace \ &= biggllbrace uin E: (q-p) int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) +lambda (p-r) int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) \ &quad {}= lambda int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{ mathfrak{g}} (z) biggrrbrace , end{aligned} \& begin{aligned}mathcal{N}^{-}&=bigllbrace uin mathcal{N}:psi ^{prime prime }_{u}(1) < 0 bigrrbrace \ &= biggllbrace uin E: (q-p) int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) + lambda (p-r) int _{ mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{ mathfrak{g}} (z) \ &quad {}< lambda int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) biggrrbrace . end{aligned} end{aligned}$$

Lemma 4

Let (u_{0}notin mathcal{N}^{0}). Then (u_{0}) is a critical point of (J_{lambda}) if (u_{0}) is a local minimizer of (J_{lambda}) on (mathcal{N}).

Proof

We remark that (u_{0}) is a solution to the optimization problem to minimize (J_{lambda}) subject to (I(u)=0), where

$$begin{aligned} I(u) =& int _{mathcal{M}} bigl( vert nabla u vert ^{p} + mu (z) vert nabla u vert ^{q} bigr) ,dv _{mathfrak{g}} (z) + int _{ mathcal{M}}V(z) vert u vert ^{p} ,dv _{mathfrak{g}} (z) \ &{}- lambda int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z), end{aligned}$$

and, since (u_{0}) is a local minimizer of (J_{lambda}) on (mathcal{N}), we have

$$ I(u_{0})=bigllangle J’_{lambda}(u_{0}),u_{0} bigrrangle . $$

(2)

Then, there exists a Lagrange multiplier (alpha in mathbb{R}) such that (J’_{lambda}(u_{0})=alpha I'(u_{0})), namely (0=langle J’_{lambda}(u_{0}), u_{0}rangle =alpha langle I'(u_{0}), u_{0}rangle ).

Furthermore, (langle I'(u_{0}), u_{0}rangle neq 0) since (u_{0}notin mathcal{N}^{0}) which implies (alpha =0) and, actually, that (u_{0}) is a critical point of (J_{lambda}). □

Lemma 5

There exists a positive constant (lambda _{0}) such that, for any (0<lambda <lambda _{0}), the functional (J_{lambda}) is bounded and coercive on (mathcal{N}).

Proof

Letting (uin E) with (Vert uVert _{E} > 1 ), we obtain

$$begin{aligned} J_{lambda}(u)geq {}& frac{1}{q} biggl( int _{mathcal{M}} vert nabla u vert ^{p} ,dv _{mathfrak{g}} (z) + int _{mathcal{M}} mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) + V_{0} int _{mathcal{M}} vert u vert ^{p} ,dv _{mathfrak{g}} (z) biggr) \ &{} -frac{lambda}{r} int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) + frac{lambda}{r^{2}} int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z), end{aligned}$$

and we know that

$$ log (s)leq frac{s^{alpha}}{alpha e} quad text{for all } alpha >0 text{ and } s>0, $$

(3)

thus

$$ J_{lambda}(u)geq frac{mu _{0}}{q} Vert u Vert _{E}^{q}- frac{lambda Vert a Vert _{infty}}{r(q-r) e} int _{mathcal{M}} vert u vert ^{q} ,dv _{mathfrak{g}} (z)$$

with (alpha =q-r).

According to Theorem 2 and Poincaré inequality, there exists a positive constant (C_{q}) such that

$$ J_{lambda}(u) geq frac{mu _{0}}{q} Vert u Vert _{E}^{q}- frac{lambda Vert a Vert _{infty}}{r(q-r) e}C_{q} Vert u Vert _{E}^{q} geq biggl( frac{mu _{0}}{q}- frac{lambda Vert a Vert _{infty}}{r(q-r) e}C_{q} biggr) Vert u Vert _{E}^{q}. $$

Choosing (0<lambda <lambda _{0}= frac{r(q-r) e}{qC_{q}Vert aVert _{infty}}) implies that (J_{lambda}) is coercive.

Moreover, we have

$$begin{aligned} J_{lambda}(u)& leq frac{1}{p} biggl(varrho (u)+ int _{mathcal{M}}V(z) vert u vert ^{p} ,dv _{mathfrak{g}} (z) -lambda int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) biggr) \ &quad {}+ frac{lambda}{r^{2}} int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) \ & =frac{lambda}{r^{2}} int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) leq frac{lambda}{r^{2}} Vert a Vert _{ infty} int _{mathcal{M}} vert u vert ^{r} ,dv _{mathfrak{g}} (z). end{aligned}$$

Thanks to Theorem 2, there exists (C_{r} > 0) such that

$$ J_{lambda}(u)leq C_{r} frac{lambda}{r^{2}} Vert a Vert _{ infty} Vert u Vert _{E}^{r}. $$

 □

Lemma 6

Let (lambda _{1} = frac{(q-p) mu _{0} }{Vert aVert _{L^{infty}}C_{1}}) where (C_{1} (r, q, mathcal{M} )) is a constant to be specified later. Then, for any λ such that (0<lambda <lambda _{1}), we have (mathcal{N}^{0}cup mathcal{N}^{-}= emptyset ) and (mathcal{N}^{+}neq emptyset ).

Proof

We proceed by contradiction to prove that (mathcal{N}^{0}cup mathcal{N}^{-}neq emptyset ).

Indeed, let (uin mathcal{N}^{0}cup mathcal{N}^{-}), then we get

$$begin{aligned}& (q-p) int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{ mathfrak{g}} (z) +lambda (p-r) int _{mathcal{M}} a(z) vert u vert ^{r} log bigl( vert u vert bigr) ,dv _{mathfrak{g}} (z) \& quad leq lambda int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{ mathfrak{g}} (z), end{aligned}$$

thus

$$ (q-p) int _{mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{ mathfrak{g}} (z) leq lambda int _{mathcal{M}} a(z) vert u vert ^{r} ,dv _{mathfrak{g}} (z) leq lambda Vert a Vert _{L^{ infty}} int _{mathcal{M}} vert u vert ^{r} ,dv _{mathfrak{g}} (z).$$

Using the fact that (L^{q}(mathcal{mathcal{M}})subset L^{r}(mathcal{mathcal{M}})) and Poincaré inequality, there exists a positive constant (C_{1} (r, q, mathcal{M} )) such that

$$ int _{mathcal{M}} vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) geq C_{1} biggl( int _{mathcal{M}} vert u vert ^{r} ,dv _{ mathfrak{g}} (z) biggr)^{frac{q}{r}},$$

hence

$$begin{aligned} (q-p) mu _{0} C_{1} biggl( int _{mathcal{M}} vert u vert ^{r} ,dv _{mathfrak{g}} (z) biggr)^{frac{q}{r}}&leq (q-p) int _{ mathcal{M}}mu (z) vert nabla u vert ^{q} ,dv _{mathfrak{g}} (z) \ &leq lambda Vert a Vert _{L^{infty}} int _{mathcal{M}} vert u vert ^{r} ,dv _{mathfrak{g}} (z), end{aligned}$$

thus

$$ biggl( int _{mathcal{M}} vert u vert ^{r} ,dv _{mathfrak{g}} (z) biggr)^{frac{q}{r}-1}leq lambda frac{ Vert a Vert _{L^{infty}} }{(q-p) mu _{0} C_{1}}, $$

and, when (lambda rightarrow 0), we have (u=0), which is a contradiction.

Now, according to Lemma 5, the set (mathcal{N}^{+}neq emptyset ). □

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