On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

with $0<q<1<p=\frac{N+2}{N-2}$, the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In [8], Wu considered the subcritical case of problem (1.2) with $\lambda u^q$ replaced by $\lambda f(x)u^q$, here $f(x)\in C(\overline{\mathrm{\Omega }})$ is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.

where Ω is a bounded smooth domain in ${\mathbb{R}}^N$ ($N⩾4$), $0<q<1<p<2^{\ast \ast }$ ($2^{\ast \ast }=\frac{N+4}{N-4}$ for $N>4$ and $2^{\ast \ast }=\mathrm{\infty }$ for $N=4$), $\lambda >0$ is a parameter, $f\in C(\overline{\mathrm{\Omega }})$ is a positive or sign-changing weight function and $h\in C(\overline{\mathrm{\Omega }})$ is a positive weight function.

For convenience and simplicity, we introduce some notations. The norm of u in $L^r(\mathrm{\Omega })$ is denoted by ${|u|}_r={({\int }_{\mathrm{\Omega }}{|u(x)|}^r)}^{1/r}$, the norm of u in $C(\overline{\mathrm{\Omega }})$ is denoted by ${|u|}_{\mathrm{\infty }}={max}_{x\in \overline{\mathrm{\Omega }}}|u(x)|$; $H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega })$ is denoted by $H(\mathrm{\Omega })$, endowed with the norm $\parallel u\parallel ={|\Delta u|}_2$; S denotes the best Sobolev constant for the embedding of $H(\mathrm{\Omega })$ in $L^{p+1}(\mathrm{\Omega })$ (see [14]); to be precise, ${|u|}_{p+1}⩽S\parallel u\parallel$ for all $u\in H(\mathrm{\Omega })$.

It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional $J_{\lambda }$ (see Rabinowitz [15]).

Note that all solutions of (1.3) are clearly in the Nehari manifold, $M_{\lambda }(\mathrm{\Omega })$. Hence, our approach to solve problem (1.3) is to analyze the structure of $M_{\lambda }(\mathrm{\Omega })$, and then to deal with the minimization problems for $J_{\lambda }$ on $M_{\lambda }^{+}(\mathrm{\Omega })$ and $M_{\lambda }^{-}(\mathrm{\Omega })$ applying the direct variational method.

The following is our main result.

Theorem 1.1 Let ${\lambda }_{\ast }=\frac{p-1}{p-q}\cdot {[\frac{1-q}{(p-q){|h|}_{\mathrm{\infty }}}]}^{\frac{1-q}{p-1}}{S}^{\frac{2(p-q)}{1-p}}{|f|}_{p^{\ast }}^{-1}$ with $p^{\ast }=\frac{p+1}{p-q}$, then problem (1.3) has at least two positive solutions for any $\lambda \in (0,{\lambda }_{\ast })$.

The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.

In this section, we prove several lemmas.

Lemma 2.1 For $\lambda \in (0,{\lambda }_{\ast })$ (where ${\lambda }_{\ast }$ is given in Theorem  1.1), we have $M_{\lambda }^0(\mathrm{\Omega })=\varphi$.

Hence, by (2.5) the desired conclusion yields. □

The proof is completed. □

The following lemma shows that the minimizers on $M_{\lambda }(\mathrm{\Omega })$ are ‘usually’ critical points for $J_{\lambda }$.

Lemma 2.3 For $\lambda \in (0,{\lambda }_{\ast })$, if $u_0$ is a local minimizer for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$, then $J_{\lambda }^{\prime }(u_0)=0$ in ${[H(\mathrm{\Omega })]}^{\ast }$.

From $u_0\in M_{\lambda }(\mathrm{\Omega })$ and Lemma 2.1, we have $〈J_{\lambda }^{\prime }(u_0),u_0〉=0$ and $〈{\psi }_{\lambda }^{\prime }(u_0),u_0〉\ne 0$. So, by (2.6)-(2.7) we get $J_{\lambda }^{\prime }(u_0)=0$ in ${[H(\mathrm{\Omega })]}^{\ast }$. □

Then we have the following lemma.

Case I. ${\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x⩽0$.

Hence, $J_{\lambda }(t^{-}u)={max}_{t⩾0}{J}_{\lambda }(tu)$.

Case II. ${\int }_{\mathrm{\Omega }}f(x){|u|}^{q+1}\mathrm{d}x>0$.

where $N(\Theta )=\{u\in H(\Theta )\mathrm{\setminus }\{0\}\mid 〈K^{\prime }(u),u〉=0\}$. Now we prove that problem (2.9) has a positive solution $w_0$ such that $K(w_0)=\beta (\Theta )>0$.

is continuous and the induced continuous map $u\to t(u)u$ defines a homeomorphism of the unit sphere of $H(\Theta )$ with $N(\Theta )$.

It is easy to verify that $g(0)=0$, $g(t)>0$ for $t>0$ small and $g(t)<0$ for $t>0$ large. Hence, ${max}_{t⩾0}g(t)$ is reached at a unique $t=t(u)$ such that $g^{\prime }(t(u))=0$ and $t(u)u\in N(\Theta )$. To prove the continuity of $t(u)$, assume that $u_n\to u$ in $H(\Theta )\mathrm{\setminus }\{0\}$. It is easy to verify that $\{t(u_n)\}$ is bounded. If a subsequence of $\{t(u_n)\}$ converges to $t_0$, it follows from (2.10) that $t_0=t(u)$ and then $t(u_n)\to t(u)$. Finally the continuous map from the unit sphere of $H(\Theta )$ to $N(\Theta )$, $u\to t(u)u$, is inverse to the retraction $u\to \frac{u}{\parallel u\parallel }$. □

where $\Gamma =\{\gamma \in C([0,1],H(\Theta ))\mid \gamma (0)=0,K(\gamma (1))<0\}$.

Lemma 2.6 $\beta (\Theta )=c_{\ast }=c>0$ is a critical value of K.

Proof From Lemma 2.5, we know that $\beta (\Theta )=c_{\ast }$. Since $K(tu)<0$ for $u\in H(\Theta )\mathrm{\setminus }\{0\}$ and t large, we obtain $c⩽c_{\ast }$. The manifold $N(\Theta )$ separates $H(\Theta )$ into two components. The component containing the origin also contains a small ball around the origin. Moreover, $K(u)⩾0$ for all u in this component, because $〈K^{\prime }(tu),u〉⩾0$, $\mathrm{\forall }t\in [0,t(u)]$. Then each $\gamma \in \Gamma$ has to cross $N(\Theta )$ and $\beta (\Theta )⩽c$. Since the embedding $H(\Theta )\hookrightarrow L^{p+1}(\Theta )$ is compact (see [14]), it is easy to prove that $c>0$ is a critical value of K and $w_0$ a positive solution corresponding to c. □

With the help of Lemma 2.6, we have the following result.

here $C(p,q)={[\frac{p-q}{(p+1)(q+1)}]}^{\frac{2}{1-q}}\cdot {[\frac{4(p+1)}{p-1}]}^{\frac{1+q}{1-q}}$.

for all $\lambda >0$. □

Next, we will use the idea of Tarantello [16] to get the following results.

for all $v\in H(\mathrm{\Omega })$.

we can get the desired results applying the implicit function theorem at the point $(1,0)$. □

for all $v\in H(\mathrm{\Omega })$.

Proof In view of Lemma 2.8, there exist $ϵ>0$ and a differentiable functional ${\xi }^{-}$ such that ${\xi }^{-}(0)=1$, ${\xi }^{-}(v)(u+v)\in M_{\lambda }(\mathrm{\Omega })$ for all $v\in B(0;ϵ)\subset H(\mathrm{\Omega })$ and we have (2.12). By use of $u\in M_{\lambda }^{-}(\mathrm{\Omega })$, we have $〈{\psi }_{\lambda }^{\prime }(u),u〉<0$. In combination with the continuity of the functions ${\psi }_{\lambda }^{\prime }$ and ${\xi }^{-}$, we get $〈{\psi }_{\lambda }^{\prime }({\xi }^{-}(v)(u+v)),{\xi }^{-}(v)(u+v)〉<0$ as ϵ sufficiently small, this implies that ${\xi }^{-}(v)(u+v)\in M_{\lambda }^{-}(\mathrm{\Omega })$. □

Firstly, we provide the existence of minimizing sequences for $J_{\lambda }$ on $M_{\lambda }(\mathrm{\Omega })$ and $M_{\lambda }^{-}(\mathrm{\Omega })$ as λ is sufficiently small.