Elliptic variational problems with mixed nonlinearities

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of elliptic equations with mixed nonlinearities in Euclidean space, extending understanding of solution behavior under various parameter and function conditions.

Contribution

It provides new results on positive solutions for elliptic equations with combined nonlinearities, considering different dimensional and parameter regimes.

Findings

Existence of positive solutions under certain conditions.

Multiple solutions are established for specific parameter ranges.

Results depend on the interplay between nonlinear terms and the domain dimension.

Abstract

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^*,q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $\lambda>0$ is a parameter.

Equations

$$-{\Delta }_{p}u+u^{p-1}=\lambda k(x)u^{r-1}-h(x)u^{q-1}.$$

$$-\Delta u=\lambda u+k(x)u^{r-1}-h(x)u^{q-1},u>0$$

$$-{\Delta }_{p}u+u^{p-1}=\lambda u^{r-1}-h(x)u^{q-1},u\ge 0,$$

$$-{\Delta }_{p}u+w(x)u^{p-1}=\lambda k(x)u^{r-1}-h(x)u^{q-1},u\ge 0,$$

$$-{\Delta }_{p}u+u^{p-1}=\lambda k(x)u^{r-1}-h(x)u^{q-1},u\ge 0$$

$$\mathscr{E}=\left\{u\in W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)}:\left\|u\right\|_{\mathscr{E}}:=\left\|u\right\|_{W^{1,\hskip 0.56905ptp}(\mathbf{R}^{N})}+\left\|u\right\|_{L^{q}_{h}(\mathbf{R}^{N})}<\infty\right\}.$$

$${\mathcal{J}}_{\lambda }(u)=\frac{1}{p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+\frac{1}{q}{\int }_{{\mathbf{R}}^N}h{\left|u\right|}^{q}dx-\frac{\lambda }{r}{\int }_{{\mathbf{R}}^N}k{\left(u^{+}\right)}^{r}dx.$$

$$\begin{split}&\int_{\mathbf{R}^{N}}k\left|u\right|^{r}\hskip 0.56905ptdx=\int_{\mathbf{R}^{N}}\left\{h\left|u\right|^{q}\right\}^{\mathfrak{x}}\left|u\right|^{r-q\mathfrak{x}}\left\{k\hskip 0.56905pth^{-\mathfrak{x}}\right\}\hskip 0.56905ptdx\\ \leq&\Big{(}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{1}}}\Big{(}\int_{\mathbf{R}^{N}}\left|u\right|^{s}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{2}}}\Big{(}\int_{\mathbf{R}^{N}}k^{r_{3}}h^{-\beta}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{3}}}\\ \leq&\,C_{1}\Big{(}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{1}}}\Big{(}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{s}{pr_{2}}}\Big{(}\int_{\mathbf{R}^{N}}k^{r_{3}}h^{-\beta}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{3}}},\end{split}$$

$${\int }_{{\mathbf{R}}^N}k{\left|u_l\right|}^{r}dx={\int }_{{\mathbf{B}}_R}k{\left|u_l\right|}^{r}dx+{\int }_{{\mathbf{B}}_R^c}k{\left|u_l\right|}^{r}dx.$$

$${\int }_{{\mathbf{B}}_R^c}k{\left|u_l\right|}^{r}dx\le C_1{\Vert k^{r_3}{h}^{-\beta }\Vert }_{1,{\mathbf{B}}_R^c}^{\frac{1}{r_3}}{\Vert u_l\Vert }_{\mathcal{E}}^r\to 0$$

$$W^{1,\hskip 0.56905ptp}\big{(}\mathbf{B}_{R}\big{)}\hookrightarrow L^{\frac{\alpha r}{\alpha-1}}\big{(}\mathbf{B}_{R}\big{)}\to L^{r}_{k}\big{(}\mathbf{B}_{R}\big{)},$$

$$\mathscr{E}\to W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)},$$

$${\int }_{{\mathbf{R}}^N}\left({\left|\nabla u_{\lambda }\right|}^p+{\left|u_{\lambda }\right|}^p\right)dx+{\int }_{{\mathbf{R}}^N}h{\left|u_{\lambda }\right|}^{q}dx\le {\lambda }^{\rho }{C}_{hk},$$

$$\Big{(}\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{\lambda}\right|^{p}+\left|u_{\lambda}\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{p-t}{p}}\leq\lambda^{\varrho}C^{\prime}_{hk}.$$

$${\int }_{{\mathbf{R}}^N}\left({\left|\nabla u_{\lambda }\right|}^p+{\left|u_{\lambda }\right|}^p\right)dx+{\int }_{{\mathbf{R}}^N}h{\left|u_{\lambda }\right|}^{q}dx=\lambda {\int }_{{\mathbf{R}}^N}k{\left|u_{\lambda }\right|}^{r}dx.$$

$$\begin{split}&\lambda\int_{\mathbf{R}^{N}}k\left|u_{\lambda}\right|^{r}\hskip 0.56905ptdx\leq\Big{(}\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{\lambda}\right|^{p}+\left|u_{\lambda}\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{5}}}\Big{(}\int_{\mathbf{R}^{N}}h\left|u_{\lambda}\right|^{q}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{4}}}\\ &\hskip 83.65108pt\times\bigg{\{}\lambda^{\rho}C^{\rho}_{1}\Big{(}\int_{\mathbf{R}^{N}}k^{r_{3}}h^{-\beta}\hskip 0.56905ptdx\Big{)}^{\frac{\rho}{r_{3}}}\bigg{\}}^{\frac{1}{\rho}}\\ \leq&\,\frac{1}{r_{5}}\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{\lambda}\right|^{p}+\left|u_{\lambda}\right|^{p}\right)\hskip 0.56905ptdx+\frac{1}{r_{4}}\int_{\mathbf{R}^{N}}h\left|u_{\lambda}\right|^{q}\hskip 0.56905ptdx+\lambda^{\rho}C_{hk}.\end{split}$$

$$\begin{split}&\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{\lambda}\right|^{p}+\left|u_{\lambda}\right|^{p}\right)\hskip 0.56905ptdx\leq\lambda^{\frac{q-t}{q-r}}C(q,r,t)\int_{\mathbf{R}^{N}}\Big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\Big{]}^{\frac{1}{q-r}}\left|u_{\lambda}\right|^{t}\hskip 0.56905ptdx\\ \leq&\,\lambda^{\frac{q-t}{q-r}}C(q,r,t)\Big{(}\int_{\mathbf{R}^{N}}\Big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{R}^{N}}\left|u_{\lambda}\right|^{\frac{\gamma t}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\,\lambda^{\frac{q-t}{q-r}}C^{\prime}_{hk}\Big{(}\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{\lambda}\right|^{p}+\left|u_{\lambda}\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{t}{p}}\end{split}$$

$$\begin{split}&\frac{\lambda}{r}\int_{\mathbf{R}^{N}}k\left(u^{+}\right)^{r}\hskip 0.56905ptdx\leq\frac{\lambda}{r}\int_{\mathbf{R}^{N}}k\left|u\right|^{r}\hskip 0.56905ptdx\leq\Big{(}\frac{r_{5}}{2p}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{5}}}\\ &\hskip 7.68222pt\times\Big{(}\frac{r_{4}}{2q}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx\Big{)}^{\frac{1}{r_{4}}}\bigg{\{}\lambda^{\rho}\Big{(}\frac{C_{1}}{r}\Big{)}^{\rho}\Big{(}\frac{2p}{r_{5}}\Big{)}^{\frac{\rho}{r_{5}}}\Big{(}\frac{2q}{r_{4}}\Big{)}^{\frac{\rho}{r_{4}}}\Big{(}\int_{\mathbf{R}^{N}}k^{r_{3}}h^{-\beta}\hskip 0.56905ptdx\Big{)}^{\frac{\rho}{r_{3}}}\bigg{\}}^{\frac{1}{\rho}}\\ \leq&\,\frac{1}{2p}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx+\frac{1}{2q}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx+\lambda^{\rho}\widetilde{C}_{hk},\end{split}$$

$${\mathcal{J}}_{\lambda }(u)\ge \frac{1}{2p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+\frac{1}{2q}{\int }_{{\mathbf{R}}^N}h{\left|u\right|}^{q}dx-{\lambda }^{\rho }{\tilde{C}}_{hk}.$$

$$\begin{split}&\frac{\lambda}{r}\int_{\mathbf{R}^{N}}k\left(u^{+}\right)^{r}\hskip 0.56905ptdx-\frac{1}{2q}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx\leq\frac{\lambda}{r}\int_{\mathbf{R}^{N}}k\left|u\right|^{r}\hskip 0.56905ptdx-\frac{1}{2q}\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx\\ \leq&\,C(q,r,t,\lambda)\Big{(}\int_{\mathbf{R}^{N}}\Big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{R}^{N}}\left|u\right|^{\frac{\gamma t}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\,C_{hk}(q,r,t,\lambda)\Big{(}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{t}{p}}\leq\frac{1}{2p}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx+C_{hk\lambda}\end{split}$$

$$\frac{\lambda }{r}{\int }_{{\mathbf{R}}^N}k{\left(u^{+}\right)}^{r}dx\le \frac{1}{2p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+\frac{1}{2q}{\int }_{{\mathbf{R}}^N}h{\left|u\right|}^{q}dx+C_{hk\lambda },$$

$$\begin{split}&\frac{\lambda}{r}\int_{\mathbf{B}^{c}_{R}}k\left(u^{+}\right)^{r}\hskip 0.56905ptdx-\frac{1}{2q}\int_{\mathbf{B}^{c}_{R}}h\left|u\right|^{q}\hskip 0.56905ptdx\\ \leq&\,C(q,r,t,\lambda)\Big{(}\int_{\mathbf{B}^{c}_{R}}\Big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{B}^{c}_{R}}\left|u\right|^{\frac{\gamma p}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\Big{(}\frac{1}{2p}-\varepsilon\Big{)}\int_{\mathbf{B}^{c}_{R}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\leq\Big{(}\frac{1}{2p}-\varepsilon\Big{)}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\end{split}$$

$$\begin{array}{rl}& \frac{\lambda }{r}{\int }_{{\mathbf{B}}_R}k{\left(u^{+}\right)}^{r}dx-\frac{1}{2q}{\int }_{{\mathbf{B}}_R}h{\left|u\right|}^{q}dx\\ \le & \epsilon {\int }_{{\mathbf{B}}_R}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+{\tilde{C}}_{hk\lambda }\le \epsilon {\int }_{{\mathbf{R}}^N}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+{\tilde{C}}_{hk\lambda }\end{array}$$

$$\left\{\begin{array}[]{ll}\mathbf{X}=\left\{x\in\mathbf{B}_{R}:k(x)<M\,\,\text{and}\,\,h(x)>\delta\right\},\\ \mathbf{Y}=\left\{x\in\mathbf{B}_{R}:k(x)<M\,\,\text{and}\,\,h(x)\leq\delta\right\}\\ \hskip 11.66563pt\varsubsetneq\mathbf{Y}^{\prime}=\left\{x\in\mathbf{B}_{R}:h(x)\leq\delta\right\},\\ \mathbf{Z}=\left\{x\in\mathbf{B}_{R}:k(x)\geq M\right\}.\end{array}\right.$$

$$\begin{split}&\frac{\lambda}{r}\int_{\mathbf{Y}\cup\mathbf{Z}}k\left(u^{+}\right)^{r}\hskip 0.56905ptdx-\frac{1}{2q}\int_{\mathbf{Y}\cup\mathbf{Z}}h\left|u\right|^{q}\hskip 0.56905ptdx\\ \leq&\,C(q,r,t,\lambda)\Big{(}\int_{\mathbf{Y}\cup\mathbf{Z}}\Big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{Y}\cup\mathbf{Z}}\left|u\right|^{\frac{\gamma p}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\,C(q,r,t,\lambda)\Big{(}\int_{\mathbf{Y}^{\prime}\cup\mathbf{Z}}\Big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{Y}^{\prime}\cup\mathbf{Z}}\left|u\right|^{\frac{\gamma p}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\,\frac{\varepsilon}{2}\int_{\mathbf{Y}^{\prime}\cup\mathbf{Z}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\leq\frac{\varepsilon}{2}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\end{split}$$

$$\begin{split}&\frac{\lambda}{r}\int_{\mathbf{X}}k\left(u^{+}\right)^{r}\hskip 0.56905ptdx-\frac{1}{2q}\int_{\mathbf{X}}h\left|u\right|^{q}\hskip 0.56905ptdx\\ \leq&\,C(q,r,t,\lambda)\Big{(}\int_{\mathbf{B}_{R}}\Big{[}\frac{k^{q-\tilde{t}}(x)}{h^{r-\tilde{t}}(x)}\Big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx\Big{)}^{\frac{1}{\gamma}}\Big{(}\int_{\mathbf{B}_{R}}\left|u\right|^{\frac{\gamma\tilde{t}}{\gamma-1}}\hskip 0.56905ptdx\Big{)}^{\frac{\gamma-1}{\gamma}}\\ \leq&\,C_{M\delta}(q,r,t,\lambda)\Big{(}\int_{\mathbf{B}_{R}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx\Big{)}^{\frac{\tilde{t}}{p}}\leq\frac{\varepsilon}{2}\int_{\mathbf{R}^{N}}\left(\left|\nabla u\right|^{p}+\left|u\right|^{p}\right)\hskip 0.56905ptdx+\widetilde{C}_{hk\lambda}.\end{split}$$

$$\underset{l\to \infty }{\lim }{\int }_{{\mathbf{R}}^N}k{\left(u_l^{+}\right)}^{r}dx={\int }_{{\mathbf{R}}^N}k{\left(u^{+}\right)}^{r}dx$$

$$\tilde{\lambda }=\underset{u\in \mathcal{E},{\Vert u^{+}\Vert }_{L_k^r({\mathbf{R}}^N)}=1}{\inf }\left\{\frac{r}{p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla u\right|}^p+{\left|u\right|}^p\right)dx+\frac{r}{q}{\int }_{{\mathbf{R}}^N}h{\left|u\right|}^{q}dx\right\};$$

$$\lambda {\int }_{{\mathbf{R}}^N}k{\left(v_{\lambda }^{+}\right)}^{r}dx>\frac{r}{p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla v_{\lambda }\right|}^p+{\left|v_{\lambda }\right|}^p\right)dx+\frac{r}{q}{\int }_{{\mathbf{R}}^N}h{\left|v_{\lambda }\right|}^{q}dx$$

$${\mathcal{J}}_{\lambda }(v_{\lambda })=\frac{1}{p}{\int }_{{\mathbf{R}}^N}\left({\left|\nabla v_{\lambda }\right|}^p+{\left|v_{\lambda }\right|}^p\right)dx+\frac{1}{q}{\int }_{{\mathbf{R}}^N}h{\left|v_{\lambda }\right|}^{q}dx-\frac{\lambda }{r}{\int }_{{\mathbf{R}}^N}k{\left(v_{\lambda }^{+}\right)}^{r}dx<0.$$

Full text

[TABLE]

Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^{*},q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $\lambda>0$ is a parameter.

2010 Mathematics Subject Classification. 35B09, 35J20, 35J92, 46E35.

Keywords. Elliptic equations, positive solutions.

1. Introduction

In 1996, Alama and Tarantello [1] investigated an elliptic problem, when $N\geq 3$, as

[TABLE]

on a bounded, smooth domain $\mathbf{\Omega}$, subject to $u=0$ on $\partial\mathbf{\Omega}$, for $h(x),k(x)>0$, $2<r<q$ and a constant $\lambda\in\mathbf{R}$, where the existence and multiplicity results of solutions to (1.1) are discussed assuming either \bigintsss_{\mathbf{\Omega}}\big{[}\frac{k^{q}(x)}{h^{r}(x)}\big{]}^{\frac{1}{q-r}}\hskip 0.56905ptdx<\infty or \bigintsss_{\mathbf{\Omega}}\big{[}\frac{k^{q-2}(x)}{h^{r-2}(x)}\big{]}^{\frac{2^{*}}{(2^{*}-2)(q-r)}}\hskip 0.56905ptdx<\infty. Notice that the latter condition is weaker than the former condition when $r<2^{*}$ and $\int_{\mathbf{\Omega}}k^{\frac{2^{*}}{2^{*}-r}}(x)\hskip 0.56905ptdx<\infty$.

It seems to the author that Chabrowski [5] is the first person who studied a similar problem to (1.1) in $\mathbf{R}^{N}$, with -\Delta_{p}\hskip 0.56905ptu=-\operatorname{div}\big{(}\left|\nabla u\right|^{p-2}\nabla u\big{)} and $p<r<q<p^{*}$, as

[TABLE]

where the existence result of solutions to (1.2) is discussed assuming that $\int_{\mathbf{R}^{N}}h^{\frac{r}{r-q}}(x)\hskip 0.56905ptdx<\infty$; some time later, Pucci and Rădulescu [10, 12] extended the above work of Chabrowski, and first studied the existence and multiplicity results of solutions to (1.2) under the same assumption. Quite recently, Autuori and Pucci [2, 3], Pucci and Zhang [11], and Rădulescu et al. [13] further studied an extension of (1.2) in $\mathbf{R}^{N}$, for $p<r<\min\{p^{*},q\}$, as

[TABLE]

with $w(x)\simeq\frac{1}{(1+\left|x\right|)^{p}}>0$111In [13], the condition $w(x)\geq w_{0}>0$ is used. and $p<N$, where the existence and multiplicity results of solutions to (1.3) are shown using either the conditions in [1] or \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-1}(x)}{h^{r-1}(x)}\big{]}^{\frac{p^{*}}{(p^{*}-1)(q-r)}}\hskip 0.56905ptdx<\infty. Again, the latter is weaker than \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q}(x)}{h^{r}(x)}\big{]}^{\frac{1}{q-r}}\hskip 0.56905ptdx<\infty if $r<p^{*}$ and $\int_{\mathbf{R}^{N}}k^{\frac{p^{*}}{p^{*}-r}}(x)\hskip 0.56905ptdx<\infty$.

In this paper, for $N\geq 2$, we study the existence of nontrivial solutions to

[TABLE]

in $\mathbf{R}^{N}$, where $1<p<q<\infty$, $p<r<\min\{p^{*},q\}$ with $p^{*}=\frac{Np}{N-p}$ if $p<N$ while $p<r<q$ if $p\geq N$, $h(x),k(x)>0$ are Lebesgue measurable functions, and $\lambda>0$ is a constant.

Our primary assumptions are as follows.

Standing Assumptions. (1-a). When $1<p<N$, k(x)\in L^{\alpha}_{loc}\big{(}\mathbf{R}^{N}\big{)} for some \alpha\in\big{(}\frac{p^{*}}{p^{*}-r},\infty\big{]}; (1-b). when $N\leq p<\infty$, k(x)\in L^{\alpha}_{loc}\big{(}\mathbf{R}^{N}\big{)} for some $\alpha\in(1,\infty]$. (2-a). When $p\leq s<r$, $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{[}\frac{r}{q-r},\infty\big{)}; (2-b). when $1<p<N$ and $p<r<s\leq p^{*}$ or when $N\leq p<\infty$ and $p<r<s<\infty$,        $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{[}0,\frac{r}{q-r}\big{]}. (3-a). When $p<s<r$, $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{[}\frac{s}{q-s},\frac{s(r-p)}{(s-p)(q-r)}\big{)}; (3-b). when $1<p<N$ and $p<r<s\leq p^{*}$ or when $N\leq p<\infty$ and $p<r<s<\infty$,        $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{(}\frac{s(r-p)}{(s-p)(q-r)},\frac{s}{q-s}\big{]}. (4-a). When $1<p<N$, \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx<\infty for some $0\leq t<p$ and \gamma\in\big{[}\frac{p^{*}}{p^{*}-t},\frac{p}{p-t}\big{]},        and \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx<\infty for some \gamma\in\big{[}\frac{p^{*}}{p^{*}-p},\infty\big{]} with h(x),k(x)\in L^{1}_{loc}\big{(}\mathbf{R}^{N}\big{)}; (4-b). when $N\leq p<\infty$, \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx<\infty for some $0\leq t<p$ and \gamma\in\big{(}1,\frac{p}{p-t}\big{]}222By abuse of notation, we include $t=0$ and $\gamma=1$ here.,        and \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\big{]}^{\frac{\gamma}{q-r}}\hskip 0.56905ptdx<\infty for some $\gamma\in(1,\infty]$ with h(x),k(x)\in L^{1}_{loc}\big{(}\mathbf{R}^{N}\big{)}. (5-a). When $1<p<N$, k(x)\in L^{\sigma}\big{(}\mathbf{R}^{N}\big{)} for some \sigma\in\big{[}\frac{p^{*}}{p^{*}-r},\infty\big{]}; (5-b). when $N\leq p<\infty$, k(x)\in L^{\sigma}\big{(}\mathbf{R}^{N}\big{)} for some $\sigma\in(1,\infty]$.

It is worthwhile to remark our hypotheses have all the known conditions regarding the integrability of the ratio function $k^{\mu}(x)/h^{\nu}(x)$ as special cases; for instance, $\beta=\frac{r}{q-r}$ or $t=0$ yields \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q}(x)}{h^{r}(x)}\big{]}^{\frac{1}{q-r}}\hskip 0.56905ptdx<\infty, $t=1$ and $\gamma=\frac{p^{*}}{p^{*}-1}$ for $1<p<N$ implies \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-1}(x)}{h^{r-1}(x)}\big{]}^{\frac{p^{*}}{(p^{*}-1)(q-r)}}\hskip 0.56905ptdx<\infty, while $t=p$ and $\gamma=\frac{p^{*}}{p^{*}-p}$ for $1<p<N$ leads to \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-p}(x)}{h^{r-p}(x)}\big{]}^{\frac{p^{*}}{(p^{*}-p)(q-r)}}\hskip 0.56905ptdx<\infty.

Notice hypotheses (1)-(2) ensure the compact embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} (see Propositions 2.1 and 2.2 below). In addition, if $\beta=0$ in (2-b), then one has assumption (5) covered expect for $\sigma=\infty$, but condition (5) is crucial in proving the existence of a second positive solution to (1.4); so, it is better to keep it separately. On the other hand, one can verify that the extremal cases in (2-b) where $\beta=0$ and $\beta=\frac{r}{q-r}$ altogether lead to certain cases in assumption (4) for appropriate indices as observed earlier; for example, the condition \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q-t}(x)}{h^{r-t}(x)}\big{]}^{\frac{p^{*}}{(p^{*}-t)(q-r)}}\hskip 0.56905ptdx<\infty is weaker than \bigintsss_{\mathbf{R}}^{N}\big{[}\frac{k^{q}(x)}{h^{r}(x)}\big{]}^{\frac{1}{q-r}}\hskip 0.56905ptdx<\infty provided $r<p^{*}$ and $\int_{\mathbf{R}^{N}}k^{\frac{p^{*}}{p^{*}-r}}(x)\hskip 0.56905ptdx<\infty$.

Finally, for $\left\|u\right\|^{q}_{L^{q}_{h}(\mathbf{R}^{N})}=\int_{\mathbf{R}^{N}}h\left|u\right|^{q}\hskip 0.56905ptdx$ (and $\left\|u\right\|^{q}_{L^{r}_{k}(\mathbf{R}^{N})}=\int_{\mathbf{R}^{N}}k\left|u\right|^{r}\hskip 0.56905ptdx$ used later), define

[TABLE]

Then, our main result of this paper reads as below.

Theorem 1.1**.**

Under our Standing Assumptions (1)-(3), or (1), (2) and (4), there is a $\lambda_{1}\geq 0$ such that equation (1.4) has at least a positive solution in $\mathscr{E}$ for every $\lambda>\lambda_{1}$. Besides, when our Standing Assumption (5) is also true, then $\lambda_{1}>0$ and equation (1.4) has at least a positive solution in $\mathscr{E}$ if and only if $\lambda\geq\lambda_{1}$ (except for $t=p$); moreover, there is a $\lambda_{2}(\geq\lambda_{1})$ such that equation (1.4) has at least two positive solutions in $\mathscr{E}$ for every $\lambda>\lambda_{2}$.

2. Proof of Theorem 1.1

In this section, we seek nontrivial positive solutions to (1.4) in $\mathscr{E}$ through identifying the critical points of the associated energy functional $\mathcal{J}_{\lambda}:\mathscr{E}\to\mathbf{R}$, defined by

[TABLE]

The first task here is to provide some compact embedding results of \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)}.

Proposition 2.1**.**

Let $1\leq p<N$ and $p\leq s<r<\min\{p^{*},q\}<\infty$, or let $N\leq p<\infty$ and $p\leq s<r<q<\infty$. Suppose $h(x),k(x)>0$ satisfy k(x)\in L^{\alpha}_{loc}\big{(}\mathbf{R}^{N}\big{)} for some \alpha\in\big{(}\frac{p^{*}}{p^{*}-r},\infty\big{]} in the former case and $\alpha\in(1,\infty]$ in the latter case, while $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{[}\frac{r}{q-r},\infty\big{)}. Then, the embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} is compact.

Proof.

Recall $\mathscr{E}$ is a subspace of W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)} by its definition. Write $\mathfrak{x}=\frac{\beta(s-r)}{s+\beta(s-q)}$, $\mathfrak{y}=\frac{r+\beta(r-q)}{s+\beta(s-q)}$ and $\mathfrak{z}=\frac{s-r}{s+\beta(s-q)}$ for an arbitrarily chosen $s\in[p,r)$, and notice $\mathfrak{x}+\mathfrak{y}+\mathfrak{z}=1$.

Now, set $r_{1}=\mathfrak{x}^{-1}$, $r_{2}=\mathfrak{y}^{-1}$ and $r_{3}=\mathfrak{z}^{-1}$ to observe

[TABLE]

with $s=r_{2}(r-q\mathfrak{x})$ for a constant $C_{1}>0$, provided $\mathfrak{x},\mathfrak{y},\mathfrak{z}>0$ simultaneously.

To see $\mathfrak{x}>0$ or $\mathfrak{z}>0$, we have $\beta>\frac{s}{q-s}$ and to see $\mathfrak{y}>0$, we have $\beta>\frac{r}{q-r}$, so that $\beta>\frac{r}{q-r}$ since $s<r$. We certainly can take $\mathfrak{y}=0$ and consequently have $\beta=\frac{r}{q-r}$.

Next, one has $\frac{q}{r_{1}}+\frac{s}{r_{2}}=q\mathfrak{x}+s\mathfrak{y}=r$; so, the embedding \mathscr{E}\to L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} is continuous. Now, let $\left\{u_{l}:l\geq 1\right\}$ be a sequence of functions in $\mathscr{E}$, with $u_{l}\rightharpoonup 0$ when $l\to\infty$ and $\left\|u_{l}\right\|_{\mathscr{E}}$ uniformly bounded. It follows that

[TABLE]

Here, and hereafter, $\mathbf{B}_{R}$ denotes the ball of radius $R$ in $\mathbf{R}^{N}$ that is centered at the origin and $\mathbf{B}^{c}_{R}=\mathbf{R}^{N}\setminus\mathbf{B}_{R}$. For the integral over $\mathbf{B}^{c}_{R}$, we apply (2.2) on $\mathbf{B}^{c}_{R}$ to derive

[TABLE]

as $R\to\infty$. For the integral over $\mathbf{B}_{R}$, our (local) hypotheses lead to

[TABLE]

because either $r\leq\frac{\alpha r}{\alpha-1}<p^{*}$ when $1\leq p<N$ or $r\leq\frac{\alpha r}{\alpha-1}<\infty$ when $N\leq p<\infty$. Hence, for a subsequence relabeled with the same index $l$, one proves $u_{l}\to 0$ in L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)}. ∎

Proposition 2.2**.**

Let $1\leq r<\min\{q,s\}<\infty$, with $p\leq s\leq p^{*}$ if $1\leq p<N$ and $p\leq s<\infty$ if $N\leq p<\infty$. Suppose further $h(x),k(x)>0$ satisfy k(x)\in L^{\alpha}_{loc}\big{(}\mathbf{R}^{N}\big{)} for some \alpha\in\big{(}\frac{p^{*}}{p^{*}-r},\infty\big{]} in the former case and $\alpha\in(1,\infty]$ in the latter case, while $\int_{\mathbf{R}^{N}}k^{\frac{s+\beta(s-q)}{s-r}}(x)h^{-\beta}(x)\hskip 0.56905ptdx<\infty$ for some \beta\in\big{[}0,\frac{r}{q-r}\big{]}. Then, the embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} is compact.

Proof.

The proof is almost identical to that of Proposition 2.1, and the only difference lies in the verification of $\mathfrak{x},\mathfrak{y},\mathfrak{z}>0$. To see $\mathfrak{x}>0$ or $\mathfrak{z}>0$, one has $\beta<\frac{s}{q-s}$ and to see $\mathfrak{y}>0$, one has $\beta<\frac{r}{q-r}$, so that $\beta<\frac{r}{q-r}$ since $s>r$. Again, taking $\mathfrak{y}=0$ yields $\beta=\frac{r}{q-r}$. Finally, for $\beta=0$, or equivalently for $\mathfrak{x}=0$, we work in the space W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)} and have

[TABLE]

which is simply a special case of Proposition 4.2 in [6] and Theorem 4.6 in [8]. ∎

Next, we make elementary observations for all solutions to equation $\eqref{eq1.4}$.

Lemma 2.3**.**

Under the Standing Assumption (3), each solution $u_{\lambda}\in\mathscr{E}$ to equation $\eqref{eq1.4}$ satisfies

[TABLE]

while under the Standing Assumption (4), each solution $u_{\lambda}\in\mathscr{E}$ to equation $\eqref{eq1.4}$ satisfies

[TABLE]

Here, $\rho,\varrho>0$ and $C_{hk},C^{\prime}_{hk}>0$ are absolute constants, independent of $\lambda,u$.

Proof.

First, one notices each solution $u_{\lambda}$ to $\eqref{eq1.4}$ (not necessarily positive) satisfies

[TABLE]

For our assumption (3), denote $r_{4}=\frac{s+\beta(s-q)}{\beta(s-r)}=r_{1}$, $r_{5}=\frac{p}{s}r_{2}=\frac{p\{s+\beta(s-q)\}}{s\{r+\beta(r-q)\}}<r_{2}$ (as $p<s$) and $\rho=\frac{r_{4}r_{5}}{r_{4}r_{5}-r_{4}-r_{5}}>r_{3}>0$ in (2.2) of Proposition 2.1 to observe, via (2.5),

[TABLE]

So, (2.3) is verified by virtue of (2.6) if $\frac{1}{r_{4}}+\frac{1}{r_{5}}<1$, which is true provided \beta\in\big{[}\frac{s}{q-s},\frac{s(r-p)}{(s-p)(q-r)}\big{)} when $s<r$ or \beta\in\big{(}\frac{s(r-p)}{(s-p)(q-r)},\frac{s}{q-s}\big{]} when $r<s$.

For our assumption (4), rewrite (2.5) by switching $\int_{\mathbf{R}^{N}}h\left|u_{\lambda}\right|^{q}\hskip 0.56905ptdx$ to the righthand side, and recall an elementary estimate h_{1}\mathfrak{u}^{\mu}-h_{2}\mathfrak{u}^{\nu}\leq C_{\mu\nu}\big{[}\frac{h_{1}^{\nu}}{h_{2}^{\mu}}\big{]}^{\frac{1}{\nu-\mu}} for $h_{1},h_{2}>0$, $\nu>\mu>0$, $\mathfrak{u}\geq 0$ and $C_{\mu\nu}>0$. Set $h_{1}=\lambda k$, $h_{2}=h$, $\mu=r-t$, $\nu=q-t$ and $\mathfrak{u}=\left|u_{\lambda}\right|$ to arrive at

[TABLE]

for $C(q,r,t)>0$; thus, (2.4) is a direct consequence of (2.7) with $\varrho=\frac{q-t}{q-r}>0$. ∎

Lemma 2.4**.**

Under the Standing Assumptions (1)-(3) or (1)-(2)&(4), $\mathcal{J}_{\lambda}$ is of class $C^{1}$, and is coercive and sequentially weakly lower semicontinuous in $\mathscr{E}$.

Proof.

The ideas follow from [9, Lemma 3.2]; so, we only stress the differences.

For our assumption (3), similar to (2.6), one has

[TABLE]

so that

[TABLE]

For our assumption (4), when $0\leq t<p$, similar to (2.7), one has

[TABLE]

for suitable constants $C(q,r,t,\lambda),C_{hk}(q,r,t,\lambda),C_{hk\lambda}>0$, which further implies

[TABLE]

so that (2.8) holds after $\lambda^{\rho}\widetilde{C}_{hk}$ being replaced by $C_{hk\lambda}$.

On the other hand, when $t=p$ in our assumption (4), one has

[TABLE]

for a sufficiently large $R>0$ and some $\varepsilon$ with $0<\varepsilon<\frac{1}{2p}$, while

[TABLE]

follows from the analysis of Lemma 2.4 in [1] seeing h(x),k(x)\in L^{1}_{loc}\big{(}\mathbf{R}^{N}\big{)} for another suitable constant $\widetilde{C}_{hk\lambda}>0$, so that we have (2.8), with $\widetilde{C}_{hk\lambda}$ replacing $\lambda^{\rho}\widetilde{C}_{hk}$, via (2.9) and (2.10) (and the latter is derived through the subsequent estimates (2.11) and (2.12)).

Here, $\widetilde{C}_{hk},C_{hk\lambda},\widetilde{C}_{hk\lambda}>0$ are absolute constants, independent of $u$.

As a matter of fact, when $t=p$ in our assumption (4), given $M,\delta>0$, set

[TABLE]

Then, $\mathscr{L}(\mathbf{Z})\to 0$ as $M\to\infty$ and $\mathscr{L}(\mathbf{Y}^{\prime})\to 0$ as $\delta\to 0$, so that

[TABLE]

provided $M$ is sufficiently large and $\delta$ is sufficiently small. Next, for each $\tilde{t}$ with $0\leq\tilde{t}<p$ and some suitable constants $C_{M\delta}(q,r,t,\lambda),\widetilde{C}_{hk\lambda}>0$, one instead derives that

[TABLE]

Since $\mathcal{J}_{\lambda}$ is coercive and $\mathscr{E}$ is reflexive (see [2, Proposition A.11]), each sequence $\left\{u_{l}:l\geq 1\right\}$ in $\mathscr{E}$ with $\mathcal{J}_{\lambda}(u_{l})$ bounded has a subsequence, using again $\left\{u_{l}:l\geq 1\right\}$, such that $u_{l}\rightharpoonup u\in\mathscr{E}$. The compact embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)}, via assumptions (1)-(2), leads to

[TABLE]

for yet another subsequence, which along with the lower semicontinuity of norms in W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)} and L^{q}_{h}\big{(}\mathbf{R}^{N}\big{)} generates the sequentially weak lower semicontinuity of $\mathcal{J}_{\lambda}$ in $\mathscr{E}$. ∎

Define

[TABLE]

then, $\tilde{\lambda}>0$. In fact, if not, then there is a sequence $\left\{u_{l}:l\geq 1\right\}$ in $\mathscr{E}$ such that $\left\|u^{+}_{l}\right\|_{L^{r}_{k}(\mathbf{R}^{N})}=1$ but $\frac{r}{p}\int_{\mathbf{R}^{N}}\left(\left|\nabla u_{l}\right|^{p}+\left|u_{l}\right|^{p}\right)\hskip 0.56905ptdx+\frac{r}{q}\int_{\mathbf{R}^{N}}h\left|u_{l}\right|^{q}\hskip 0.56905ptdx\to 0$; this would yield $\left\|u_{l}\right\|_{\mathscr{E}}\to 0$, contradicting the compact embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} in view of $\left\|u_{l}\right\|_{L^{r}_{k}(\mathbf{R}^{N})}\geq\left\|u^{+}_{l}\right\|_{L^{r}_{k}(\mathbf{R}^{N})}=1$.

Denote by $\lambda^{*}$ the supermum of $\lambda$ such that (1.4) has no nontrivial positive solution for each $\mu<\lambda$, and denote by $\lambda^{**}$ the infimum of $\lambda$ such that (1.4) has at least one nontrivial positive solution at $\lambda$. Then, we have $0\leq\lambda^{*}=\lambda^{**}\leq\tilde{\lambda}$. Actually, for each $\lambda>\tilde{\lambda}$,

[TABLE]

follows, with some $v_{\lambda}\in\mathscr{E}$, through homogeneity, which can be rewritten as

[TABLE]

This, together with Lemma 2.4, leads to $\mathcal{J}_{\lambda}(u_{\lambda})=\inf\limits_{u\in\mathscr{E}}\mathcal{J}_{\lambda}(u)\leq\mathcal{J}_{\lambda}(v_{\lambda})<0$ for an $u_{\lambda}\geq 0$ in $\mathscr{E}$ since $\mathcal{J}_{\lambda}(\left|u_{\lambda}\right|)\leq\mathcal{J}_{\lambda}(u_{\lambda})$, which is a nontrivial positive solution to equation (1.4). So, one has $\lambda^{**}\leq\tilde{\lambda}$. On the other hand, if $\lambda^{*}>\lambda^{**}$, one would find a $\lambda^{\prime}\in[\lambda^{**},\lambda^{*})$ such that equation (1.4) has at least one nontrivial positive solution at $\lambda^{\prime}$ according to the definition of $\lambda^{**}$ - this however is against the definition of $\lambda^{*}$; if $\lambda^{*}<\lambda^{**}$, one would find a $\lambda^{\prime}\in(\lambda^{*},\lambda^{**}]$ and then a $\mu^{\prime}(<\lambda^{\prime})$ such that equation (1.4) has at least one nontrivial positive solution at $\mu^{\prime}$ according to the definition of $\lambda^{*}$ - this however is against the definition of $\lambda^{**}$. So, $\lambda^{*}=\lambda^{**}$.

Write $\lambda_{1}:=\lambda^{*}=\lambda^{**}$ in the sequel.

Proposition 2.5**.**

Under the Standing Assumptions (1)-(3) or (1)-(2)&(4), $\lambda_{1}\geq 0$ and equation (1.4) has a nontrivial positive solution $u_{\lambda}\geq 0$ in $\mathscr{E}$ for every $\lambda>\lambda_{1}$.

Proof.

By definition, $\lambda_{1}=\lambda^{*}$; so, if $u_{\lambda}\geq 0$ is a nontrivial positive solution to (1.4) in $\mathscr{E}$, then $\lambda\geq\lambda_{1}$. Now, we verify (1.4) has at least a nontrivial solution $u_{\lambda}\geq 0$ in $\mathscr{E}$ for each $\lambda>\lambda_{1}$ via Struwe [14, Theorem 2.4]; see also [2, Theorem 4.2] and [9, Proposition 3.3].

By definition, $\lambda_{1}=\lambda^{**}$; so, one finds a $\mu\in[\lambda_{1},\lambda)$ at which (1.4) has a nontrivial solution $u_{\mu}\geq 0$ in $\mathscr{E}$ that is a subsolution to (1.4) at $\lambda$. Consider the constrained minimizing problem $\inf\limits_{u\in\mathscr{M}}\mathcal{J}_{\lambda}(u)$ for $\mathscr{M}=\left\{u\in\mathscr{E}:u\geq u_{\mu}\geq 0\right\}$. Since $\mathscr{M}$ is closed and convex, it is weakly closed in $\mathscr{E}$. Thus, Lemma 2.4 yields an $u_{\lambda}(\geq u_{\mu})$ in $\mathscr{M}$ with $\mathcal{J}_{\lambda}(u_{\lambda})=\inf\limits_{u\in\mathscr{M}}\mathcal{J}_{\lambda}(u)$. Take \varphi\in C^{1}_{c}\big{(}\mathbf{R}^{N}\big{)}; set $\varphi_{\varepsilon}=\max\{0,u_{\mu}-u_{\lambda}+\varepsilon\varphi\}\geq 0$ and $v_{\varepsilon}=\varphi_{\varepsilon}+u_{\lambda}-\varepsilon\varphi\hskip 0.56905pt(\geq u_{\mu})$ in $\mathscr{M}$ for some $\varepsilon>0$. Then, we have $\mathcal{J}_{\lambda}^{\prime}(u_{\lambda})(u_{\lambda})\leq\mathcal{J}_{\lambda}^{\prime}(u_{\lambda})(v_{\varepsilon})$, which further leads to

[TABLE]

Put $\mathbf{\Omega}_{\varepsilon}=\left\{x\in\mathbf{R}^{N}:\varphi_{\varepsilon}(x)>0\right\}=\left\{x\in\mathbf{R}^{N}:u_{\lambda}(x)-u_{\mu}(x)<\varepsilon\varphi(x)\right\}\varsubsetneq\operatorname{supp}(\varphi^{+})$. As $u_{\mu}$ is a subsolution to (1.4) at $\lambda$ and $\varphi_{\varepsilon}\geq 0$, one has $\mathcal{J}_{\lambda}^{\prime}(u_{\mu})(\varphi_{\varepsilon})\leq 0$ so that

[TABLE]

as $\varepsilon\to 0^{+}$, seeing $0<\varphi_{\varepsilon}\leq\varepsilon\left|\varphi\right|$ on $\mathbf{\Omega}_{\varepsilon}$. This combined with (2.15) yields $\mathcal{J}_{\lambda}^{\prime}(u_{\lambda})(\varphi)\leq 0$ for every \varphi\in C^{1}_{c}\big{(}\mathbf{R}^{N}\big{)}; so, it further implies $\mathcal{J}_{\lambda}^{\prime}(u_{\lambda})(-\varphi)\leq 0$. Via density, $\mathcal{J}_{\lambda}^{\prime}(u_{\lambda})(v)=0$ for all $v\in\mathscr{E}$. Therefore, $u_{\lambda}(\geq u_{\mu}\geq 0)$ is a nontrivial solution to equation (1.4) at $\lambda$. ∎

Proposition 2.5 obviously provides the proof to the first assertion of Theorem 1.1, and below we shall discuss the second twofold assertion of Theorem 1.1.

Lemma 2.6**.**

Under the Standing Assumption (5), there exists an absolute constant $C_{k}>0$ such that each nontrivial solution $u_{\lambda}\in\mathscr{E}$ to equation $\eqref{eq1.4}$ satisfies

[TABLE]

Besides, when the Standing Assumption (3) or (4) also holds, then there is another absolute constant $\widehat{C}_{hk}>0$ such that $\lambda_{1}\geq\widehat{C}_{hk}>0$.

Proof.

First, when our assumption (5) is true, one observes

[TABLE]

Combining (2.17) with (2.5), one has \big{(}\int_{\mathbf{R}^{N}}k\left|u_{\lambda}\right|^{r}\hskip 0.56905ptdx\big{)}^{\frac{p}{r}}\leq\lambda\hskip 0.56905ptC_{k}\int_{\mathbf{R}^{N}}k\left|u_{\lambda}\right|^{r}\hskip 0.56905ptdx for all nontrivial solutions $u_{\lambda}$ to (1.4) (not necessarily positive), which in turn yields (2.16).

Now, when our assumption (3) also holds, by (2.3), (2.16) and (2.17), $(\lambda\hskip 0.56905ptC_{k})^{\frac{p}{p-r}}\leq C_{k}\lambda^{\rho}C_{hk}$ follows so that $\lambda\geq\breve{C}_{hk}>0$ since $p<r$, which in particular yields $\lambda_{1}\geq\breve{C}_{hk}>0$. On the other hand, when our assumption (4) also holds for $t=p$, (2.4) automatically implies $\lambda_{1}\geq\breve{C}^{\prime}_{hk}>0$; if $0\leq t<p$ in (4), then by (2.4), (2.16) and (2.17), $(\lambda\hskip 0.56905ptC_{k})^{\frac{p}{p-r}}\leq C_{k}(\lambda^{\varrho}C^{\prime}_{hk})^{\frac{p}{p-t}}$ follows so that likewise $\lambda_{1}\geq\breve{C}^{\prime}_{hk}>0$. Thus, the proof is done for \widehat{C}_{hk}=\min\big{\{}\breve{C}_{hk},\breve{C}^{\prime}_{hk}\big{\}}>0. ∎

Proposition 2.7**.**

Under the Standing Assumptions (1)-(3)&(5) or (1)-(2)&(4)-(5), we have $\lambda_{1}>0$ and equation (1.4) has a nontrivial positive solution $u_{\lambda}$ in $\mathscr{E}$ if and only if $\lambda\geq\lambda_{1}$ (except for $t=p$); moreover, for $\lambda_{2}:=\tilde{\lambda}\hskip 0.56905pt(\geq\lambda_{1})$ as introduced in (2.14), equation (1.4) has at least two distinct nontrivial positive solutions $u_{\lambda}\neq\tilde{u}_{\lambda}$ in $\mathscr{E}$ for every $\lambda>\lambda_{2}$.

Proof.

Recall $\lambda_{1}>0$ has been proven. So, we only need to show (1.4) has a nontrivial positive solution at $\lambda_{1}$. Assume $\left\{\lambda(l)>\lambda_{1}:l\geq 1\right\}$ decrease to $\lambda_{1}$, with $\left\{u_{\lambda(l)}\geq 0:l\geq 1\right\}$ an associated sequence of nontrivial solutions to (1.4), which is bounded in $\mathscr{E}$ by Lemma 2.3.

Actually, (2.3) provides the boundedness of $\left\{u_{\lambda(l)}\geq 0:l\geq 1\right\}$ in $\mathscr{E}$ immediately while (2.4) implies that of $\left\{u_{\lambda(l)}\geq 0:l\geq 1\right\}$ in W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)} for $0\leq t<p$ in (4); rewrite (2.5) by switching only one half of $\int_{\mathbf{R}^{N}}h\left|u_{\lambda}\right|^{q}\hskip 0.56905ptdx$ to the righthand side (but keeping the other half) and reuse the analysis of (2.7) to observe the boundedness of $\left\{u_{\lambda(l)}\geq 0:l\geq 1\right\}$ in L^{q}_{h}\big{(}\mathbf{R}^{N}\big{)}.

Thus, there exists a subsequence $\left\{u_{\lambda(l)}:l\geq 1\right\}$, using the same index $\lambda(l)$, with $u_{\lambda(l)}\rightharpoonup\omega$ in $\mathscr{E}$ and $u_{\lambda(l)}\to\omega$ in L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} for some function $\omega\in\mathscr{E}$ such that $u_{\lambda(l)}\to\omega\geq 0$ a.e. on $\mathbf{R}^{N}$. Then, one has (2.13) for $u_{\lambda(l)},\omega\geq 0$ (instead of $u_{l},u$) and

[TABLE]

see for example [2, Lemma 3.4]. Keep in mind $\mathcal{J}_{\lambda(l)}^{\prime}(u_{\lambda(l)})=0$; that is,

[TABLE]

As $\mathcal{J}_{\lambda}^{\prime}(\omega)$ is a continuous, linear functional in $\mathscr{E}$,

[TABLE]

follows when $l\to\infty$, from which, together with [7, Lemma 3.2], one concludes $u_{\lambda(l)}\to\omega$ in $\mathscr{E}$. This particularly implies $\nabla u_{\lambda(l)}\to\nabla\omega$ a.e. on $\mathbf{R}^{N}$, so that we also have

[TABLE]

Recall $\omega\geq 0$. Upon letting $l\to\infty$ on both sides of (2.18), for all $v\in\mathscr{E}$, we have

[TABLE]

Because (2.16) yields $\int_{\mathbf{R}^{N}}k\omega^{r}\hskip 0.56905ptdx\geq(\lambda(1)\hskip 0.56905ptC_{k})^{\frac{r}{p-r}}>0$ in view of the compact embedding \mathscr{E}\hookrightarrow L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)}, $u_{\lambda_{1}}:=\omega\geq 0$ is a nontrivial positive solution to equation (1.4) in $\mathscr{E}$.

On the other hand, from the discussions in [10, Lemma 3] or [12, Lemma 3], one knows any other solution $\tilde{u}_{\lambda}\geq 0$ to (1.4) at $\lambda\hskip 0.56905pt(>\lambda_{2})$, if it exists, should satisfy $\tilde{u}_{\lambda}\leq u_{\lambda}$ with $\mathcal{J}_{\lambda}(u_{\lambda})<0$. Furthermore, it is easily seen from (2.17) that

[TABLE]

provided 0<\left\|u\right\|_{W^{1,\hskip 0.56905ptp}(\mathbf{R}^{N})}<\min\Big{\{}\left\|u_{\lambda}\right\|_{W^{1,\hskip 0.56905ptp}(\mathbf{R}^{N})},\big{(}\frac{r}{\lambda p}C^{-\frac{r}{p}}_{k}\big{)}^{\frac{1}{r-p}}\Big{\}}. Observe that the mountain pass structure by (2.21) is on W^{1,\hskip 0.56905ptp}\big{(}\mathbf{R}^{N}\big{)} rather than on $\mathscr{E}$. Thus, applying the mountain pass theorem of Candela and Palmieri [4, Theorem 2.5] (see [2, Theorem A.3] as well for a different proof), there exists a sequence of functions $\left\{\tilde{u}_{l}:l\geq 1\right\}$ in $\mathscr{E}$ that satisfy $\mathcal{J}_{\lambda}(\tilde{u}_{l})\to c>0$ and $\mathcal{J}_{\lambda}^{\prime}(\tilde{u}_{l})\to 0$ in the dual space of $\mathscr{E}$ when $l\to\infty$, where we set $c=\inf\limits_{h\in\mathscr{H}}\max\limits_{z\in[0,1]}\mathcal{J}_{\lambda}(h(z))>0$ for \mathscr{H}=\left\{h\in C\big{(}[0,1];\mathscr{E}\big{)}:h(0)=0,\,h(1)=u_{\lambda}\right\}. As a result, we find a subsequence $\left\{\tilde{u}_{l}:l\geq 1\right\}$, using the same notation, such that $\tilde{u}_{l}\rightharpoonup\xi$ in $\mathscr{E}$ and $\tilde{u}_{l}\to\xi$ in L^{r}_{k}\big{(}\mathbf{R}^{N}\big{)} for some function $\xi\in\mathscr{E}$ with $\tilde{u}_{l}\to\xi$ a.e. on $\mathbf{R}^{N}$. The same discussion as done ahead of (2.20) then leads to $\tilde{u}_{l}\to\xi$ in $\mathscr{E}$. In particular, we have $\mathcal{J}_{\lambda}^{\prime}(\xi)v=0$ for all $v\in\mathscr{E}$; upon using $v=\xi^{-}\in\mathscr{E}$ as a test function, one realizes $\xi^{-}\equiv 0$ and $\xi\geq 0$. As a consequence, $\tilde{u}_{\lambda}:=\xi\geq 0$ is another nontrivial positive solution to equation (1.4) in $\mathscr{E}$ that is different from $u_{\lambda}$, since $\mathcal{J}_{\lambda}(\tilde{u}_{\lambda})=c>0$. ∎