Quasilinear problems under local Landesman–Lazer condition

Abstract

This paper presents results on the existence and multiplicity of solutions for quasilinear problems in bounded domains involving the p-Laplacian operator under local versions of the Landesman–Lazer condition. The main results do not require any growth restriction at infinity on the nonlinear term which may change sign. The existence of solutions is established by combining variational methods, truncation arguments and approximation techniques based on a compactness result for the inverse of the p-Laplacian operator. These results also establish the intervals of the projection of the solution on the direction of the first eigenfunction of the p-Laplacian operator. This fact is used to provide the existence of multiple solutions when the local Landesman–Lazer condition is satisfied on disjoint intervals.

Appendix

Proof of Theorem 2.6:

By Stampacchia method [33], there is $M=M(\Vert g\Vert _\sigma )$ such that

$$\begin{aligned} \Vert u\Vert _\infty \le M. \end{aligned}$$

(6.1)

Next, given $k \ge 0$ and a ball $B_\rho$ of $\mathbb {R}^N$, $\rho >0$, we consider $\eta (x)=\xi ^p|u-k|^+$, where $\xi :B_\rho \rightarrow [0,1]$ is a smooth function of compact support on $B_\rho$. Setting $A_{k, \rho }=\{x \in B_\rho \cap \Omega ; u(x)>k\}$ and considering that $\eta \in W_0^{1,p}(\Omega )$, from (2.23), we get

$$\begin{aligned} \int _{A_{k, \rho }} \xi ^p |\nabla u|^p dx+p\int _{A_{k, \rho }} |u-k| \xi ^{p-1} |\nabla u|^{p-2} \nabla u \cdot \nabla \xi dx=\int _{A_{k, \rho }} g(x)\xi ^p (u-k) dx. \end{aligned}$$

Consequently, by Hölder inequality,

$$\begin{aligned} \int _{A_{k, \rho }} \xi ^p |\nabla u|^p dx \le p\int _{A_{k, \rho }} \xi ^{p-1} |\nabla u|^{p-1} |u-k| |\nabla \xi | dx+\Vert u\Vert _\infty \Vert g\Vert _\sigma |A_{k, \rho }|^{1-\frac{1}{\sigma }}. \end{aligned}$$

Next, using the Young inequality, we get

$$\begin{aligned} p\xi ^{p-1}|\nabla u|^{p-1} |u-k||\nabla \xi | \le \frac{1}{2}|u-k|^p|\nabla \xi |^p+2^{p-1}(p-1)^{p-1}|u-k|^p|\nabla \xi |^p. \end{aligned}$$

Combining the above inequalities, we obtain

$$\begin{aligned} \int _{A_{k, \rho }} \xi ^p |\nabla u|^p dx \le 2^p (p-1)^{p-1}\int _{A_{k, \rho }}|u-k|^p|\nabla \xi |^p dx + 2\Vert u\Vert _\infty \Vert g\Vert _\sigma |A_{k, \rho }|^{1-\frac{1}{\sigma }}.\nonumber \\ \end{aligned}$$

(6.2)

Now, given $\delta >0$, we let $k \ge 0$ be such that $\sup _{B_\rho \cap \Omega } [u(x)-\delta ] \le k$. From these values of k, we take $\xi$ such that $\xi (x)=1$ for every $x \in B_{\rho -\mu \rho }$, for $0<\mu <1$, in such a way that $|\nabla \xi |<\frac{c}{\mu \rho }$. Then, from (6.2) we may write

$$\begin{aligned} \int _{A_{k, (1-\mu )\rho }} |\nabla u|^p dx\le & {} \frac{2^p(p-1)^p c^p}{(\mu \rho )^p} \sup _{A_{k, \rho }}|u-k|^p |A_{k, \rho }| +2\Vert u\Vert _\infty \Vert g\Vert _\sigma |A_{k, \rho }|^{1-\frac{1}{\sigma }}\\= & {} \Big [\frac{2^p (p-1)^{p-1} c^p}{\mu ^p\rho ^p}|A_{k, \rho }|^{\frac{1}{\sigma }}\sup _{A_{k, \rho }}|u-k|^p+2\Vert u\Vert _\infty \Vert g\Vert _\sigma \Big ] |A_{k, \rho }|^{1-\frac{1}{\sigma }}\\\le & {} \Big [\frac{2^p (p-1)^{p-1} c}{\mu ^p\rho ^p}|c_N|^{\frac{1}{\sigma }} \rho ^{\frac{N}{\sigma }}\sup _{A_{k, \rho }}|u-k|^p+2\Vert u\Vert _\infty \Vert g\Vert _\sigma \Big ] |A_{k, \rho }|^{1-\frac{1}{\sigma }} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \int _{A_{k, (1-\mu )\rho }} |\nabla u|^p dx \le \gamma \Big [\frac{\max |u-k|^p}{\mu ^p\rho ^{p(1-\frac{N}{p\sigma })}}+1\Big ]|A_{k, \rho }|^{1-\frac{p}{\sigma p}}, \end{aligned}$$

where $\gamma =\max \{2\Vert u\Vert _\infty \Vert g\Vert _\sigma , {c_N}^{\frac{1}{\sigma }} 2^p(p-1)^{p-1}\}$.

Observing that we obtain the same estimate for the function $-u(x)$ for every $k>\sup _{B_\rho \cap \Omega }[-u(x)-\delta ]$, by taking $\eta (x)=\xi ^p(-u-k)^+$, we may assert that $u \in \mathcal {B}_p(\overline{\Omega }, M, \gamma , \delta , \frac{1}{\sigma p})$, where $\mathcal {B}_p$ is as defined in [22, p. 90].

Since $\sigma p>N$, we may invoke Theorem 7.1 in [22] to find that there is $c>0$ and $\alpha \in (0, 1)$ such that $\Vert u\Vert _{C^{0, \alpha }(\overline{\Omega })} \le c$ with the constants c and $\alpha$ depending on $p, M, \gamma ,\delta , \sigma$ and $\Omega$. In view of (6.1), this concludes the proof of Theorem 2.6. $\square$